Section 2 Vortices in Fluids

#### **Chapter 3**

## Searching of Individual Vortices in Experimental Data

*Daniel Duda*

#### **Abstract**

The turbulent flows consist of many interacting vortices of all scales, which all together self-organize being responsible for the statistical properties of turbulence. This chapter describes the searching of individual vortices in velocity fields obtained experimentally by Particle Image Velocimetry (PIV) method. The vortex model is directly fitted to the velocity field minimazing the energy of the residual. The zero-th step (which does not a priori use the vortex model) shows the velocity profile of vortices. In the cases dominated by a single vortex, the profile matches the classical solutions, while in turbulent flow field, the profile displays velocity decrease faster than 1*=r*. The vortices fitted to measured velocity field past a grid are able to describe around 50 % of fluctuation energy by using 15 individual vortices, and by using 100 vortices, the fluctuating field is reconstructed by 75 %. The found vortices are studied statistically for different distances and velocities.

**Keywords:** vortex, turbulence, Particle Image Velocimetry, grid turbulence, individual vortex searching algorithm, vortex model

#### **1. Introduction**

Contemporary exploration of turbulent flows focuses on statistical characteristics [1] such as study of distributions [2, 3], Fourier analysis [4], correlations [5, 6], or the Proper Orthogonal Decompositions [7–9]. The success of statistical approach is declared by the large applicability of numerical simulations, which are able to perfectly match the experimental data. Although it is possible to predict the statistical development of turbulent flow, this is still far from *understanding* the turbulence. The turbulence consists of vortices [10] and other coherent structures [11] whose multi-body interactions are responsible for the life-like behavior—the flow can be *infected* by turbulence [12], and it dies when it is not fed [13]; turbulence fastens the energy transfer from low-entropy energy source to large-entropy energy (heat) by decreasing its own entropy via self-organization and the rise of coherent structures.

The importance of individual vortices to the turbulent statistics is best shown by the problem of *quantum turbulence* [14, 15], which consists of *quantized vortices* [16], which fulfill the Helmholtz circulation theorems [17]—their circulation is constant

and equal to <sup>Γ</sup>1qv <sup>¼</sup> *<sup>κ</sup>* <sup>¼</sup> <sup>2</sup>*π*ℏ*=m*<sup>4</sup> <sup>≈</sup> <sup>9</sup>*:*<sup>997</sup> � <sup>10</sup>�8m2*=*s, (*m*<sup>4</sup> is the mass of single helium 4 atom, it applies 2 � *m*<sup>3</sup> for helium 3 as it is a fermion); thus the vortices cannot end anywhere in the fluid, only at the fluid domain boundary, or they can form closed loops. The energy cascade can be realized only via vortex interactions, reconnections [16], and the helical Kelvin waves on the quantized vortices leading to phonon emission due to nonlinear interactions [18]. This nature of turbulence made of a tangle of identical vortices instead of different vortices as it is in Richardson cascade leads to polynomial velocity distribution [19] instead of almost-Gaussian distribution observed in classical turbulence [3, 20]. Despite this fact, the large-scale observation of superfluid flows shows the same picture as the classical flows do [21, 22]. The transition between both regimes depends on the length scale [2]. The interacting tangle of quantized vortices builds up the turbulence, whose structure is classical on large scale.

The fluid simulation by using the quantized vortices [23] is able to reconstruct the velocity spectra [18] and overall topology [24]. This method is applied in classical turbulence among others by the group of Ilia Marchevsky [25–27].

The behavior of individual vortices in experiment is studied by many groups; however, it is often limited to the case of some single vortex or vortex system dominating the flow. Among others, let us mention the work of Ben-Gida [28], who detected vortices in a wake past accelerating hydrofoil in stably stratified or mixed water. He used the maxima of *λ*<sup>2</sup> criterion [29]. De Gregorio [30] observed the tip vortex of helicopter rotor blade, and for its detection used the Γ<sup>2</sup> criterion [31]. They measured the vortex velocity profile and found that it is similar to the Vatistas model (discussed later here); they studied the development of the tip vortex and observed the interactions of tip vortices of various blades and various turn ages downstream the helicopter jet. Graftieaux et al. [31] developed the the functions Γ<sup>1</sup> and Γ<sup>2</sup> for the study of swirling flow in a duct. They detected a single vortex in each snapshot and in average field measuring the distribution of the distance of average and instantaneous vortex center. Their scalar function used for vortex detection is effectively similar to smoothed circulation over some neightborhood; therefore it nicely solves the issue of all experimental data: the noise; on the other hand, it introduces a new artificial parameter of the detection: the neighborhood area. Kolář [32] developed probably the most accurate criterion for identifying the three components of velocity gradient tensor—the shear, strain, and rotation. But his method needs a large number of transformations in each point. Maciel et al. [33] noticed that eigen axes of the velocity gradient tensor might do the same job.

In this chapter, the method is based on direct fitting of the instantaneous velocity field by some vortex model with scalar criterion used for the prefit only. In the next section, the available vortex models are introduced, then the velocity profiles on experimental data are shown introducing a new vortex model. Later the prefit function and the fitting procedure are shown, and at the end, some results obtained in the grid turbulence are presented.

#### **1.1 Vortex profiles**

A principal disadvantage of any fitting algorithm is the need of *a priori* knowledge of the functional dependence of the data, in our case, to know the vortex model fitted to the data. There have been a lot of different vortex models developed in the past. The basic idea of a vortex model is the circulation-free potential vortex, whose entire circulation is focused inside an infinitesimal topological singularity the vortex filament. Everywhere else, the vorticity *ω* is zero.

*Searching of Individual Vortices in Experimental Data DOI: http://dx.doi.org/10.5772/intechopen.101491*

$$u\_{\theta}^{\rm PV}(r) = \frac{\Gamma}{2\pi r} \tag{1}$$

The vorticity *ω* of this potential vortex is a scalar in the discussed simple twodimensional case: *<sup>ω</sup>* <sup>¼</sup> *<sup>∂</sup><sup>v</sup> <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup><sup>u</sup> ∂y* � � <sup>¼</sup> <sup>Γ</sup> 2*π* <sup>1</sup>� *<sup>x</sup>*2þ*y*<sup>2</sup> ð Þ�*x*�ð Þ <sup>2</sup>*<sup>x</sup> <sup>x</sup>*2þ*y*<sup>2</sup> ð Þ<sup>2</sup> � �1� *<sup>x</sup>*2þ*y*<sup>2</sup> ð Þþ*y*�ð Þ <sup>2</sup>*<sup>y</sup> <sup>x</sup>*2þ*y*<sup>2</sup> ð Þ<sup>2</sup> � � ¼ <sup>Γ</sup> <sup>2</sup>*<sup>π</sup>* � <sup>1</sup>�<sup>1</sup> *x*2þ*y*<sup>2</sup>

Among the infinite velocity of undefined direction in the center, the large velocity gradients smoothen the flow in a way, that there is minimum relative motion at small scales leading to the *solid-body rotation* with tangential velocity linearly increasing with the distance from the center

$$
\mu\_{\theta}^{\text{SBR}}(r) = \frac{\Gamma}{2\pi R} \cdot \frac{r}{R} \tag{2}
$$

and vorticity *ω* being constant everywhere

$$\rho = \left(\frac{\partial v}{\partial \mathbf{x}} - \frac{\partial u}{\partial \mathbf{y}}\right) = \frac{\Gamma}{2\pi R^2} \left(\frac{\partial \mathbf{x}}{\partial \mathbf{x}} - \frac{\partial (-\mathbf{y})}{\partial \mathbf{y}}\right) = \frac{\Gamma}{\pi R^2}$$

Simple connection of these two ideas is called Rankine vortex. A new parameter of the vortex is introduced: the vortex core radius *R* (in solid body rotation vortex (2), *R* played only the unit role: *r=R* is dimensionless distance, Γ*=*2*πR* is tangential velocity at dimensionless distance *r=R* ¼ 1). The fluid in this vortex rotates as a solid body inside the sharply bounded vortex core, while it orbits without internal rotation as a potential vortex outside of the circle bounded by *R*

$$u\_{\theta}^{\text{RV}}(r) = \frac{\Gamma}{2\pi R} \cdot \begin{cases} r/R & \text{for } r < R \\ R/r & \text{for } r > R \end{cases} \tag{3}$$

Generally, there are not much sharp changes in the nature; therefore, a smooth solution is introduced by Oseen

$$u\_{\theta}^{\rm OV}(r) = \frac{\Gamma}{2\pi R} \cdot \frac{R}{r} \cdot \left(\mathbf{1} - e^{-\left(r/R\right)^2}\right) \tag{4}$$

This is one of the exact solutions of Navier-Stokes equations containing the temporal evolution as well, and it is called Lamb-Oseen vortex and then the core scales as *<sup>R</sup>* � ffiffi *<sup>t</sup>* <sup>p</sup> with time. However, we focus on descriptive analysis of instantaneous two-dimensional velocity fields observed experimentally without the temporal development. There exists more possible exact solutions of Navier-Stokes equations, let us mention at least the Burgers vortex, the Kerr-Dold vortex, or the Amromin vortex [34] with turbulent vortex core and potential envelope.

Mathematical simplification of Oseen vortex is suggested by Kaufmann [35] and later discovered independently by Scully et al. [36]. It uses just the first term of Taylor expansion of the exponential in the Oseen vortex, Eq. (4), as it is shown by Bhagwat and Leishman [37], and it is generalized by Vatistas [38].

$$u\_{\theta}^{\rm PV}(r) = \frac{\Gamma}{2\pi R} \cdot \frac{r/R}{\left[1 + \left(r/R\right)^{2n}\right]^{\frac{1}{n}}},\tag{5}$$

which equals to Kaufmann vortex for *n* ¼ 1, and it converges to Rankine vortex for *n* ! ∞.

#### **Figure 1.**

*(a) The tangential velocity profile of discussed vortex models. The velocity is normalized by the circumferential velocity G* ¼ Γ*=*2*πR, the distance is normalized by the vortex core radius R. (b) The profiles of theoretical vorticity of the discussed vortex models. The mainstream of vortex models converges to hyperbolic velocity decay* � 1*=r for large r thus having zero vorticity in the far field. Faster velocity decay is redeemed by a skirt of opposite vorticity around the core, see curves denoted "Taylor" and "VNPE."*

All the vortex models mentioned up to here display the hyperbolic decrease of tangential velocity with distance, *<sup>u</sup><sup>θ</sup>* � *<sup>r</sup>*�1, see **Figure 1**. Such a vortex has infinite energy! No matter, which profile is found in its core. Let us integrate the kinetic energy of the orbiting fluid since some distance *A* large enough to eliminate the different core descriptions:

$$E = \int\_{A}^{\infty} \frac{1}{2} \boldsymbol{u}^2(r) \cdot 2\boldsymbol{\pi}r \cdot \mathbf{d}r = \frac{1}{2} \left(\frac{\Gamma}{2\pi R}\right)^2 2\pi \int\_{A}^{\infty} \left(\frac{R}{r}\right)^2 r \mathbf{d}r = \pi \left(\frac{\Gamma}{2\pi R}\right)^2 R^2 \left[\ln\frac{r}{R}\right]\_{A}^{\infty} = \infty \tag{6}$$

independently on *A* or other details near the core. This divergence is often solved by declaring some maximum size *B* of the area influenced by the vortex, which is the size of the experimental cell. It could be the size of a laboratory or the circumference of a planet. Anyway, it is an arbitrary parameter the total energy depends on. It signifies that the distant regions have the same weight as the near regions. This is a very uncomfortable property.

A faster decay of tangential velocity can be found in the Taylor vortex [39].

$$u\_{\theta}^{\rm TV}(r) = \frac{\Gamma}{2\pi R} \cdot \frac{r}{R} \cdot e^{-\frac{4}{2}(r/R)^2},\tag{7}$$

which is obtained as the first order of *Laguerre polynomials* solution for vorticity, whose zero-th order is the already mentioned Oseen vortex. The detailed mathematics can be found in the book [40], specifically, the Section 6.2.1., and it will be not reproduced here. The velocity decays quite fast and thus the energy converges

$$E = \frac{1}{2} \int\_0^\infty u^2(r) 2\pi r \mathrm{d}r = \pi G^2 R^2 \int \varkappa^3 e^{-\varkappa^2} \mathrm{d}\varkappa = \frac{\pi}{2} G^2 R^2,\tag{8}$$

where<sup>1</sup> *<sup>G</sup>* <sup>¼</sup> <sup>Γ</sup> <sup>2</sup>*π<sup>R</sup>* represents an characterisitc vortex core velocity and *<sup>x</sup>* <sup>¼</sup> *<sup>r</sup> <sup>R</sup>* is the dimensionless distance.

The other hand of faster velocity decay is a skirt of vorticity opposite to that in the core. Let us apply the vorticity operator in cylindrical coordinates

$$\rho\_x(r) = \left(\nabla \times \vec{u}\right)\_x = \frac{1}{r} \left(\frac{\partial r u\_\theta}{\partial r} - \frac{\partial u\_r}{\partial \theta}\right) = \frac{1}{r} \frac{\partial}{\partial r} r G \frac{r}{R} e^{-r^2/2R^2} = \frac{G}{R} e^{-r^2/2R^2} \left(2 - \frac{r^2}{R^2}\right), \tag{9}$$

which changes the sign at *<sup>r</sup>=<sup>R</sup>* <sup>¼</sup> ffiffi 2 <sup>p</sup> , the opposite vorticity value reaches its maximum at *r=R* ¼ 2, and then it decays toward zero. The skirt of opposite vorticity is a property of any profile with tangential velocity decay faster than 1*=r*, as the profile 1*=r* is the limiting case for zero vorticity, see **Figure 1**.

#### **2. Experimental setup and methods**

#### **2.1 Particle Image Velocimetry**

The experimental data were obtained by using the standard method of Particle Image Velocimetry (PIV) [41], which is already a standard tool in hydrodynamic research in air or water and even in superfluid helium [42] as well as in high-speed applications [43]. Contrary to the pressure probes, hot wire anemometry, or laser Doppler anemometry, the result of this method is an instantaneous two-dimensional velocity field [44],2 which opens the exploration of the turbulent flows topology [8, 45, 46]. It is based on the optical observation of small particles [47] carried by the fluid. The particles are illuminated by a double-pulsed laser in order to capture their movement during the time between pulses. There exists even a time-resolved PIV, which uses fast laser and camera, and thus it is able to capture the temporal development and measure, e.g., the temporal spectra [48]. Our system at the University of West Bohemia in Pilsen belongs to the slow ones with repeating frequency 7*:*4Hz; therefore, only the statistical properties can be studied with quite good spatial resolution 64 � 64 grid points sampled on a 4Mpix (2048 � 2048pix2 ) camera images.

#### **2.2 Observed velocity profiles**

Let us look at the experimental data. To get the velocity profile of a vortex, it is needed to know vortex parameters: position, radius, and circulation or the effective circumferential velocity *G* ¼ Γ*=*2*πR* respectively. The listed parameters are results of the fitting procedure; however, the fitting procedure needs to use some vortex model to minimize its energy and, therefore, the result is already a product of the used vortex model. To avoid this back-loop effect, only the *prefit* is used. This function is explained later; it uses the spatial distribution of modified *Q*-invariant of the velocity gradient tensor and does not need any vortex model explicitly. The vortex velocity profile is obtained as an ensemble average of measured velocity profiles across the vortex; the spatial coordinate is normalized by the vortex radius *R* and the velocity by the vortex circumferential velocity *G*. The standard deviation of such ensemble is displayed as a shadow area in **Figures 2**–**6**.

<sup>1</sup> The integral Ð *x*<sup>3</sup>*e*�*x*<sup>2</sup> <sup>d</sup>*<sup>x</sup>* is solved by substituting *<sup>ξ</sup>* <sup>¼</sup> *<sup>x</sup>*2, then d*<sup>ξ</sup>* <sup>¼</sup> <sup>2</sup>*x*d*<sup>x</sup>* and integral is <sup>1</sup> 2 Ð *ξe*�*<sup>ξ</sup>*d*ξ*; perpartes we get � <sup>1</sup> <sup>2</sup> *<sup>ξ</sup>e*�*<sup>ξ</sup>* � <sup>Ð</sup> *<sup>e</sup>*�*<sup>ξ</sup>*d*<sup>ξ</sup>* � � ¼ � <sup>1</sup> <sup>2</sup> *<sup>e</sup>*�*<sup>ξ</sup>*ð Þ *<sup>ξ</sup>* <sup>þ</sup> <sup>1</sup> , i.e. � <sup>1</sup> <sup>2</sup> *<sup>e</sup>*�*x*<sup>2</sup> *<sup>x</sup>*ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> .

<sup>2</sup> To obtain the full velocity gradient tensor, Regunath and coworkers had to use 2 laser systems of different colors with slightly shifted planes [44].

#### **Figure 2.**

*(a) Example of instantaneous velocity field measured at a plane perpendicular to the axis of a pump sucking the fluid out. There is found a single vortex in each snapshot, no fitting is used, only the prefit based on the Q criterion. (b) The velocity profile across the found vortices. The profile line is adapted to the position of each vortex, it is rescaled to each vortex radius and each velocity is normalized by the circumferential velocity G, then it is averaged; standard deviation of the ensemble is displayed as a transparent shadow. The theoretical profiles of Taylor vortex (dash-dotted) and Oseen vortex (dashed) are displayed as well.*

**Figure 2(a)** shows the experimental data measured in a plane perpendicular to *suction vortex* formed near the inlet to a pump pumping water from reservoir. The flow field is dominated by this single vortex, which has strong divergence component, it slightly moves around the center, and other parameters vary as well. **Figure 2(b)** shows that the vortex profile in this case roughly follows the Oseen vortex; however, in one direction (toward the left-hand side in the figure), the velocity decays even slower than the Oseen vortex model predicts. This is caused probably by the reservoir geometry. This data were measured by prof. Uruba.

**Figure 3** shows the secondary flow in a corner (bottom and left edge of the figure) of a channel, the main flow is perpendicular to the measured plane. The

#### **Figure 3.**

*Example of instantaneous velocity field measured in a plane perpendicular to the main flow through a channel close to the channel corner (physical corner is in the bottom left corner of the figure). When the boundary layers are laminar, a single symmetry-breaking vortex is formed close to the channel corner. (b) Vertical profiles of horizontal velocity and horizontal profiles of vertical velocity across the vortex.*

#### *Searching of Individual Vortices in Experimental Data DOI: http://dx.doi.org/10.5772/intechopen.101491*

#### **Figure 4.**

*(a) Example of instantaneous fluctuating velocity field (fluctuating in respect to the instantaneous field average) with five vortices in each snapshot, no fitting is used. The radius of the vortex is calculated by using the number of grid points contributing to the corresponding patch of Q-criterion. (b) Profiles of the found vortices. The set denoted small contains approx.* 10% *of vortices with the smallest radius, the large set consists of approx* 10% *of the largest vortices. The theoretical profiles of Taylor vortex (dash-dotted), Oseen vortex (dashed), and a vortex with non-potential envelope (dotted) are displayed as well.*

displayed vortex forms, when the boundary layers are laminar, this vortex spontaneously brakes the symmetry, and it leads to faster transition to turbulence of the boundary layers at higher velocities. More details about this measurement can be found in our previous publications [49, 50]. In this case, the vortex profile is pushed toward the Taylor profile, which is caused by the presence of solid wall and thus zero velocity at the left and bottom side. In the upper direction, there is observed even an overshoot of the profile caused by the stream supplying the vortex from the central flow.

**Figure 4** shows the turbulent flow behind a grid; the distance is 200mm, i.e., <sup>12</sup>*:*8*M*, *<sup>M</sup>* is the mesh parameter, Reynolds number is 3*:*<sup>1</sup> 103 . The main flow points from left to right and the convective velocity component is subtracted. More details about this experiment can be found in our previous publication [51]. In this case, the flow field is not dominated by a single vortex; instead, there are more vortices of similar level. A consequence is that the standard deviation is much more massive than in the previous cases. The averaged profile displays velocity decay faster than the potential envelope ( *<sup>r</sup>*1), but not as fast as the Taylor vortex model (7).

A similar velocity profile can be seen in a very different case—the jet flow, see **Figure 5**, which shows the data measured in a plane perpendicular to the jet axis at distance of one nozzle diameter past the nozzle. The jet-generating device misses the flow straightener; therefore the jet core contains turbulence originating in the fan; more details can be found in our conference contribution [52]. The vortices prefitted within the jet core (depicted by the blue rectangle in **Figure 5**) display slightly faster velocity decay than the vortices elsewhere, i.e., mainly in the shear layer.

A highly turbulent flow emerges in the steam turbines; the data measured in a model axial air turbine are shown in **Figure 6**. Here, the strong advection in the axial direction (from left to right in the figure) is subtracted, the rest shows a wide horizontal strip of lower turbulence, which is caused by the rotor jet (fluid passing the interblade channel), this structure overlays a less apparent structure of wakes past rotor wheel, which display as strips of wilder flow in top-bottom direction.

#### **Figure 5.**

*(a) Example of instantaneous in-plane velocity field perpendicular to a turbulent jet axis with five vortices in each snapshot, no fitting is used. (b) Profiles of the found vortices. The set is approximately separated into vortices in the jet core and elsewhere according to the blue rectangle in panel (a).*

#### **Figure 6.**

*(a) Example of instantaneous velocity field in the axial tangential plane inside an axial turbine past the first stage (stator + rotor); the convective velocity in axial direction (from left to right) is subtracted. (b) Profiles of the found vortices. The profiles in axial direction display strong overshoot caused by the pattern of wakes past rotor wheel (such wakes pass the field of view from top to bottom with small left-right drift as the rotor wheel rotates from bottom to top in the field of view perspective).*

This pattern would be better apparent in an averaged image, but here the instantaneous field is shown. More detailed description of this interesting flow can be found in our article [53]. The vortices in this case display a strong asymmetry—in tangential direction (up-down in the figure), their peak velocity is significantly smaller than the peak velocity in axial direction (left-right). In the axial direction, there is a strong "overshoot" of velocity decay, which is caused by the alternating velocity pattern.

The observation made in very different cases does not support the generally accepted hypothesis of potential envelope around the vortex. This envelope forms, when there is only single vortex dominating the flow (see **Figure 2** with single

suction vortex), in the case of turbulent field consisting of multiple vortices, the velocity decay is faster, but not as fast as in the Taylor vortex model (7). Therefore we venture to offer another model of vortex with non-potential envelope

$$u\_{\theta}^{\text{VNPE}}(r) = \frac{\Gamma}{2\pi R} \cdot \frac{r/R}{\left(1 + \frac{1}{4}\left(r/R\right)^2\right)^2} \tag{10}$$

which decays as *r*�<sup>3</sup> at large *r* and at smaller *r*, it roughly follows the Oseen vortex model, see **Figure 1**. It is important to note that this vortex is not a solution of Navier-Stokes equations! It is based on the observations only, and there is no any theoretical argument for it.

Similarly as the Taylor vortex, this model displays a skirt of opposite vorticity as well. The vorticity profile is

$$\alpha\_{\mathbf{z}}(r) = \mathbf{G} \frac{2 - \frac{1}{2} \left(r/R\right)^2}{\left(\mathbf{1} + \frac{1}{4} \left(r/R\right)^2\right)^3}. \tag{11}$$

It reaches zero at the distance

$$0 = 2 - \frac{1}{4} \left(\frac{r\_0}{R}\right)^2 \Rightarrow \frac{r\_0}{R} = 2,\tag{12}$$

see **Figure 1**; since this distance, the vorticity approaches zero from opposite direction as � *<sup>r</sup>*�<sup>4</sup> for large *<sup>r</sup>*. The energy of this vortex model is finite even in unbounded domain:

$$E = \frac{1}{2} \int\_0^\infty \left[ G \frac{r/R}{\left(1 + \frac{\left(r/R\right)^2}{4}\right)^2} \right]^2 \cdot 2\pi r \mathrm{d}r = \frac{4}{3} \pi G^2 R^2. \tag{13}$$

#### **2.3 Vortex prefit**

Any general fitting algorithm falls into some local minimum. This minimum does not need to be the really wanted results, it can be just a small dimple in a wall of huge valley. To avoid this effect, one can (i) modify the fitting algorithm to see larger surroundings of the point, e.g., by using *simulated annealing* [54, 55], or (ii) just start close to the result. The second possibility solves another small issue—the starting point of the fitting algorithm. In the single particular case solved here, it means to find a peak of appropriate scalar variable, which would signify the presence of vortex.

The prefit is sketched in **Figure 7**: the starting point is the spatial scalar field of ffiffiffiffiffiffi *Qd* <sup>p</sup> � sgn*ω*, where sgn*<sup>ω</sup>* <sup>¼</sup> *<sup>ω</sup>* j j *<sup>ω</sup>* is the sign of vorticity. *Qd* is the *<sup>Q</sup>*-invariant with subtracted divergence. Alternatively, any scalar with sparse non-zero values could be used (i.e., not simply the vorticity). Then the separated patches of non-zero signal are detected, see panel (c) of **Figure 6**. The vortex is built up by using the most energetic patch (label 3 in **Figure 7(c)**); the vortex position is the center of mass of the patch, the vortex radius *<sup>R</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffi *n=π* p , where *n* is the number of points of the patch (note that the unit of *R* is the grid point). The circumferential velocity *G* is calculated as the average of tangential projection of the measured velocities at eight locations around the vortex in the distance *R* from its center (crosses in **Figure 7(d)**).

#### **Figure 7.**

*Steps of prefit: (a) calculate the scalar field of* ffiffiffiffiffiffiffi *Q*<sup>þ</sup> *d* q sgn *ω, which produces a lot of zeros, thus the areas with non-zero value (panel (b)) are sparse and thus the percolation does not occur. (c) The patches or individual continuous areas of non-zero* ffiffiffiffiffiffiffi *Q*<sup>þ</sup> *d* q sgn*ω are identified and the one with the largest energy (label three in this case) serves for generating the prefitted vortex. Note that the patches numbered 26 and 30 would merge if the sign of local vorticity was not used to separate the opposite orientations. (d) The prefitted vortex with estimated radius from the number of points in the patch; the crosses around the vortex show the eight positions of velocity estimation.*

The just described procedure does not use the vortex model; therefore it is suitable for velocity profile estimation as has been done in the previous section. On the other hand, vortex parameters are only estimated; therefore the fitting is needed to adapt the vortex parameters to the actual velocity field.

#### **2.4 Vortex fitting**

A single vortex is described by four fitting parameters: the position *x* and *y*, core radius *R*, and circumferential velocity *G*, which is easier to use than the circulation Γ ¼ 2*πG*. Of course, this set of parameters describes only the cases, when the vortex tube crosses the measured plane perpendicularly; other angles might produce deformation from the ideal circular shape. But, as John von Neumann said: *With four parameters I can fit an elephant, and with five I can make him wiggle his trunk* [56]. Therefore, it is preferred to avoid using too many fitting parameters; the listed set is considered to be a minimum. This issue will be a true challenge in the case of instantaneous volumetric data in the future.

The used fitting algorithm is called *Amoeba* [55] or *Downhill simplex method* or *Nelder-Mead* by its inventors [57]. The energy of residual velocity field is calculated for each variant serving as a score. Here comes the need of specific vortex model discussed earlier. The algorithm selects single movement from a closed set of movements in the parameter space in order to keep away from areas with high residual energy and converging to some local minimum. The algorithm is in much more detail described in the book [55].

Once the energy residual of a single vortex reaches local minimum, this vortex is subtracted from the velocity field. Then the entire procedure is repeated by using the residual velocity field as the input.

**Figure 8** shows the results of fitting a single instantaneous velocity field by depicted number of vortices. It is clearly visible that the energy decreases as the field is approximated by more and more vortices. It can be seen as a kind of decomposition, although its effectivity is poor in comparison with pure mathematical approaches, e.g., the Proper Orthogonal Decomposition [8, 9]. On the other hand, it describes the fluctuating velocity field by using objects with clear physical interpretation, while the physical interpretation of POD modes is not straightforward [58]. Still, a question remains here: whatever the found vortices are real. To be specific, in **Figure 8**, a large vortex can be seen even in the first set, the core of this *Searching of Individual Vortices in Experimental Data DOI: http://dx.doi.org/10.5772/intechopen.101491*

#### **Figure 8.**

*Vortices fitted in a single instantaneous velocity field measured past a grid; the same example field as in previous figures. (a) The input velocity field, (b, c, d, e) the velocity field calculated from theoretical vortex profiles. (f, g, h, ch) The residual field, i.e., input field minus the field of found vortices.*

vortex spans out of the field of view, thus no one knows, if the vortex was still there in the case that areas were measured. For example, a simple advective motion can be explained by a pair of huge vortices up and down the measured area. Of course, that is unphysical. As the number of vortices increases, even smaller and smaller vortices are added converging to a situation, that each single noise vector is described by a single vortex. This limit is unphysical as well, but where is the boundary?

**Figure 9** shows the decrease of effectivity of this procedure—as the number of vortices increases, there remains structures less and less similar to a vortex in the instantaneous velocity field. While the first vortex typically covers around 10% of the energy of input fluctuating velocity field (in this case). The convergence of the energy of the rest gets slower, and it becomes to be quite ineffective to describe 75% of the fluctuations by the simple vortices described here. The parameters of found vortices develop as well, see **Figure 10**, which shows the probability density functions of vortex core radii and circumferential velocities. The vortices found later are typically smaller and have smaller circumferential velocity (the positive and negative values count together in the logarithmic plot of **Figure 10(c)** and **(d)**).

#### **Figure 9.**

*(a) Energy of the residual velocity field after subtracting the nth vortex as a function of number of vortices. The area represents the standard deviation of the ensemble of 1476 snapshots. (b) The energy "saved" by nth vortex, again, the area represents the standard deviation of the ensemble.*

#### *Vortex Dynamics - From Physical to Mathematical Aspects*

**Figure 10.**

*Probability density functions (PDFs) of core radii R (a and b) and circumferential velocities G (c and d) for several number of vortices searched in a single instantaneous velocity field—Black: Single vortex in each field, maroon: 5 vortices, yellow: 10 vortices, green: 30 vortices and blue: 100 vortices. The PDF can be weighted by the number of vortices (i.e., one vortex, one vote) or by the energy, panels (b and d).*

#### **3. Results**

The aim of this chapter is mainly to describe the ideas of the developed algorithm. The results and their physical interpretation need more effort in the future. In this section, some ways of result analysis are shown based on the distribution study. As a example case, the grid turbulence is selected, because this is a deeply explored canonical case, see [59] and many more experimental data, e.g., in [4, 60–65].

When exploring the vortices in grid turbulence in dependence on the distance behind the grid or on the Reynolds number, the *effectivity* of vortex fitting remains almost constant. **Figure 11** shows, that the maximum population is around 1 in all cases. There is slow decrease of the number of vortices with low effectivity (i.e., structures with large radius or large velocity causing large theoretical energy, *Et <sup>R</sup>*<sup>2</sup> *G*2 , which do not correspond to the energy saved), and there exist cases with saved energy larger than the theoretical one; however, this distribution decreases much faster.

The vortex core radii in **Figure 12(a)** do not seem to depend on the distance *x* past the grid, although it is known that the characteristic turbulent length scales (Kolmogorov and Integral one) typically increase with distance. At the lowest distance, there can be observed a weak wavening of the distribution. The distribution dependence on Reynolds number is weak as well (**Figure 12(b)**), although a very fine change of vortex core radii scaling at radii larger maximal population. Honestly speaking, the similarity of the distributions is suspicious, and it has to be proven in the future that the shape of the radii distribution is not affected by the measurement spatial resolution (the studied datasets have all the same spatial resolution).

#### *Searching of Individual Vortices in Experimental Data DOI: http://dx.doi.org/10.5772/intechopen.101491*

#### **Figure 11.**

*Probability density functions (PDFs) of the vortex effectivity, i.e., the ratio of energy saved by the probed vortex and the theoretical energy of the vortex. Left panel (a) shows the data at different distances behind the grid, the mesh-based Reynolds number is* <sup>3</sup>*:*<sup>1</sup> � 103*; panel (b) shows the data at different Reynolds number, the distance <sup>x</sup>=<sup>M</sup>* <sup>¼</sup> <sup>12</sup>*:*8*. The "k" in the legend plays for* �103*.*

**Figure 12.**

*Probability density functions (PDFs) of core radii R found in fields of view in several distances past the grid, panel (a); and at several Reynolds numbers, panel (b). The dotted lines highlight scalings of R*�<sup>3</sup> *and R*�<sup>2</sup> *the observed data lie in between. It seems that the scaling exponent slightly decreases with Re.*

The circumferential velocities *G* ¼ Γ*=*2*πR* of the vortices move toward smaller values with increasing distance, see **Figure 13(a)**. This effect is clearly caused by the decreasing turbulence intensity [51] as the vortices are searched within the fluctuating velocity field. **Figure 13(b)** shows that the velocity normalized by the wind tunnel velocity of maximum population increases. At lower velocities, the PDF decrease with increasing *G* displays two regimes, first it decreases slower, then faster, while at higher velocities, only the fast decay is observable. It has to be mentioned, in the light of observations in **Figure 9**, that this effect can be caused by the number of fitted vortices, which do not need to be appropriate for the actual datasets. It is quite difficult to distinguish the effects of the method and the physical phenomena.

The distance to nearest other vortex seems to be unaffected by the grid distance and flow velocity, see **Figure 14**. But the absolute values of the nearest vortex cannot have some physical sense, as this quantity is the first one dependent on the number of searched vortices, thus the vortex density. But the non-changing shape of this distribution suggests that there is nothing like evolution pattern of vortices or vortex lattice.

**Figure 13.**

*Probability density functions (PDF) of core circumferential velocities G for several distances past the grid, panel (a); and several wind tunnel velocities changing the Reynolds number, panel (b).*

#### **Figure 14.**

*Probability density functions (PDF) of the distance to nearest vortex for several distances past the grid, panel (a); and several Reynolds numbers, panel (b).*

#### **4. Conclusions**

The turbulent flows consist of many interacting vortices of all scales, which all together self-organize being responsible for the statistical properties of turbulence. In this contribution, the algorithm for detection of individual vortices via direct fitting of measured velocity field has been presented. It has been shown via the zeroth step of fitting that the velocity profile of vortex in turbulent flow decreases faster than the generally accepted models suggest. This is advantageous, because the energy of vortex with velocity decrease faster than 1*=r* converges. On the other hand, it has a "skirt" of vorticity opposite to the center one. The vortices found in grid turbulence display average radius decreasing with distance and Reynolds number, while the scaling at larger *R* seems to not depend on those parameters. The effective circumferential velocity *G* ¼ Γ*=*2*πR* decreases with distance and increases with Reynolds number (faster than expected linear). The algorithm is still under development and mainly the physical interpretation of the results needs more work in the future studying and comparing results of different flow cases.

#### **Acknowledgements**

Great thanks belong to my colleagues: Václav Uruba and Vitalii Yanovych.

The research was supported from ERDF under project LoStr No. CZ.02.1.01/0.0/ 0.0/16\_026/0008389. The publication was supported by project CZ.02.2.69/0.0/ 0.0/18\_054/0014627.

### **Nomenclature**


### **Abbreviations**


*Vortex Dynamics - From Physical to Mathematical Aspects*

### **Author details**

Daniel Duda University of West Bohemia in Pilsen, Pilsen, Czech Republic

\*Address all correspondence to: dudad@kke.zcu.cz

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Searching of Individual Vortices in Experimental Data DOI: http://dx.doi.org/10.5772/intechopen.101491*

#### **References**

[1] Uriel Frisch and Andre Nikolaevich Kolmogorov. Turbulence: The Legacy of AN Kolmogorov. Cambridge: Cambridge University Press; 1995

[2] La Mantia M, Švančara P, Duda D, Skrbek L. Small-scale universality of particle dynamics in quantum turbulence. Physical Review B. 2016;**94**(18)

[3] Tabeling P, Zocchi G, Belin F, Maurer J, Willaime H. Probability density functions, skewness, and flatness in large reynolds number turbulence. Physical Review E. 1996;**53**:1613-1621

[4] Burgoin M et al. Investigation of the small-scale statistics of turbulence in the modane s1ma wind tunnel. CEAS Aeronautical Journal. 2018;**9**(2):269-281

[5] Azevedo R, Roja-Solórzano LR, Leal JB. Turbulent structures, integral length scale and turbulent kinetic energy (tke) dissipation rate in compound channel flow. Flow Measurement and Instrumentation. 2017;**57**:10-19

[6] Schulz-DuBois EO, Rehberg I. Structure function in lieu of correlation function. Applied Physics. 1981;**24**: 323-329

[7] Kirkpatrick S, Gelatt CD, Vecchi MP. Optimization by simulated annealing. Science. 1983;**220**:671-680

[8] Uruba V. Decomposition methods for a piv data analysis with application to a boundary layer separation dynamics. *Transactions of the VŠB – Technical University of Ostrava* Mechanical Series. 2010;**56**:157-162

[9] Uruba V. Energy and entropy in turbulence decompositions. Entropy. 2019;**21**(2):124

[10] Richardson LF. Atmospheric diffusion shown on a distanceneighbour graph. Proceedings of the Royal Society A. 1926;**110**:709-737

[11] Fiedler HE. Coherent structures in turbulent flows. Progress in Aerospace Sciences. 1988;**25**(3):231-269

[12] Barkley D, Song B, Mukund V, Lemoult G, Avila M, Hof B. The rise of fully turbulent flow. Nature. 2015; **526**(7574):550-553

[13] Valente PC, Vassilicos JC. The decay of turbulence generated by a class of multiscale grids. Journal of Fluid Mechanics. 2011;**687**:300-340

[14] Barenghi CF, Skrbek L, Sreenivasan KR. Introduction to quantum turbulence. Proceedings of National Academy of Sciences of the United States of America. 2014;**111**: 4647-4652

[15] Vinen WF. An introduction to quantum turbulence. Journal of Low Temperature Physics. 2006;**145**(1–4): 7-24

[16] Fonda E, Meichle DP, Ouellette NT, Hormoz S, Lathrop DP. Direct observation of Kelvin waves excited by quantized vortex reconnection. Proceedings of the National Academy of Sciences. 2014;**111**(Supplement\_1): 4707-4710

[17] Helmholtz H. Über integrale der hydrodynamischen gleichungen, welche den wirbelbewegungen entsprechen. Journal für die reine und angewandte Mathematik. 1858;**55**:25-55

[18] Baggaley AW, Laurie J, Barenghi CF. Vortex-density fluctuations, energy spectra, and vortical regions in superfluid turbulence. Physical Review Letters. 2012;**109**(20):205304

[19] La Mantia M, Duda D, Rotter M, Skrbek L. Velocity statistics in quantum turbulence. In: Procedia IUTAM. Vol. 9. 2013. pp. 79-85

[20] Staicu AD. Intermittency in Turbulence. Eidhoven: University of Technology Eidhoven; 2002

[21] Duda D, Švančara P, La Mantia M, Rotter M, Skrbek L. Visualization of viscous and quantum flows of liquid he 4 due to an oscillating cylinder of rectangular cross section. Physical Review B—Condensed Matter and Materials Physics. 2015;**92**(6)

[22] Duda D, Yanovych V, Uruba V. An experimental study of turbulent mixing in channel flow past a grid. PRO. 2020; **8**(11):1-17

[23] Hänninen R, Baggaley AW. Vortex filament method as a tool for computational visualization of quantum turbulence. Proceedings of the National Academy of Sciences of the United States of America. 2014;**111**(Suppl. 1): 4667-4674

[24] Varga E, Babuin S, V. S. L'vov, A. Pomyalov, and L. Skrbek. Transition to quantum turbulence and streamwise inhomogeneity of vortex tangle in thermal counterflow. Journal of Low Temperature Physics. 2017;**187**(5–6):531-537

[25] De Gregorio F, Visingardi A, Iuso G. An experimental-numerical investigation of the wake structure of a hovering rotor by piv combined with a *γ*<sup>2</sup> vortex detection criterion. Energies. 2021;**14**(9):2613

[26] Marchevsky IK, Shcheglov GA, Dergachev SA. On the algorithms for vortex element evolution modelling in 3d fully lagrangian vortex loops method. In Topical Problems of Fluid Mechanics. 2020;**2020**:152-159

[27] Kaufmann W. Über die ausbreitung kreiszylindrischer wirbel in zähen

(viskosen) flüssigkeiten. Ingenieur-Archiv. 1962;**31**(1):1-9

[28] Ben-Gida H, Liberzon A, Gurka R. A stratified wake of a hydrofoil accelerating from rest. Experimental Thermal and Fluid Science. 2016;**70**: 366-380, 105374 p.

[29] Jeong J, Hussain F. On the identification of a vortex. Journal of Fluid Mechanics. 1995;**285**:69-94

[30] Dergachev SA, Marchevsky IK, Shcheglov GA. Flow simulation around 3d bodies by using lagrangian vortex loops method with boundary condition satisfaction with respect to tangential velocity components. Aerospace Science and Technology. 2019;**94**:105374

[31] Graftieaux L, Michard M, Grosjean N. Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows. Measurement Science and Technology. 2001;**12**(9):1422-1429

[32] Koschatzky V, Moore PD, Westerweel J, Scarano F, Boersma BJ. High speed piv applied to aerodynamic noise investigation. Experiments in Fluids. 2011;**50**(4):863-876

[33] Maciel Y, Robitaille M, Rahgozar S. A method for characterizing crosssections of vortices in turbulent flows. International Journal of Heat and Fluid Flow. 2012;**37**:177-188, 118108 p.

[34] Amromin E. Analysis of vortex core in steady turbulent flow. Physics of Fluids. 2007;**19**:118108

[35] Keshavarzi A, Melville B, Ball J. Three-dimensional analysis of coherent turbulent flow structure around a single circular bridge pier. Environmental Fluid Mechanics. 2014;**14**:821-847

[36] Scully MP, Sullivan JP. Helicopter rotor wake geometry and airloads and *Searching of Individual Vortices in Experimental Data DOI: http://dx.doi.org/10.5772/intechopen.101491*

development of laser doppler velocimeter for use in helicopter rotor wakes. In: Technical Report. MIT; 1972. Available from: http://citeseerx.ist.psu. edu/viewdoc/summary?doi= 10.1.1.982.2439

[37] Bhagwat MJ, Leishman JG. Generalized viscous vortex model for application to free-vortex wake and aeroacoustic calculations. Annual Forum Proceedings-American Helicopter Society. 2002;**58**:2042-2057

[38] Vatistas GH, Kozel V, Mih WC. A simpler model for concentrated vortices. Experiments in Fluids. 1991;**11**(1):73-76

[39] Taylor GI. On the Dissipation of Eddies. ACA/R&M-598. London: H.M. Stationery Office; 1918

[40] Wu JZ, Ma HY, Zhou MD. Vorticity and Vortex Dynamics. Berlin Heidelberg New York: Springer; 2006

[41] Tropea C, Yarin A, Foss JF. Springer Handbook of Experimental Fluid Mechanics. Berlin Heidelberg: Springer; 2007

[42] La Mantia M, Skrbek L. Quantum turbulence visualized by particle dynamics. Physical Review B— Condensed Matter and Materials Physics. 2014;**90**(1):1-7

[43] Kurian T, Fransson JHM. Grid-generated turbulence revisited. Fluid Dynamics Research. 2009;**41**(2): 021403

[44] Regunath GS, Zimmerman WB, Tesař V, Hewakandamby BN. Experimental investigation of helicity in turbulent swirling jet using dual-plane dye laser PIV technique. Experiments in Fluids. 2008;**45**(6):973-986

[45] Agrawal A. Measurement of spectrum with particle image velocimetry. Experiments in Fluids. 2005;**39**(5):836-840

[46] Romano GP. Large and small scales in a turbulent orifice round jet: Reynolds number effects and departures from isotropy. International Journal of Heat and Fluid Flow. 2020;**83**: 108571

[47] Jašková D, Kotek M, Horálek R, Horčička J, Kopecký V. Ehd sprays as a seeding agens for piv system measurements. In: ILASS – Europe 2010, 23rd Annual Conference on Liquid Atomization and Spray Systems; 23 September 2010; Brno, Czech Republic. p. 2010

[48] Jiang MT, Law AW-K, Lai ACH. Turbulence characteristics of 45 inclined dense jets. Environmental Fluid Mechanics. 2018;**19**:1-28

[49] Bém J, Duda D, Kovařík J, Yanovych V, Uruba V. Visualization of secondary flow in a corner of a channel. In: AIP Conference Proceedings. Vol. 2189. 2019. pp. 020003-1-020003-6

[50] Duda D, Jelínek T, Milčák P, Němec M, Uruba V, Yanovych V, et al. Experimental investigation of the unsteady stator/rotor wake characteristics downstream of an axial air turbine. International Journal of Turbomachinery, Propulsion and Power. 2021;**6**(3)

[51] Duda D. Preliminary piv measurement of an air jet. AIP Conference Proceedings. 2018;**2047**: 020001

[52] Duda D, Bém J, Yanovych V, Pavlíček P, Uruba V. Secondary flow of second kind in a short channel observed by piv. European Journal of Mechanics, B/Fluids. 2020;**79**:444-453

[53] Duda D, La Mantia M, Skrbek L. Streaming flow due to a quartz tuning fork oscillating in normal and superfluid he 4. Physical Review B. 2017;**96**(2): 024519

[54] Kolář V. Vortex identification: New requirements and limitations. International Journal of Heat and Fluid Flow. 2007;**28**(4):638-652

[55] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge: Cambridge University Press; 2007

[56] Meyer J, Khairy K, Howard J. Drawing an elephant with four complex parameters. American Journal of Physics. 2010;**78**:648-649

[57] Nelder JA, Mead R. A simplex method for function minimization. The Computer Journal. 1965;**7**(4): 308-313

[58] Uruba V, Hladík O, Jonáš P. Dynamics of secondary vortices in turbulent channel flow. Journal of Physics: Conference Series. 2011;**318**: 062021

[59] Kuzmina K, Marchevsky I, Soldatova I. The high-accuracy numerical scheme for the boundary integral equation solution in 2d lagrangian vortex method with semianalytical vortex elements contribution accounting. In: Topical Problems of Fluid Mechanics 2020. Prague: Czech Academy of Sciences; 2020. pp. 122-129

[60] Comte-Bellot G, Corrsin S. The use of a contraction to improve the isotropy of grid-generated turbulence. Journal of Fluid Mechanics. 1966;**25**:657-682

[61] Wierciński Z, Grzelak J. The decay power law in turbulence. Transactions of the Institute of Fluid-flow Machinery. 2015;**130**:93-107

[62] Jonáš P, Mazur O, Uruba V. On the receptivity of the by-pass transition to the length scale of the outer stream turbulence. European Journal of Mechanics, B/Fluids. 2000;**19**(5): 707-722

[63] Mohamed MS, LaRue JC. The decay power law in grid-generated turbulence. Journal of Fluid Mechanics. 1990;**219**: 195-214

[64] Roach PE. The generation of nearly isotropic turbulence by means of grids. International Journal of Heat and Fluid Flow. 1987;**8**:82-92

[65] Warhaft Z, Lumley JL. An experimental study of the decay of temperature fluctuations in grid generated turbulence. Journal of Fluid Mechanics. 1978;**88**:659-684

#### **Chapter 4**

## Vortex Dynamics in Complex Fluids

*Naoto Ohmura, Hayato Masuda and Steven Wang*

#### **Abstract**

The present chapter provides an overview of vortex dynamics in complex fluids by taking examples of Taylor vortex flow. As complex fluids, non-Newtonian fluid is taken up. The effects of these complex fluids on the dynamic behavior of vortex flow fields are discussed. When a non-Newtonian shear flow is used in Taylor vortex flow, an anomalous flow instability is observed, which also affects heat and mass transfer characteristics. Hence, the effect of shear-thinning on vortex dynamics including heat transfer is mainly referred. This chapter also refers to the concept of new vortex dynamics for chemical process intensification technologies that apply these unique vortex dynamics in complex fluids in Conclusions.

**Keywords:** Taylor vortex flow, complex fluid, non-Newtonian fluid, heat transfer, process intensification

#### **1. Introduction**

Historically, innovative processes have been created using organized vortices. For example, in Japan, Kiyomasa Kato, a Sengoku daimyo (Japanese territorial lord in the Sengoku period) in Kumamoto Prefecture, made a canal (called "hanaguri canal") with a partition (baffle) having a semicircular hole at the bottom as shown in **Figure 1**. The flow velocity of the water flowing through the hole in the lower part of the partition increases due to the effect of the contraction of the flow, and a strong circulating vortex is formed in the water channel divided by the partition. By intensifying the flow in the canal, water can be supplied to about 95 ha of land in nine villages in the downstream without piling up volcanic ash or earth and sand, and the harvest has increased about three times. Based on this idea by Kiyomasa Kato, in order to solve the particle sedimentation problem in oscillatory baffled reactors (OBR) which is one of the hopeful process intensification techniques, our group [1] succeeded in preventing the particle sedimentation to the bottom of the reactor and obtaining extremely monodispersed particles in a calcium carbonate crystallization process by changing from a normal baffle with a hole in the center to a snout-type baffle as shown in **Figure 2**.

In addition, the function of vortex flow is not only to intensify the previously noticed transport phenomena such as mixing, heat transfer, and mass transfer, but also to have a new function that has not been previously noticed, such as classification and separation of particles. Ohmura et al. [2] found that particles with different

**Figure 1.** *Schematic of Hanaguri Canal.*

**Figure 2.**

*Comparison of performance of an oscillatory baffled crystallizer between using normal and Hanaguri-type baffles.*

sizes move on different streamlines within a Taylor cell and proposed that this could be applied to a particle classification device. Kim et al. [3] applied this idea to a continuous crystallizer and proposed a device for granulating particles of different sizes while classifying them. Wang et al. [4] also proposed a novel solid–liquid separation system that breaks the conventional stereotype of mixing equipment by applying the particle clustering phenomenon in isolated mixing regions in stirring tanks. In this way, vortices with a systematic structure have very attractive properties, such as solid accumulation, mixing and reaction enhancement, particle classification, and mass transport. If we can understand the characteristics of this organized vortex structure and manipulate it freely, we may be able to develop innovative chemical processes.

In many industrial processes, such as chemical, food, and mineral processes, the fluids handled are not only simple homogeneous Newtonian fluids, but also often complex fluids, such as non-Newtonian fluids, multi-phase fluids with highly dispersed phases, and viscoelastic fluids. Therefore, in order to apply the new "vortex dynamics" currently being constructed to process intensification technologies and implement it in society, it is necessary to develop the concept of new "vortex dynamics" from simple fluids to complex fluids. According to the abovementioned background, the present chapter provides an overview of vortex dynamics in complex fluids by taking examples of Taylor vortex flow.

#### **2. Vortex dynamics with non-Newtonian fluids**

A non-Newtonian fluid property causes a multiple fluid motion. These motions are quite interesting from fundamental and practical viewpoints. Especially, in vortex flow systems, fluid elements experience curved streamlines. In polymeric fluid systems, the polymer molecule chain does not line along curved stream lines, and consequently, hoop stress in a normal direction occurs. As a result, coupling normal stresses and curved streamlines causes elastic instabilities [5]. These instabilities are observed in various flows, e.g., Poiseuille flow [6], microchannel flow [7], and swirling flow [8]. Many polymeric fluids show not only viscoelastic behavior but also shear-thinning behavior. The shear-thinning property causes the viscosity distribution accompanied by the shear-rate distribution in the fluid system. Coelho and Pinho [9] showed that the shear-thinning affects the flow transition of vortex shedding in a cylinder flow. Ascanio et al. [10] reported that the mixing process of shear-thinning fluids under a time-periodic flow field is different from that of Newtonian fluid. Thus, vortex dynamics in non-Newtonian fluid systems is far from complete.

To investigate the effect of non-Newtonian property on vortex dynamics in more detail, many researchers have been utilizing Taylor–Couette flow, which is one of the most canonical flow systems in fluid mechanics, with non-Newtonian fluids [11–14]. Taylor–Couette flow is the flow between coaxial cylinders with the inner one rotating. This flow shows a cascade transition from laminar Couette flow to fully turbulent wavy vortex flow with the increase in circumferential Reynolds number (*Re*). When the value of *Re* exceeds the critical *Re* (*Re*cr), Taylor vortex flow firstly appears. As mentioned above, many researchers have been studied the Taylor–Couette flow with non-Newtonian fluids. For example, Muller et al. [11] and Larson et al. [12] revealed that the elastic instability occurs in Taylor–Couette flow and organized flow modes based on Deborah number (*De*), which the ratio of a characteristic relaxation time of the fluid to a characteristic residence time in the flow geometry [5]. **Figure 3** shows laminar Taylor–Couette flow with Newtonian (40 wt% glycerol aqueous solution) and viscoelastic fluid (0.75 wt% sodium polyacrylate aqueous solution).

The flow pattern was visualized by adding a small amount of Kalliroscope AQ-1000 flakes. As shown in **Figure 3**, the cellular structure of Taylor vortices seems to be complicated in the viscoelastic fluid even at the relatively low *Re*. The detailed mechanism is found in their papers [11–14]. Other interesting point is an

#### **Figure 4.**

*Viscosity distribution in the annular space obtained by numerical simulation [15]. The fluid was assumed to be a shear-thinning fluid.*

enlarged vortex structure by shear-thinning property. Escudier et al. [15] found that the cellular vortex is axially stretched and the vortex eye (the location of zero axial velocity in the vortex interior) is radially shifted toward the center body.

However, the first Taylor–Couette instability has not been fully understood yet in non-Newtonian fluid systems. One of the reasons is the discrepancy between *Re*cr reported by several researchers for non-Newtonian fluids. Alibenyahia et al. [16] reviewed the discrepancy; Jastrebski et al. [17] reported *Re*cr decreased with the shear-thinning property, on the other hand, Caton et al. [18] found the opposite tendency. Actually, this discrepancy is explained by the difference in how to define the effective Reynolds number, *Re*eff, in their papers. In non-Newtonian fluids, how to define *Re* is quite complicated because the viscosity locally varies as shown in **Figure 4** [19]. Practically, *Re*eff based on the effective viscosity in the system should be discussed. Several researchers have been trying to define more rational *Re*eff in various flow systems, e.g., rising bubble flow in shear-thickening fluid [20], Rayleigh–Bénard convection with shear-thinning fluids [21], and non-Newtonian fluid flow past a circular cylinder [22].

We previously proposed a new definition of *Re*eff based on the effective viscosity (*η*eff), which is obtained by numerical simulation. *η*eff is calculated by averaging the locally distributed viscosity using a weight of dissipation function as follows [23]:

$$\eta\_{\rm eff} = \sum\_{i=1}^{N} \dot{\gamma}\_i^2 \eta\_i \Delta V\_i / \sum\_{i=1}^{N} \dot{\gamma}\_i^2 \Delta V\_i,\tag{1}$$

where *N* is the total mesh number, *η<sup>i</sup>* [Pa�s] is the local viscosity, *γ*\_*<sup>i</sup>* [1/s] is the local shear rate, and Δ*Vi* [m3 ] is the local volume for each cell. It should be noted that *η*eff is obtained using numerical simulation. The computational domain is shown in **Figure 5**. The governing equations are as follows:

$$\nabla \cdot \mathbf{u} = \mathbf{0},\tag{2}$$

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{\nabla p}{\rho} + \frac{1}{\rho} \nabla \cdot (2\eta \mathbf{D}) + \mathbf{g},\tag{3}$$

**Figure 5.** *Computational domain [22].* R*<sup>i</sup> and* R*<sup>o</sup> are the radii of inner and outer cylinders, respectively.*

#### *Vortex Dynamics in Complex Fluids DOI: http://dx.doi.org/10.5772/intechopen.101423*

where **u** [m/s] is the velocity, *p* [Pa] is the pressure, *ρ* [kg/m<sup>3</sup> ] is the density, *η* [Pa�s] is the viscosity depending on the shear rate, **<sup>D</sup>** (= (∇**<sup>u</sup>** <sup>+</sup> <sup>∇</sup>**u**<sup>T</sup> ) / 2) [1/s] is the rate of deformation tensor, **g** [m/s<sup>2</sup> ] is the gravitational acceleration. The rheological property is characterized by Carreau model as follows [24]:

$$\eta = \eta\_0 \left[ \mathbf{1} + (\boldsymbol{\beta} \cdot \dot{\boldsymbol{\gamma}})^2 \right]^{(n-1)/2},\tag{4}$$

where *η*<sup>0</sup> [Pa�s] is the zero shear-rate viscosity, *γ*\_ [1/s] is the shear rate, *β* [s] is the characteristic time, and *n* [�] is the power index, which indicates the slope of decreasing viscosity with shear rate. In the case of *n* < 1, the fluid shows the shearthinning behavior. The detailed information of numerical procedure is written in our paper [23].

**Figure 6** shows the critical value of *Re*eff for various shear-thinning fluids as a function of gap ratio *R*<sup>i</sup> / *R*o. The theoretical *Re*cr for Newtonian fluids derived by Taylor [25] was denoted by the dashed line in **Figure 6**. It is found that the critical *Re*eff for shear-thinning fluids was in agreement with the theoretical value at *R*<sup>i</sup> / *R*<sup>o</sup> > 0.7. Thus, *Re*eff defined based on *η*eff by Eq. (1) is rational as a practical basis. The effect of shear-thinning property on the vortex structure is also interesting from the viewpoint of fluid dynamics. **Figure 7** shows the number of pairs of Taylor

**Figure 6.** Re*cr for various shear thinning [18].*

**Figure 7.** *Variation in the number of pairs of Taylor vortices [23].*

cells, *N*, as a function of *Re*eff at the aspect ratio *Γ* = 20 [26]. In all fluid systems, *N* tended to increase with *Re*eff. This tendency agrees with reports by other researchers [27]. Furthermore, the shear-thinning property seems to make Taylor cells large because *N* decreases with the shear-thinning property at the same degree of *Re*eff. This tendency was remarkable in the case of *n* = 0.3. This means that the shearthinning property axially enlarges Taylor cells. Although the detailed mechanism of enlarging Taylor cells is under consideration, it will be clarified by numerical simulation of development process of Taylor vortices.

We also introduce heat transfer characteristics of Taylor–Couette flow with shear-thinning fluids. In addition to Eqs. (2) and (3), energy equation was solved:

$$\frac{\partial}{\partial t} \left( \rho \mathbf{C}\_{\mathrm{p}} \right) + \nabla \cdot \left( \rho \mathbf{C}\_{\mathrm{p}} T \mathbf{u} \right) = \nabla \cdot (\kappa \nabla T), \tag{5}$$

where *C*<sup>p</sup> [J/kg�K] is the specific heat capacity,*T* [K] is the temperature, and *κ* [J/m�s�K] is the thermal conductivity. **Figure 8** shows the axial variation in the local Nusselt number, *Nu*L, at the surface of the outer cylinder at *Re*eff = 158 [26]. The *Nu*<sup>L</sup> at the surface of the outer cylinder was calculated as follows:

$$N\mu\_{\rm L} = \frac{2hd}{\kappa},\tag{6}$$

where *h* is a local heat transfer coefficient. As clearly shown in **Figure 6**, *Nu*<sup>L</sup> decreases with the increase in the shear-thinning property. This decrease is explained by increasing the thickness of velocity boundary layer for shear-thinning fluid systems (**Figure 9**). Generally speaking, it is said that the shear-thinning property improves heat transfer performance at same *Re* [28, 29]. This is because the viscosity reduction by the shear-thinning property is not adequately reflected in *Re* used in papers. In other words, the actual flow condition is underestimated in the case of shear-thinning fluids. Thus, the heat transfer performance is not accurately compared between Newtonian and shear-thinning fluids unless *Re*eff is used for representation of flow condition.

#### **Figure 8.**

*Axial variation in the local Nusselt number (*Nu*L) along the surface of the outer cylinder at* Re*eff = 158 [23].* λ*eff is the wavelength of Taylor cells.*

*Vortex Dynamics in Complex Fluids DOI: http://dx.doi.org/10.5772/intechopen.101423*

**Figure 9.** *Dependence of the dimensionless thickness of the velocity boundary layer [23].*

#### **3. Conclusions**

In this section, we mainly refer the effect of shear-thinning on vortex dynamics including heat transfer. However, the viscoelastic property further complicates vortex dynamics as shown in **Figure 3**. In the future, vortex dynamics and transport phenomena in viscoelastic fluid systems should be investigated in more detail. In this case, it is considered to be important to construct a mathematical model by multi-scale analysis focusing on the interaction among scales of microstructure (molecular structure of polymers, micelles, particles, etc.), mesostructure (entanglement of polymer, particle aggregation, etc.), and macrostructure (vortex flow) of complicated fluid. For example, when a polymer solution flows in a micro channel having a sharp contraction part, an unsteady vortex called viscoelastic turbulence is generated in a corner part of the contraction part at higher Weissenberg number [30]. When the scale of the microchannel becomes small, the scale of the flow can be compared with the scale of the polymer. Since the influence of the elasticity derived from the deformation of the polymer itself on the flow becomes large, there is a possibility that the dynamic characteristics of the vortex generated in the contraction part can be controlled by the channel shape. In order to construct a methodology of controlling the viscoelastic vortex, a multi-scale simulation combined with molecular dynamics and computational fluid dynamics may be important.

As this viscoelastic vortex example shows, the field in which the vortex occurs affects the characteristics of the vortex. In the case of a Taylor vortex flow system, for example, the structure and dynamic characteristics of the vortices largely depend on the surface properties. It has been reported that heat transfer is enhanced by processing regular unevenness in the circumferential direction on the outer cylinder surface [31]. In the case of conical Taylor vortex flow, our previous work [32] successfully reproduced the phenomenon that the vortices move upward spontaneously under specific conditions by numerical analysis, and it was found that mass transfer was enhanced in polymer fluid system. In this way, it is possible to control the characteristics of the vortex flow by a structurally organized (having low entropy or fractal) nonuniform field rather than simply a random (highentropy) nonuniform field. Therefore, in order to systematize a new vortex dynamics for freely manipulating vortices, it is necessary to quantitatively express the heterogeneity by introducing the concept of entropy and fractal and to clarify the relationship between the structure of the field and the characteristics of vortices.

### **Acknowledgements**

This work was supported by the KAKENHI Grant-in-Aid for Scientific Research (A) JP18H03853 and the Fostering Joint International Research (B) JP19KK0127.

### **Conflict of interest**

The authors declare no conflict of interest.

#### **Author details**

Naoto Ohmura<sup>1</sup> \*, Hayato Masuda<sup>2</sup> and Steven Wang<sup>3</sup>


\*Address all correspondence to: ohmura@kobe-u.ac.jp

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Vortex Dynamics in Complex Fluids DOI: http://dx.doi.org/10.5772/intechopen.101423*

### **References**

[1] Amano K, Horie T, Ohmura N, Watabe Y. Analysis of fluid dynamics in an oscillatory baffled reactor for continuous crystallization. In: Proceedings of the 6th International Workshop on Process Intensification (IWPI2018). Taipei: National Taiwan University; 2018. pp. 106-107

[2] Ohmura N, Suemasu T, Asamura Y. Particle classification in Taylor vortex flow with an axial flow. Journal of Physics: Conference Series. 2005;**14**: 64-71. DOI: 10.1088/1742-6596/14/009

[3] Kim JS, Kim DH, Gu B, Kim DY, Yang DR. Simulation of Taylor-Couette reactor for particle classification using CFD. Journal of Crystal Growth. 2013; **373**:106-110. DOI: 10.1016/j. crysgro.2012.12.006

[4] Wang S, Metcalfe G, Stewart RL, Wu J, Ohmura N, Feng X, et al. Solidliquid separation by particle-flowinstability. Energy & Environmental Science. 2014;**7**:3982-3988. DOI: 10.1039/c4ee02841d

[5] Pakdel P, McKinley GH. Elastic instability and curved streamlines. Physical Review Letters. 1996;**77**: 2459-2462. DOI: 10.1103/ PhysRevLett.77.2459

[6] Joo YL, Shaqfeh ESG. Viscoelastic Poiseuille flow through a curved channel: A new elastic instability. Physics of Fluid A: Fluid Dynamics. 1991;**3**:2043-2046. DOI: 10.1063/1.857886

[7] Hong SO, Cooper-White JJ, Kim JM. Inertio-elastic mixing in a straight microchannel with side wells. Applied Physics Letters. 2016;**108**:014103. DOI: 10.1063/1.4939552

[8] Yao G, Yang H, Zhao J, Wen D. Experimental study on flow and heat transfer enhancement by elastic instability in swirling flow. International Journal of Thermal Sciences. 2020;**157**: 106504. DOI: 10.1016/j. ijthermalsci.2020.106504

[9] Coelho PM, Pinho F. Vortex shedding in cylinder flow of shearthinning fluids. I. Identification and demarcation of flow regime. Journal of Non-Newtonian Fluid Mechanics. 2003; **110**:110143-110176. DOI: 10.1016/ S0377-0257(03)00007-7

[10] Ascanio G, Foucault S, Tanguy PA. Time-periodic mixing of shear-thinning fluids. Chemical Engineering Research and Design. 2004;**82**:1199-1203. DOI: 10.1205/cerd.82.9.1199.44155

[11] Muller SJ, Larson RG, Shaqfeh ESG. A purely elastic transition in Taylor-Couette flow. Rheologica Acta. 1989;**28**: 499-503. DOI: 10.1007/BF01332920

[12] Larson R, Shaqfeh E, Muller S. A purely elastic instability in Taylor– Couette flow. Journal of Fluid Mechanics. 1990;**218**:573-600. DOI: 10.1017/S0022112090001124

[13] Groisman A, Steinberg V. Couette-Taylor flow in a dilute polymer solution. Physical Review Letters. 1996;**77**: 1480-1483. DOI: 10.1103/ PhysRevLett.77.1480

[14] Cagney N, Lacassagne T, Balabani S. Taylor–Couette flow of polymer solutions with shear-thinning and viscoelastic rheology. Journal of Fluid Mechanics. 2020;**905**:A28. DOI: 10.1017/jfm.2020.701

[15] Escudier MP, Gouldson IW, Jones DM. Taylor vortices in Newtonian and shear-thinning liquids. Proceedings of The Royal Society A. 1995;**449**: 155-176. DOI: 10.1098/rspa.1995.0037

[16] Alibenyahia B, Lemaitre C, Nouar C, Ait-Messaoudene. Revisiting the stability of circular Couette flow of shear-thinning fluids. Journal of Non-Newtonian Fluid Mechanics. 2012; **183-184**:37-51. DOI: 10.1016/j. jnnfm.2012.06.002

[17] Jastrzębski M, Zaidani HA, Wroņski S. Stability of Couette flow of liquids with power law viscosity. Rheologica Acta. 1992;**31**:264-273. DOI: 10.1007/BF00366505

[18] Caton F. Linear stability of circular Couette flow of inelastic viscoplastic fluids. Journal of Non-Newtonian Fluid Mechanics. 2006;**134**:148-154. DOI: 10.1016/j.jnnfm.2006.02.003

[19] Masuda H, Horie T, Hubacz R, Ohmura N, Shimoyamada N. Process development of starch hydrolysis using mixing characteristics of Taylor vortices. Bioscience, Biotechnology, and Biochemistry. 2017;**81**:755-761. DOI: 10.1080/09168451.2017.1282806

[20] Ohta M, Kimura S, Furukawa T, Yoshida Y, Sussman M. Numerical simulations of a bubble rising through a shear-thickening fluid. Journal of Chemical Engineering of Japan. 2012;**45**: 713-720. DOI: 10.1252/jcej.12we041

[21] Jenny M, Plaut E, Briard A. Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids. Journal of Non-Newtonian Fluid Mechanics. 2015;**219**:19-34. DOI: 10.1016/j.jnnfm.2015.03.002

[22] Ohta M, Toyooka T, Matsukuma. Numerical simulations of Carreaumodel fluid flows past a circular cylinder. Asia-Pacific Journal of Chemical Engineering. 2020;**15**:e2527. DOI: 10.1002/apj.2527

[23] Masuda H, Horie T, Hubacz R, Ohta M, Ohmura N. Prediction of onset of Taylor-Couette instability for shearthinning fluids. Rheologica Acta. 2017;**56**: 73-84. DOI: 10.1007/s00397-016-0987-7

[24] Carreau PJ. Rheological equations from molecular network theories. Transactions of the Society of Rheology. 1972;**16**:99-127. DOI: 10.1122/1.549276

[25] Taylor GI. Stability of a viscous liquid contained between two rotating cylinders. Philosophical Transactions of the Royal Society A. 1923;**223**:289-343. DOI: 10.1098/rsta.1923.0008

[26] Masuda H, Shimoyamada M, Ohmura N. Heat transfer characteristics of Taylor vortex flow with shearthinning fluids. International Journal of Heat and Mass Transfer. 2019;**130**: 274-281. DOI: 10.1016/j. ijheatmasstransfer.2018.10.095

[27] Neitzel GP. Numerical computation of time-dependent Taylor-vortex flows in finite-length geometries. Journal of Fluid Mechanics. 1984;**141**:51-66. DOI: 10.1017/S0022112084000732

[28] Izadpanah E, Rabiee MB, Sadeghi H, Talebi S. Effect of rotating and oscillating blade on the heat transfer enhancement of non-Newtonian fluid flow in a channel. Applied Thermal Engineering. 2017;**113**:1277-1282. DOI: 10.1016/j. applthermaleng.2016.11.124

[29] Crespí-Llorens D, Vicente P, Viedma A. Experimental study of heat transfer to non-Newtonian fluids inside a scraped surface heat exchanger using a generalization method. International Journal of Heat and Mass Transfer. 2018;**118**:75-87. DOI: 10.1016/j. ijheatmasstransfer.2017.10.115

[30] Rodd LE, Cooper-White JJ, Boger DV, McKinley GH. Role of the elasticity number in the entry flow of dilute polymer solutions in microfabricated contraction geometries. Journal of Non-Newtonian Fluid Mechanics. 2007;**143**:170-191. DOI: 10.1016/j.jnnfm.2007.02.006

[31] Nouri-Borujerdi A, Nakhchi ME. Optimization of the heat transfer

*Vortex Dynamics in Complex Fluids DOI: http://dx.doi.org/10.5772/intechopen.101423*

coefficient and pressure drop of Taylor-Couette-Poiseuille flows between an inner rotating cylinder and an outer grooved stationary cylinder. International Journal of Heat and Mass Transfer. 2017;**108**:1449-1459. DOI: 10.1016/j.ijheatmasstransfer. 2017.01.014

[32] Masuda H, Iyota H, Ohmura N. Global convection characteristics of conical Taylor–Couette flow with shearthinning fluids. Chemical Engineering & Technology. 2021;**44**:2049-2055. DOI: 10.1002/ceat.202100236

#### **Chapter 5**

## Vortex Analysis and Fluid Transport in Time-Dependent Flows

*Stefania Espa, Maria Grazia Badas and Simon Cabanes*

#### **Abstract**

In this contribution, we present a set of procedures developed to identify fluid flow structures and characterize their space-time evolution in time-dependent flows. In particular, we consider two different contests of importance in applied fluid mechanics: 1) large-scale almost 2D atmospheric and oceanic flows and 2) flow inside the left ventricle in the human blood circulation. For both cases, we designed an ad hoc experimental model to reproduce and deeply investigate the considered phenomena. We will focus on the post-processing of high-resolution velocity data sets obtained via laboratory experiments by measuring the flow field using a technique based on image analysis. We show how the proposed methodologies represent a valid tool suitable for extracting the main patterns and quantify fluid transport in complex flows from both Eulerian and Lagrangian perspectives.

**Keywords:** pattern identification, laboratory experiments, image analysis, rotating turbulence, flow in the left ventricle

#### **1. Introduction**

In most of the fluid flows of interest in nature and technology (i.e., geophysical flows, blood flow in the human circulation as well as flows in turbomachinery and around vehicles) the presence of turbulence in normally observed; therefore, their reproducibility and repeatability have always represented a crucial issue. In this regard, it is widely recognized that laboratory experiments represent a valid tool for the simulation and investigation of complex fluid flows under controlled conditions. With the improvement of measuring techniques, the possibility of acquiring huge high-resolution data sets in space and time is continually increasing. It is then fundamental to consider procedures suitable for a proper analysis of these data aimed at the definition and the characterization of the main flow pattern and of their evolution. In this contribution, we consider two examples of different contests of importance in applied fluid mechanics: 1) β-plane turbulence in the framework of large-scale almost 2D atmospheric and oceanic flows and 2) effect of artificial valves on the flow in the left ventricle in the framework of an in vitro model of human blood circulation. In both cases, the complexity of the flow arises from the embedded non-linear phenomena i.e., interaction of structures at different scales, the interplay between vortices waves and turbulence, anisotropy in the energy

transfers, and in transport phenomena. Due to chaotic advection, the Lagrangian motion of passive particles can be very complex even in regular, i.e., non-turbulent, flow fields [1] as in the situations here discussed in which we considered almost 2D and time-periodic velocity fields. The chapter is organized as follows. In Section 2, we describe the case studies and the considered experimental apparatus. Theory, its application to the experiments, and the different post-processing methodology are described in Section 3, Section 4 contains some results. We discuss and give our conclusions in Section 5.

#### **2. Material and methods**

We provide below the description of the experimental models designed to reproduce: 1) turbulent flows affected by a *β*-effect, 2) the flow downstream a natural/artificial valve in the left ventricle as well as an overview of the technique used to measure the velocity fields.

#### **2.1 Rotating turbulent flows with a** *β***-effect**

In rotating turbulent flows, the latitudinal variation of the Coriolis parameter, the so-called *β*-effect, may redirect the upward energy flux towards the zonal modes thus inducing the anisotropization of the inverse energy cascade, typically observed in large-scale geophysical flows. Due to the combined effects of planetary rotation, topographical constraints, and fluid stratification, these circulations can be assumed quasi-two-dimensional to the first degree of approximation. Actually, the anisotropic inverse energy cascade represents one of the leading causes for the formation and maintenance of jet-like structures along the zonal direction, the socalled zonation [2–5] observed in the atmospheres of the Giant Planets and in the terrestrial oceans. These environments are characterized by the existence of a banded structure, i.e., eastward and westward zonal flows, as well as by the coexistence of turbulence and waves on all scales [6].

In this contest, in addition to the characteristic scales of 2D turbulence [7] associated with the small-scale forcing *kf* and the large-scale friction *kfr*, two more wavenumbers have to be considered: the Rhines wavenumber *kRh* and the transitional wavenumber *kβ*. The Rhines wavenumber is defined as the scale at which the velocity root-mean-square *URMS* is equal to the phase speed of Rossby waves *kRh* = (*β*/2*URMS*) 1/2 [2] and can be related to the meridional size of the jet. If the flow is continuously forced at small scales and at a constant rate *ε*, the balance between the eddy characteristic time and the Rossby wave period exists in correspondence of *k<sup>β</sup>* = (*β* <sup>3</sup> /*ε*) 1/5, i.e., the so-called anisotropic transitional wavenumber which characterizes the threshold of the inverse cascade anisotropization [8]. The ratio between the transitional wavenumber and the Rhines wavenumber provides the nondimensional number, *R<sup>β</sup>* = *kβ*/*kRh*, known as the zonostrophy index. This represents a key parameter in turbulent flows subjected to a *β*-effect, since it discerns different flow regimes of the so-called *β*-plane turbulence. Indeed *R<sup>β</sup>* < 1.5 pertains to flows with strong large-scale friction (friction dominated regime), in the range 1.5 < *R<sup>β</sup>* < 2.5 a flow shows a transitional behavior, and for *R<sup>β</sup>* > 2.5 a flow develops within the regime of zonostrophic turbulence [8].

To deeply investigate these features, we carried out several experimental campaigns in a rotating tank facility available at the Hydraulics Laboratory of the Sapienza University of Rome. As reported in previous papers [9, 10], the

*Vortex Analysis and Fluid Transport in Time-Dependent Flows DOI: http://dx.doi.org/10.5772/intechopen.105196*

experimental setup consists of a square tank 1 m in diameter placed on a rotating table whose imposed rotation is counter-clockwise in order to emulate flows in the Northern hemisphere of a planet. To simulate the dynamics associated with the latitudinal variation of the Coriolis parameter in the Polar Regions, we consider the effects induced by the parabolic shape assumed by the free surface of a rotating fluid. In fact, it is represented by a quadratic variation in *r*, being *r* the radial distance from the pole and assuming the pole as the reference point (polar *β*-plane or *γ* plane approximation) [10, 11]. In this model, the center of the tank (i.e., the point of maximum depression of the fluid surface) represents the pole, while the periphery of the domain corresponds to lower latitudes.

In particular, a local Cartesian frame of reference at the midlatitude of the tank (*rm* = *R*/2; where *R* is the radius of the tank) was considered to evaluate the strength of the *β* term in each experiment [9]. We run a huge set of experiments by changing the main parameters of the flow, i.e. the rotation rate of the system, the fluid thickness, the amount of energy introduced into the system as well as the forcing characteristics [12–15]. Here, we focus on the analysis of the flow induced by a localized forcing, i.e. the formation of a single eastward/westward jet. To this aim, we consider an electromagnetic forcing obtained with the Lorentz force arising from the interaction of a horizontal electric field and a vertical magnetic field.

We perform a set of runs in which the magnets are located along an arc of latitude in the range 180° < *φ* < 360° at a distance *r* = 17 cm from the pole; the considered angular velocity and fluid depth at rest are *Ω* = 3rads<sup>1</sup> and *H0* = 4 cm, respectively. To force the flow, we considered the same orientation of polarity chosen such as to introduce an eastward/westward momentum and facilitate the formation of an eastward/westward zonal jet; in fact, the stationary position of the magnets locked the jet's location. In each of these runs, we vary the intensity of the current in the range 2A ≤ *I* ≤ 6A; the forcing was continuously applied for all the duration of the experiments.

#### **2.2 Flow in the left ventricle in the human blood circulation**

The overall functionality of the heart pump is strongly related to the intraventricular flow features. Complexity in the ventricular flow is mainly due to fluid-wall interactions and turbulence onset in correspondence of the boundaries, threedimensionality, and asymmetry in the pattern development. Here, the focus is on the investigation of the flow in the left ventricle (LV) during a cardiac cycle: it consists of an intense jet forming downstream of the mitral valve and in the development of the related coherent structures i.e., a vortex ring, which grows up during the systole, impinges on the ventricle walls and vanishes almost completely during the systole. A deeper analysis of the flow pattern evolution has shown on one hand that the observed flow structure appears to be favorable to ejection through the aortic valve during the systole [16] and on the other hand the mutual relationships between the formation and development of coherent structures in the LV and its functionality. Actually, one of the main reasons for the deviation from physiological conditions is represented by the replacement of the mitral valve with a prosthetic one, which obviously causes deep modifications in the hemodynamics and, consequently, in the associated flow pattern [17–19].

We reproduce in the laboratory the ventricular flow by means of a pulse duplicator widely described in previous papers [19–21], below we summarize its working principle. A flexible, transparent sack made of silicone rubber (wall

thickness 0.7 mm) simulates the LV allowing at the same time for the optical access. The model ventricle is fixed on a circular plate, 56 mm in diameter, and connected to a constant-head tank by means of two Plexiglas conduits. Along the outlet (aortic) conduit a check valve was mounted, whereas different types of valves were placed on the inlet (mitral) orifice.

We consider three different scenarios: a) the inlet was designed in order to obtain a uniform velocity profile at the orifice mimicking physiological conditions, b) a monoleaflet (Bjork–Shiley monostrut) in mitral position 3) a bileaflet bicarbon prosthetic valve in mitral position; both valves were 31 mm in nominal diameter. The model of the LV was placed in a rectangular tank with Plexiglas (transparent) walls; its volume changed according to the motion of the piston, placed on the side of the tank. The piston was driven by a linear motor, controlled by means of a speed-feedback servo-control. The motion assigned to the linear motor was tuned to reproduce the volume change by clinical data acquired in vivo by echo-cardiography on a healthy subject [20].

#### **2.3 Measuring technique**

Two-dimensional velocity fields are measured by means of an image analysis technique called Feature Tracking, FT [22, 23]. The measurement chain can be summarized in the following steps: 1) identification of a proper measurement plane in the fluid domain; 2) seeding of the working fluid with a passive tracer; 3) illumination of the measurement plane previously identified; 4) image acquisition; 5) image pre-processing of the acquired images; 6) particle detection and temporal tracking to isolate particles and track them in consecutive frames; 7) data postprocessing to obtain the relevant flow parameters. Obviously, flow images are acquired at a certain space–time resolution, depending on the characteristic time and length scales of the investigated phenomena, the details for each apparatus are provided in the corresponding subsection.

Pre-processing includes the sequence of operations carried out to improve the quality of acquired images for the subsequent core of the processing phase. Basically, the procedure implies the background removal as well as the removal of parts of the image which are not significant for the flow analysis as for instance regions close to the boundaries. In fact, the glares due to the interaction between the lighting system and the domain walls may affect the processing algorithm.

FT is a multi-frame algorithm based on the assumption of image light intensity conservation in space and time between two successive frames and in the neighborhood of the seeding particles; this assumption holds for small time intervals. The algorithm essentially considers measures of correlation windows between successive frames and evaluates displacements by considering the best correspondence (in terms of a defined matching measure) of selected interrogation windows between subsequent images. Sparse velocity vectors are then obtained by dividing the displacement by the time interval between two frames; FT then provides a Lagrangian description of the velocity field. These sparse data can be interpolated on a regular grid through a resampling procedure allowing for the reconstruction of the instantaneous and time-averaged Eulerian velocity fields as well. The advantage of having at the same time both the Lagrangian and the Eulerian description of the flow is evident; in addition, if compared to other tracking algorithms, FT is not constrained by low seeding density, so it provides accurate displacement vectors even when the number of tracer particles within each image is very large [22].

#### **3. Data analysis**

#### **3.1 Traveling waves and eddies**

As mentioned before, in these jet flows waves and eddies co-exist; to highlight the propagation of the traveling structures in the physical space, we consider both a measure based on Hovmöller diagrams and the theoretical phase speed of the Rossby wave.

As for the former, we map the time evolution of the stream function *ψ* as a function of *φ* at different radius, and in particular in correspondence of *r* = *rMS*, i.e., the radius where the radial shear is a maximum. The diagrams may show linear features with negative or positive slopes, indicating westward/eastward propagating structures; the propagating structures could be waves or eddies, or both. To calculate the propagation velocity *Vp* in correspondence of a radius *r* we estimate the slope Δ*φ=*Δ*t* of the contour lines in the azimuthal diagrams:

$$V\_p = r \frac{\Delta \rho}{\Delta t} \tag{1}$$

then the net speed of the propagating structures is evaluated by subtracting the mean zonal velocity from *Vp*:

$$\left| V\_{pn} = V\_p - \left< V\_x \right>\_{\varrho} \right. \tag{2}$$

where h i *Vz <sup>φ</sup>* is the mean zonal velocity averaged over a range of φ corresponding to the forced sector and time interval of �300 s.

As for the theoretical speed, we have shown in [14] how to derive the dispersion relation of a linear Rossby wave in polar coordinates; here, we reported the final expression:

$$V\_t = U - \beta \frac{R^2}{a^2} \tag{3}$$

being R the radius of the device (in this case the radius of the circle inscribed in the square tank), *U* the average zonal velocity in correspondence of the chosen radius *r,* and *α* the coefficient of the Bessel Fourier decomposition, depending on the geometry of the system and on the width of the forced sector and the characteristic of the forcing.

In oceanography, one of the most popular methods used to detect coherent longlived coherent structures, such as mesoscale eddies, is based on the estimation of the Okubo-Weiss parameter [24, 25]. This quantity describes the relative dominance of deformation with respect to rotation of the flow and it is defined as:

$$OW = \mathfrak{s}\_n - \mathfrak{s}\_s - \alpha^2 \tag{4}$$

where *sn* ¼ *ux* � *vy* and *ss* ¼ *vx* þ *uy* are the normal and shear components of strain, respectively, *ω* ¼ *vx* � *uy* is vorticity. The subscripts ()*<sup>x</sup>* and ()*<sup>y</sup>* indicate partial differentiation of the horizontal velocities (*u* and *v*) in the *x* and *y* directions, respectively. In order to distinguish regions characterized by different topology within the flow domain, one has first to fix a positive threshold OW0 of the Okubo-Weiss parameter. Then, according to it, the domain can be divided into zones corresponding to vortex cores (OW < �OW0), organized structures surrounding vortex cores (OW > OW0), and the background field (|OW| ≤ OW0). A value typically assumed for the threshold is OW0 = 0.2*σOW*, where *σOW* is the standard deviation of OW parameter [26, 27].

#### **3.2 Finite-time Lyapunov exponents and Lagrangian coherent structures**

Finite-Time Lyapunov Exponents (FTLE) represents a powerful tool suitable to track coherent structures and to unveil their connections to energetic and mixing processes, in fact, it has been used extensively in different contexts, including biological and geophysical flows [28, 29]. Basically, the FTLE measure the maximum linearized growth rate of the distance among initially adjacent particles tracked over a finite integration time. In brief, the computation of FTLE follows from the definition of the flow mapΦð Þ *<sup>x</sup> <sup>t</sup>*þ*T*<sup>∗</sup> *<sup>t</sup>* over a finite time interval *T\**:

$$\Phi(\mathfrak{x})\_t^{t+T^\*}: \qquad \mathfrak{x}(t) \to \mathfrak{x}(t+T^\*) \tag{5}$$

mapping a material point x(t) at time t to its position at t + *T\** along its trajectory. After linearization, the amount of stretching about a trajectory is defined in terms of the Cauchy-Green deformation tensor by the matrix:

$$
\Delta = \left(\frac{d\Phi(\mathbf{x})\_t^{t+T^\*}}{dt}\right)^2 \tag{6}
$$

Since the maximum stretching occurs when the initial separation is aligned with the maximum eigenvalue of Δ, the FTLE is defined as:

$$\sigma(\mathbf{x}, t, T^\*) = \frac{1}{|T^\*|} \ln \sqrt{\lambda\_{\text{max}}} \tag{7}$$

Where *λmax* is the maximum eigenvalue of Δ and ffiffiffiffiffiffiffiffiffi *<sup>λ</sup>max* <sup>p</sup> corresponds to the maximum stretching factor. In particular, if a positive time interval is considered, the FTLEs measure separation forward in time, thus identifying repelling structures. On the contrary, if negative time intervals are considered, FTLEs measure separation backward in time, thus highlighting attracting structures [28, 30].

In addition, Lagrangian Coherent Structures (LCS) can be inferred from FTLE, [31]. LCS analysis represents a very powerful tool in cardiovascular fluid dynamics [32]; allowing for the identification of stagnant fluid areas, which are associated with an increased risk of thrombus as well as with blood cell damage. In addition, it helps to discern the regions directly affected by the vortices within the fluid domain and, possibly, their, modifications related to pathologies. FTLE investigation was successfully applied to the analysis of data sets obtained from both numerical simulations [27] and in vitro study [33] of a mechanical heart valve, as well as for the in vitro investigation of coherent structures educed from two-dimensional velocity fields in a LV model [21]. Recently, FTLE is also being used in the analysis of data sets collected in vivo [34, 35] and have been recognized as one of the main methods for the analysis of Lagrangian transport in blood flows [36, 37].

#### **4. Results and discussion**

#### **4.1 Rotating flows affected by a β-effect**

Before running each experiment, the fluid surface is seeded with styrene particles (mean size *dm* = 50 μm) acting as passive tracers and the fluid surface is lighted with two lateral lamps. The rotation rate of the table is then raised up to the chosen value and, once the solid body rotation is established, the forcing is switched on, and flow

*Vortex Analysis and Fluid Transport in Time-Dependent Flows DOI: http://dx.doi.org/10.5772/intechopen.105196*

images are acquired at 20fps by a high-resolution video camera (1023x1240 pixels) co-rotating with the system, perpendicular to the tank and with the optical axis parallel to the rotation axis. FT allows to reconstruct particle trajectories i.e., to provide a description of the flow in a Lagrangian framework; once the instantaneous sparse velocity vectors have been detected, they are interpolated onto a regular grid. In this case, it was convenient to choose a polar coordinate (*r*, *φ*) system with the pole corresponding to the center of the tank: the azimuthal direction *φ* identifies points with the same fluid depth (the so-called zonal direction) and at constant radius *r*. Sparse data have then been rearranged on a polar grid with 120 radii and 60 circles using a standard cubic spline interpolation procedure. The non-dimensional parameters of importance in our model are: the aspect ratio i.e., the ratio between the horizontal and vertical dimension of the flow domain *H0*/*L* < <1 (shallow fluid); the Ekman number Ek = *ν* /*Ω H*<sup>2</sup> <sup>0</sup> is order O(10<sup>4</sup> ), ν is the kinematic viscosity; the Rossby number Ro = *U*/2*ΩL* order O(10<sup>3</sup> ), *U* is the velocity scale; the Reynolds number Re = *Ulv*/*ν* is order O(102 ), *lv* is the characteristic length scale of the eddies. We summarize here some of the main results obtained in the characterization of eastward/westward flows, hereafter indicated as EW and WW case.

#### *4.1.1 Waves and eddies propagation*

In **Figures 1** and **2** we plot the instantaneous and time-averaged flow fields obtained in one run (I = 4A) of the experiments WW and EW; the plots are shown hereafter refer to experiments performed using the same forcing amount. **Figure 1** clearly shows a meandering jet squeezed between westward propagating eddies in the instantaneous flow field; on the contrary, the averaged field reveals strong alternating zonal jets and no eddies. These experimental features resemble ocean observations that highlight numerous westward propagating eddies on short time scales [12]. At the difference, the eastward jet is not associated with eddy shedding and traveling structures and the instantaneous and averaged flow appear to be rather similar (**Figure 2**).

To characterize the traveling structure observed in the WW case, we map the velocity stream function *ψ* as a function of time *t* and longitude *φ* in correspondence of the radius of maximum radial shear, *rMS* (**Figure 3**). In fact, in [14] we were able to demonstrate that the best match between the theoretical and experimental estimation of the speed of propagating structures is found in the correspondence of this radius. To emphasize this aspect, we plot in **Figure 4**, from left to right: the

**Figure 1.**

*Instantaneous (left) and time mean (right) normalized stream function superimposed on the streamlines for a WW flow.*

#### **Figure 2.**

*Instantaneous (left) and time mean (right) normalized stream function superimposed on the streamlines for a EW flow.*

#### **Figure 3.**

*Azimuthal Hovmöller diagram of the stream function* ψ *with the radius of the maximum radial shear chosen as the reference radius, WW flow.*

#### **Figure 4.**

*From left to right: Azimuthal Hovmöller diagram of the stream function* ψ *with the radius of the maximum radial shear chosen as the reference radius; radial Hovmöller diagram of* ψ *averaged azimuthally in the range forced sector; radial shear profiles of the azimuthal velocity* Vz *averaged azimuthally in the same sector and in time.*

azimuthal (*φ*-*t*) Hovmöller diagram of *ψ*, the radial *t*-*r* Hovmöller diagram of *ψ*, the radial profiles of the mean radial shear *d VZ <sup>φ</sup>=dr* and the mean azimuthal velocity *VZ <sup>φ</sup>*. The profiles show a maximum of the mean zonal velocity in correspondence of the radius *rC* corresponding to the jet centerline while the maximum of the mean shear is at a radius *rMS* > *rC*. We recall that the definition of the jet boundaries and of its time evolution is crucial in the definition of the barriers to meridional transport [5, 13].

*Vortex Analysis and Fluid Transport in Time-Dependent Flows DOI: http://dx.doi.org/10.5772/intechopen.105196*

**Figure 5.**

*Instantaneous fields of vorticity field* ζ *(left) and Okubo–Weiss parameter Qow (right); velocity field superimposed (blue arrows) for a WW flow.*

As discussed in Section 3.1, by measuring the slope of the lines of the same color, we were able to estimate experimentally the speed of the propagating structures relative to the zonal flow with Eq. (2); we then calculate the theoretical speed using Eq. (3) and compare the obtained values. The comparison shows that the relative error, i.e. the ratio between the measured and the expected speed, is minimum in correspondence of *rMS* (O(10<sup>2</sup> )).

In order to compare our method to evaluate the eddies propagation speed with a method widely used in the applications we also applied an OW-based method to our experimental data sets. At first, we evaluated OW parameter through Eq. (4) at each time instant. Then, using a threshold of OW0 = 0.5*σOW* we identified the vortex cores and their surrounding area. We refined the detection by combining the OW parameter with the physical properties of the flow field (high vorticity areas, velocity vectors) and by applying geometrical constraints. An example is provided in **Figure 5** in which we show a snapshot of the vorticity field *ζ* (left) and of the Okubo–Weiss parameter QOW (right) superimposed on the corresponding velocity field. In the vorticity map, regions of dark blue (dark red) identify strong anticyclonic (cyclonic) circulation. In the Qow field, dark blue identifies regions where vorticity is much stronger than strain (i.e., eddy cores), and dark red where strain is much greater than vorticity.

Finally, once identified the coherent vortices, we detected the center of each structure and tracked them in the considered time interval. We found values of the propagating speed close to the ones found through the Hovmöller diagrams. We conclude that waves and propagating eddies coexist in the zonal pattern and confirm their duality nature [14]. The application of the same procedure of analysis overall the EW experiments is actually in progress [38].

#### *4.1.2 Characteristic scales and flow regime*

The estimation of flow characteristic length scales is crucial to identify the flow regime in rotating turbulent flows with a *β*-effect. To this aim, as discussed in Section 2.1, we calculate the Rhines number, *kRH*, the transitional wave number, *kβ,* and their ratio *Rβ*. The results for a set of WW and EW experiments are reported in **Table 1**.

According to the classification provided in [8] we conclude that all our experiments reproduced flows in a transitional regime.


**Table 1.**

*Characteristic scales and zonostrophy index estimated from experimental data.*

#### **4.2 Flow patterns in the left ventricle downstream of prosthetic valves**

To perform flow measurements in the LV, the vertical symmetry plane aligned with the mitral and aortic valve axes is illuminated by a 12 W, infrared laser. The working fluid inside the ventricle (distilled water) is seeded with neutrally buoyant particles (*dm* � 30 μm). A high-speed digital camera (250 frames/s, 1280 � 1024 pixel resolution) is triggered by the motor to frame the time evolution of the phenomenon for *N* cardiac cycles. The acquired images are processed by means of a FT algorithm and velocity fields on a regular grid 51 � 51 are obtained for the considered time interval. Two-dimensional Eulerian velocity data were then phase averaged over *N* = 50 cycles. Here, we discuss two groups of experiments performed considering a period *T* = 6 s and stroke volumes *SV1* = 64 ml, *SV2* = 80 ml. We briefly recall that during the cardiac cycle the flow rate change according to the considered law [21]: the fluid enters the LV through the mitral valve during the diastole (0.00 *T* – 0.75 *T*) and is ejected out through the aortic valve during the systole (0.75 *T* – 1.00 *T*). Two peaks separated by the diastasis characterize the diastole phase: the first is called E-wave and corresponds to the dilation of the ventricle, and the second, called A-wave, is due to the contraction of the left atrium.

For the dynamic similarity, we consider the Reynolds number *Re* ¼ *UD=υ* and the Womersley number *Wo* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *D*2 *=Tυ* q ; respectively equal to *Re1* = 8322; *Re2* = 10403; *Wo* = 22.8; in both cases within the physiological range. Here, *D* is the maximum diameter of the ventricle, *U* the peak velocity through the mitral orifice, *ν* the kinematic viscosity of the working fluid i.e., distilled water.

We use the public domain code NEWMAN [39] to compute the FTLEs from the planar velocity dataset above described, for the details see [40]. We remark that FTLE fields are computed from 2D measurements even if it is well known that the observed phenomenon is 3D; indeed, as the measurement plane is a plane of symmetry the assumption of two-dimensionality is quite acceptable.

**Figure 6** shows backward FTLE at the end of the E-wave for the three simulated conditions. Backward FTLE ridges correspond to the front of the diastolic jet, sharply separating the fluid which just entered the ventricle from the receiving fluid.

The analysis of the FTLE patterns throughout the cardiac cycle (not shown here) highlights how in the physiological configuration the observed coherent structures appear to be optimal for the systolic function. Indeed, the modifications in the transmitral flow due to the presence of a prosthetic valve deeply impact on the interaction between the coherent structures generated during the first phase of the diastole and the incoming jet during the second diastolic phase. We observed that

*Vortex Analysis and Fluid Transport in Time-Dependent Flows DOI: http://dx.doi.org/10.5772/intechopen.105196*

**Figure 6.**

*Velocity fields and backward FTLE at the end of E-wave (the small inset shows the current time in the cardiac cycle as a black dot): Left: Physiological configuration, center: Monoleaflet valve, and right: Bileaflet valve.*

while the flow generated by a bileaflet valve preserves most of the beneficial features of the top hat inflow, downstream of a monoleafleat one the strong jet forming at the end of the diastole prevents the permanence of large coherent structures within the LV (**Figure 7**).

In order to complete the FTLE analysis, we reconstruct the trajectories of a number (O(104 )) of synthetic fluid particles entering the ventricle through the mitral orifice during the LV filling by numerically integrating the experimental velocity fields; for each run, synthetic particles were released during each time step of the diastolic waves from the mitral orifice section and were subsequently tracked during the cardiac cycle. The aim was to further clarify the role of LCS by overlapping the

**Figure 7.** *Same as above in correspondence of the systolic peak.*

**Figure 8.** *Syntetic particles overlapped on FTLE maps at the end of the a wave.*

**Figure 9.**

*Synthetic particles entered through the mitral orifice during diastolic waves colored according to the shear stress cumulated until the end of the A-wave (the color bar values correspond to the non-dimensional maximum shear stress).*

particle positions on the FTLE maps, and to verify if and how LCS may act as pseudo-barriers for transport and mixing. An example is reported in **Figure 8**.

We finally compute the shear stress experienced by the particles along their trajectories in order to emphasize the differences among the simulated conditions and to clarify the possible implications on the hemodynamics. Results corresponding to the end of the A wave are shown in **Figure 9**.

The plots show that, in case (a) the stress magnitudes induced by the smoother flow pattern are lower than values measured in case (b) and (c). In fact, while in physiological conditions particles characterized by the highest shear are washed out by the systolic wave, in presence of prosthetic valves they tend to be advected towards regions of the LV not affected by the systolic ejection (see **Figure 3**).

#### **5. Conclusions**

In this work, we review a set of methodologies suitable for the characterization of time-periodic complex flows; in particular, here, the focus is on rotating flows affected by a *β*-effect and blood flow in the left ventricle. The interest in deepening these contexts depends on their importance from both an applicative and a methodological point of view. Indeed, we consider almost 2D and time-periodic flows in which, due to chaotic advection, the Lagrangian motion of passive particles can be very complex even in regular, i.e., non-turbulent, Eulerian flow fields. We believe that the obtained results show, on one hand, that the designed experimental models prove suitable to reproduce the investigated phenomena, and on the other hand confirm that the proposed methodologies represent valid and powerful tools for identifying and characterizing the main flow patterns in their space–time evolution.

#### **Acknowledgements**

The authors would like to thank the Sapienza University of Rome (Research program SAPIEXCELLENCE SPC: 2021-1136-1451-173491), the European Union's Horizon 2020 research and innovation program (Marie Sklodowska–Curie Grant Agreement No. 797012) and the Italian Ministry of Research (project PRIN 2017 A889FP).

*Vortex Analysis and Fluid Transport in Time-Dependent Flows DOI: http://dx.doi.org/10.5772/intechopen.105196*

#### **Conflict of interest**

The authors declare no conflict of interest.

#### **Author details**

Stefania Espa<sup>1</sup> \*, Maria Grazia Badas<sup>2</sup> and Simon Cabanes<sup>1</sup>


\*Address all correspondence to: stefania.espa@uniroma1.it

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Crisanti A, Falcioni M, Paladin G, Vulpiani A. Lagrangian chaos:Transport, mixing and diffusion in fluids. La Rivista del Nuovo Cimento. 1991;**14**: 1-80. DOI: 10.1007/BF02811193

[2] Rhines PB. Waves and turbulence on a beta-plane. Journal of Fluid Mechanics. 1975;**69**:417-443. 69(3), 417-443. DOI: 10.1017/S0022 112075001504

[3] Cho JYK, Polvani LM. The emergence of jets and vortices in freely evolving, shallow-water turbulence on sphere. Physics of Fluids. 1996;**8**: 1531-1552. DOI: 10.1063/1.868929

[4] Maximenko NA, Bang B, Sasaki H. Observational evidence of alternating zonal jets in the World Ocean. Geophysical Research Letters. 2005;**32**: L12607. DOI: 10.1029/2005GL022728

[5] Galperin B, Read PL (Eds.). Zonal Jets: Phenomenology, Genesis, and Physics. Cambridge University Press; 2019. p. 550. DOI: 10.1017/ 9781107358225

[6] Galperin B, Sukoriansky S, Dikovskaya N. Geophysical flows with anisotropic turbulence and dispersive waves: Flows with a β-effect. Ocean Dynamics. 2010;**60**:427-441. DOI: 10.1007/s10236-010-0278-2

[7] Kraichnan RH. Inertial ranges in twodimensional turbulence. Physics of Fluids. 1967;**10**:1417. DOI: 10.1063/ 1.1762301

[8] Sukoriansky S, Dikovskaya N, Galperin B. On the arrest of the inverse energy cascade and the Rhines scale. Journal of the Atmospheric Sciences. 2007;**64**:3312-3327. DOI: 10.1175/ JAS4013.1

[9] Espa S, Di Nitto G, Cenedese A. The emergence of zonal jets in forced

rotating shallow water turbulence: A laboratory study. Europhysics Letters. 2010;**92**:34006. DOI: 10.1209/ 0295-5075/92/34006

[10] Di Nitto G, Espa S, Cenedese A. Simulating zonation in geophysical flows by laboratory experiments. Physics of Fluids. 2013;**25**:086602. DOI: 10.1063/1.4817540

[11] Derzho OG, Afanasyev YD. Rotating dipolar gyres on a β-plane. Physics of Fluids. 2008;**20**:036603. DOI: 10.1063/ 1.2890083

[12] Galperin B, Hoemann J, Espa S, Di Nitto G. Anisotropic turbulence and Rossby waves in an easterly jet: An experimental study. Geophysical Research Letters. 2014;**41**:6237-6243. DOI: 10.1002/2014gl060767

[13] Galperin B, Hoemann J, Espa S, Di Nitto G, Lacorata G. Anisotropic macroturbulence and diffusion associated with a westward zonal jet: From laboratory to planetary atmospheres and oceans. Physical Review E. 2016;**94**:063102. DOI: 10.1103/PhysRevE.94.063102

[14] Espa S, Cabanes S, King GP, Di Nitto G, Galperin B. Eddy-wave duality in a rotating flow. Physics of Fluids. 2020;**32**:076604. DOI: 10.1063/ 5.0006206

[15] Cabanes S, Espa S, Galperin B, Young RMB, Read PL. Revealing the intensity of turbulent energy transfer in planetary atmospheres. Journal of Geophysical Research. 2020;**47**:2316. DOI: 10.1029/2020GL088685

[16] Pedrizzetti G, Domenichini F. Nature optimizes the swirling flow in the human left ventricle. Physical Review Letters. 2005;**95**:108101. DOI: 10.1103/PhysRevLett.95.108101

*Vortex Analysis and Fluid Transport in Time-Dependent Flows DOI: http://dx.doi.org/10.5772/intechopen.105196*

[17] Yoganathan AP, He Z, Casey Jones S. Fluid mechanics of heart valves. Annual Review of Biomedical Engineering. 2004;**6**:331-362. DOI: 10.1146/annurev.bioeng.6. 040803.140111

[18] Sotiropoulos F, Le TB, Gilmanov A. Fluid mechanics of heart valves and their replacements. Annual Review of Fluid Mechanics. 2016;**48**:259-283. DOI: 10.1146/annurev-fluid-122414-034314

[19] Querzoli G, Cenedese A, Fortini S. Effect of the prosthetic mitral valve on vortex dynamics and turbulence of the left ventricular flow. Physics of Fluids. 2010;**22**:041901. DOI: 10.1063/ 1.3371720

[20] Fortini S, Querzoli G, Espa S, Cenedese A. Three-dimensional structure of the flow inside the left ventricle of the human heart. Experiments in Fluids. 2013;**54**:1-9. DOI: 10.1007/s00348-013-1609-0

[21] Espa S, Badas MG, Fortini S, Querzoli G, Cenedese A. A Lagrangian investigation of the flow inside the left ventricle. European Journal of Mechanics - B/Fluids. 2012;**35**:9-19. DOI: 10.1016/j.euromechflu.2012.01.015

[22] Moroni M, Cenedese A. Comparison among feature tracking and more consolidated velocimetry image analysis techniques in a fully developed turbulent channel flow. Measurement Science and Technology. 2005;**16**:2307. DOI: b10.1088/0957-0233/16/11/025

[23] Funiciello F, Moroni M, Piromallo C, Faccenna C, Cenedese A, Bui HA. Mapping mantle flow during retreating subduction: Laboratory models analyzed by feature tracking. Journal of Geophysical Research. 2006; **111**:B03402-B03412. DOI: 10.1029/ 2005GL025390

[24] Okubo A. Horizontal dispersion of floatable particles in the vicinity of

velocity singularities such as convergences. Deep Sea Research. 1970; **17**:445-454. DOI: 10.1016/0011-7471 (70)90059-8

[25] Weiss J. The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D. 1991;**48**: 273-294. DOI: 10.1016/0167-2789(91) 90088-Q

[26] Elhmaïdi D, Provenzale A, Babiano A. Elementary topology of two dimensional turbulence from a Lagrangian viewpoint and singleparticle dispersion. Journal of Fluid Mechanics. 1993;**257**:533-558. DOI: 10.1017/S0022112093003192

[27] Pasquero C, Provenzale A, Babiano A. Parameterization of dispersion in two-dimensional turbulence. Journal of Fluid Mechanics. 2001;**439**:279-303. DOI: 10.1017/ S0022112001004499

[28] Shadden SC, Lekien F, Marsden JE. Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in twodimensional aperiodic flows. Physica D: Nonlinear Phenomena. 2005;**212**: 271-304. DOI: 10.1016/j.physd.2005. 10.007

[29] Beron-Vera FJ, Olascoaga M, Goni G. Oceanic mesoscale eddies as revealed by Lagrangian coherent structures. Geophysical Research Letters. 2008;**35**:12. DOI: 10.1029/ 2008GL033957

[30] Haller G. Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D. 2001;**149**:248-277. DOI: 10.1016/S0167-2789(00)00199-8

[31] Haller G. Lagrangian coherent structures. Annual Review of Fluid Mechanics. 2015;**47**:137-162. DOI: 10.1146/annurev-fluid-010313-141322 [32] Shadden SC, Taylor CA. Characterization of coherent structures in the cardiovascular system. Annals of Biomedical Engineering. 2008;**36**(7): 1152-1162. DOI: 10.1007/s10439-008- 9502-3

[33] Miron P, Vétel J, Garon A. On the use of the finite-time Lyapunov exponent to reveal complex flow physics in the wake of a mechanical valve. Experiments in Fluids. 2014;**55**: 1-15. DOI: 10.1007/s00348-014-1814-5

[34] Hendabadi S, Bermejo J, Benito Yotti R, Fernández-Avilés F, del Álamo JC, Shadden SC. Topology of blood transport in the human left ventricle by novel processing of Doppler echocardiography. Annals of Biomedical Engineering. 2013;**41**:2603-2616. DOI: 10.1007/s10439-013-0853-z

[35] Bermejo J, Benito Y, Alhama M, Yotti R, Martínez-Legazpi P, Del Villar CP, et al. Intraventricular vortex properties in nonischemic dilated cardiomyopathy. AJP - Heart and Circulatory Physiology. 2014;**306**: H718-H729. DOI: 10.1152/ajpheart. 00697.2013

[36] Badas MG, Domenichini F, Querzoli G. Quantification of the blood mixing in the left ventricle using finite time Lyapunov exponents. Meccanica. 2017;**52**(3):529-544. DOI: 10.1007/ s11012-016-0364-8

[37] Di Labbio G, Vétel J, Kadem L. Material transport in the left ventricle with aortic valve regurgitation. Physical Review Fluids. 2021;**6**:059901. DOI: 10.1103/PhysRevFluids.6.059901

[38] Espa S, Cabanes S. Eddies and waves in a rotating flow: An experimental study. In: Proceedings of the 39th IAHR World Congress 19–24 June 2022, Granada, Spain

[39] Du Toit PC, Marsden JE. Horseshoes in hurricanes. Journal of Fixed Point

Theory and Applications 2010;**7**(2):351- 384. DOI: 10.1007/s11784-010-0028-6.

[40] Badas MG, Espa S, Fortini S, Querzoli G. 3D finite time Lyapunov exponents in a left ventricle laboratory model. EPJ Web of Conferences. 2015; **926**:02004. DOI: 10.1051/epjconf/ 20159202004

#### **Chapter 6**

## Relaxation Dynamics of Point Vortices

*Ken Sawada and Takashi Suzuki*

#### **Abstract**

We study a model describing relaxation dynamics of point vortices, from quasistationary state to the stationary state. It takes the form of a mean field equation of Brownian point vortices derived from Chavanis, and is formulated by our previous work as a limit equation of the patch model studied by Robert-Someria. This model is subject to the micro-canonical statistic laws; conservation of energy, that of mass, and increasing of the entropy. We study the existence and nonexistence of the global-in-time solution. It is known that this profile is controlled by a bound of the negative inverse temperature. Here we prove a rigorous result for radially symmetric case. Hence *E=M*<sup>2</sup> large and small imply the global-in-time and blowup in finite time of the solution, respectively. Where E and M denote the total energy and the total mass, respectively.

**Keywords:** point vortex, quasi-equilibrium, relaxation dynamics

#### **1. Introduction**

Our purpose is to study the system

$$\begin{aligned} \rho\_t + \nabla \cdot a\nabla^\perp \varphi &= \nabla \cdot (\nabla a + \beta a \nabla \varphi) & \text{in } \Omega \times (0, T), \\ \frac{\partial \rho}{\partial \nu} + \beta a \frac{\partial \varphi}{\partial \nu} \Big|\_{\partial \Omega} &= 0, \quad \left. \rho \right|\_{t=0} = \rho\_0(\mathbf{x}) \end{aligned} \tag{1}$$

with

$$-\Delta \boldsymbol{\mu} = \boldsymbol{\omega} \quad \text{in } \Omega, \quad \boldsymbol{\psi}|\_{\partial \Omega} = 0, \quad \boldsymbol{\beta} = -\frac{\int\_{\Omega} \nabla \boldsymbol{\alpha} \cdot \nabla \boldsymbol{\psi}}{\int\_{\Omega} \boldsymbol{\alpha} \left| \nabla \boldsymbol{\psi} \right|^{2}},\tag{2}$$

where Ω ⊂ **R**<sup>2</sup> is a bounded domain with smooth boundary ∂Ω, *ν* is the outer unit normal vector on ∂Ω, and

$$\nabla = \begin{pmatrix} \frac{\partial}{\partial \mathbf{x}\_1} \\ \frac{\partial}{\partial \mathbf{x}\_2} \end{pmatrix}, \quad \nabla^\perp = \begin{pmatrix} \frac{\partial}{\partial \mathbf{x}\_2} \\ -\frac{\partial}{\partial \mathbf{x}\_1} \end{pmatrix}, \quad \mathbf{x} = (\mathbf{x}\_1, \mathbf{x}\_2). \tag{3}$$

The unknown *ω* ¼ *ω*ð Þ *x*, *t* ∈ **R** stands for a mean field limit of many point vortices,

*Vortex Dynamics - From Physical to Mathematical Aspects*

$$a\rho(\mathbf{x},t)d\mathbf{x} = \sum\_{i=1}^{N} a\_i \delta\_{\mathbf{x}\_i(t)}(d\mathbf{x}).\tag{4}$$

It was derived, first, for Brownian point vortices by [1, 2], with *β* ¼ *β*ð Þ*t* standing for the inverse temperature. Then, [3, 4] reached it by the Lynden-Bell theory [5] of relaxation dynamics, that is, as a model describing the movement of the mean field of many point vortices, from quasi-stationary state to the stationary state. This model is consistent to the Onsager theory [6–12] on stationary states and also the patch model proposed by [13, 14], that is,

$$\rho a(\mathbf{x}, t) = \sum\_{i=1}^{N\_p} \sigma\_i \mathbf{1}\_{\Omega\_i(t)}(\mathbf{x}), \tag{5}$$

where *Np*, *σi*, and Ω*i*ð Þ*t* denote the number of patches, the vorticity of the *i*-th patch, and the domain of the *i*-th patch, respectively [15–17].

This chapter is concerned on the one-sided case of

$$
\alpha\_0 = \alpha\_0(\mathfrak{x}) > 0. \tag{6}
$$

If this initial value is smooth, there is a unique classical solution to (1)–(4) local in time, denoted by *ω* ¼ *ω*ð Þ *x*, *t* , with the maximal existence time *T* ¼ *T*max ∈ ð � 0, þ∞ . More precisely, the strong maximum principle to (1) guaranttes

$$
\alpha = \alpha(\mathfrak{x}, t) > 0 \quad \text{on } \mathfrak{Q} \times \quad [0, T). \tag{7}
$$

Then, the Hopf lemma to the Poisson equation in (2) ensures

$$\left.\frac{\partial\Psi}{\partial\nu}\right|\_{\partial\Omega} < 0,\tag{8}$$

and hence the well-definedness of

$$-\beta = \frac{\int\_{\Omega} \nabla \boldsymbol{\omega} \cdot \nabla \boldsymbol{\varphi}}{\int\_{\Omega} \boldsymbol{\alpha} \left| \nabla \boldsymbol{\varphi} \right|^{2}}.\tag{9}$$

We confirm that system (1)–(3) satisfies the requirements of isolated system of thermodynamics. First, the mass conservation is derived from (1) as

$$\frac{d}{dt}\Big|\_{\Omega} \, \alpha = 0,\tag{10}$$

because

$$\left. \nu \cdot \nabla^{\perp} \psi \right|\_{\partial \Omega} = 0 \tag{11}$$

holds by (2). Second, the energy conservation follows as

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\|\nabla\boldsymbol{\nu}\|\_{2}^{2} &= (\nabla\boldsymbol{\mu}, \nabla\boldsymbol{\nu}\_{t}) = (a\_{t}, \boldsymbol{\mu}) \\ &= \left(a\nabla^{\perp}\boldsymbol{\mu}, \nabla\boldsymbol{\nu}\right) - (\nabla\boldsymbol{\alpha} + \beta a\nabla\boldsymbol{\nu}, \nabla\boldsymbol{\nu}) \\ &= -\left(\nabla\boldsymbol{\alpha}, \nabla\boldsymbol{\nu}\right) - \beta \int\_{\Omega} \boldsymbol{\alpha} \left|\nabla\boldsymbol{\nu}\right|^{2} = \mathbf{0} \end{aligned} \tag{12}$$

*Relaxation Dynamics of Point Vortices DOI: http://dx.doi.org/10.5772/intechopen.100585*

by (1) and (2), because

$$
\nabla^{\perp} \boldsymbol{\mu} \cdot \nabla \boldsymbol{\mu} = \mathbf{0}, \tag{13}
$$

where , ð Þ denotes the *<sup>L</sup>*<sup>2</sup> inner product. Third, the entropy increasing is achieved, writing (1) as

$$\rho\_l = \nabla \cdot \boldsymbol{\omega} \left( -\nabla^{\perp} \boldsymbol{\varphi} + \nabla (\log \boldsymbol{\omega} + \beta \boldsymbol{\nu}) \right), \quad \frac{\partial}{\partial \boldsymbol{\nu}} (\log \boldsymbol{\omega} + \beta \boldsymbol{\nu}) \bigg|\_{\partial \Omega} = \mathbf{0}. \tag{14}$$

In fact, it then follows that

$$\int\_{\Omega} \rho\_{\mathbb{H}} (\log \boldsymbol{\omega} + \beta \boldsymbol{\mu}) = \int\_{\Omega} \boldsymbol{\omega} \nabla^{\perp} \boldsymbol{\upmu} \cdot \nabla (\log \boldsymbol{\upmu} + \beta \boldsymbol{\upmu}) - \boldsymbol{\upalpha} |\nabla (\log \boldsymbol{\upalpha} + \beta \boldsymbol{\upmu})|^{2} \, d\mathbf{x} \tag{15}$$

with

$$\begin{aligned} \int\_{\Omega} \boldsymbol{\omega} \nabla^{\perp} \boldsymbol{\upmu} \cdot \nabla (\log \boldsymbol{\omega} + \beta \boldsymbol{\upmu}) &= \int\_{\Omega} \nabla \boldsymbol{\upmu} \cdot \nabla^{\perp} \boldsymbol{\upmu} \\\\ &= \int\_{\partial \Omega} \boldsymbol{\upmu} \cdot \nabla^{\perp} \boldsymbol{\upmu} - \int\_{\Omega} \boldsymbol{\upalpha} \nabla \cdot \left( \nabla^{\perp} \boldsymbol{\upmu} \right) = \mathbf{0} \end{aligned} \tag{16}$$

from (11) and

$$
\nabla^{\perp} \cdot \nabla = \nabla \cdot \nabla^{\perp} = \mathbf{0}.\tag{17}
$$

Since

$$\int\_{\Omega} \rho\_l \log \boldsymbol{\omega} = \frac{d}{dt} \int\_{\Omega} \boldsymbol{\alpha} (\log \boldsymbol{\omega} - 1), \quad \int\_{\Omega} \rho\_l \boldsymbol{\nu} = \frac{1}{2} \frac{d}{dt} \| \nabla \boldsymbol{\nu} \|\_{2}^{2} = \mathbf{0},\tag{18}$$

We thus end up with the mass conservation

$$M = \int\_{\mathfrak{Q}} a \,, \tag{19}$$

the energy conservation

$$E = \|\nabla \boldsymbol{\mu}\|\_2^2 = (\boldsymbol{\mu}, \boldsymbol{\alpha}),\tag{20}$$

and the entropy increasing

$$\frac{d}{dt}\int\_{\Omega} \rho(\log \omega - 1) = -\int\_{\Omega} \rho |\nabla(\log \omega + \beta \wp)|^2 \le 0. \tag{21}$$

Henceforth, *C*>0 stands for a generic constant. In the previous work [4] we studied radially symmetric solutions and obtained a criterion for the existence of the solution global in time. Here, we refine the result as follows, where *B*ð Þ 0, 1 denotes the unit ball.

**Theorem 1** *Let*

$$\Omega = B(0, \mathbf{1}), \quad \alpha\_0 = \alpha\_0(r), \ \ a\_{0r} < 0, \quad 0 < r = |\mathbf{x}| \le 1. \tag{22}$$

*Then there is C*<sup>0</sup> >0 *such that*

$$\|C\_0\| \|o\_0\|\|\_2^3 \le \underline{Eo} \Rightarrow T = +\infty,\ \|o(\cdot, t)\|\_\infty \le \mathcal{C},\ t \ge 0,\tag{23}$$

*where*

$$
\underline{\boldsymbol{\alpha}} = \min\_{\overline{\boldsymbol{\alpha}}} \boldsymbol{w}\_0 > \mathbf{0}. \tag{24}
$$

**Theorem 2** *Under the assumption of* (22) *there is δ*<sup>0</sup> > 0 *such that*

$$\frac{E}{M^2} < \delta\_0 \Rightarrow T < +\infty. \tag{25}$$

**Remark 1** *Since*

$$\begin{split} \|\boldsymbol{\varrho}\|\_{\Omega} \|\_{2}^{3} &= \left( \int\_{\Omega} \boldsymbol{\varrho}\_{0}^{2} \right)^{3/2} \geq \left( \underline{\boldsymbol{\omega}}^{2/3} \int\_{\Omega} \boldsymbol{\varrho}\_{0}^{4/3} \right)^{3/2} \\ &= \underline{\boldsymbol{\omega}} \Big( \int\_{\Omega} \boldsymbol{\omega}\_{0}^{4/3} \Big)^{3/2} \geq \underline{\boldsymbol{\omega}} |\boldsymbol{\Omega}|^{-1/2} \left( \int\_{\Omega} \boldsymbol{\omega}\_{0} \right)^{2} = \underline{\boldsymbol{\alpha}} |\boldsymbol{\Omega}|^{-1/2} \mathbf{M}^{2} \end{split} \tag{26}$$

*the assumption (23) implies*

$$\frac{E}{M^2} \ge \mathcal{C}\_0 |\Omega|^{-1/2}. \tag{27}$$

*Therefore, roughly, the conditions E=M*<sup>2</sup> <sup>≫</sup> <sup>1</sup> *and E=M*<sup>2</sup> <sup>≪</sup> <sup>1</sup> *imply T* ¼ þ<sup>∞</sup> *and T* < þ ∞*, respectively.*

**Remark 2** *The assumption* (22) *implies*

$$
\beta = \beta(t) < 0, \quad 0 \le t < T,\tag{28}
$$

*and then we obtain Theorem 1. In other words, the conclusion of this theorem arises from* (28)*, without* (22).

**Remark 3** *Since*

$$\frac{E}{M^2} = \frac{\int\_{\Omega} \left| \nabla \boldsymbol{\mu} \right|^2}{\left( \int\_{\Omega} \boldsymbol{\alpha} \right)^2} \tag{29}$$

*it holds that*

$$\frac{E}{M^2} = \|\nabla c\|\_2^2, \quad c = \frac{\left(-\Delta\right)^{-1} \alpha\_0}{\int\_{\Omega} \alpha\_0} = \frac{\mu\_0}{\int\_{\partial\Omega} - \frac{\partial \mu\_0}{\partial \nu}},\tag{30}$$

*where <sup>ψ</sup>*<sup>0</sup> ¼ �ð Þ <sup>Δ</sup> �<sup>1</sup> *ω*0.

The system (1)–(4) thus obays a profile of the micro-canonical ensemble. In a system associated with the canonical ensemble, the inverse temperature *β* is a constant in (1) independent of *t*, with the third equality in (2) elimiated:

$$\begin{aligned} \alpha \boldsymbol{\nu} + \nabla \cdot \boldsymbol{\alpha} \nabla^{\perp} \boldsymbol{\nu} &= \nabla \cdot (\nabla \boldsymbol{\alpha} + \beta \boldsymbol{\alpha} \nabla \boldsymbol{\nu}), & \frac{\partial \boldsymbol{\alpha}}{\partial \boldsymbol{\nu}} + \beta \boldsymbol{\alpha} \frac{\partial \boldsymbol{\mu}}{\partial \boldsymbol{\nu}} \bigg|\_{\partial \Omega} &= \mathbf{0}, \quad \boldsymbol{\alpha}|\_{t=0} = \boldsymbol{\alpha}\_0(\mathbf{x}) > \mathbf{0} \\ \boldsymbol{\nu} - \Delta \boldsymbol{\nu} &= \boldsymbol{\alpha}, \quad \boldsymbol{\nu}|\_{\partial \Omega} = \mathbf{0}. \end{aligned} \tag{31}$$

Then there arise the mass conservation

$$\frac{d}{dt}\int\_{\Omega} \rho = 0,\tag{32}$$

and the free energy decreasing

$$\frac{d}{dt}\int\_{\Omega} \rho (\log \boldsymbol{\omega} - \mathbf{1}) + \frac{\beta}{2} |\nabla \boldsymbol{\varphi}|^2 \, d\mathbf{x} = -\int\_{\Omega} \rho |\nabla (\log \boldsymbol{\omega} + \beta \boldsymbol{\varphi})|^2 \, \boldsymbol{\xi} \, d\mathbf{x} \tag{33}$$

The system (31) without vortex term,

$$\begin{aligned} \rho\_t &= \nabla \cdot (\nabla \omega + \beta \omega \nabla \varphi), \quad \left. \frac{\partial \omega}{\partial \nu} + \beta \omega \frac{\partial \nu}{\partial \nu} \right|\_{\partial \Omega} = \mathbf{0}, \quad \left. \rho \right|\_{t=0} = \omega\_0(\mathbf{x}) > \mathbf{0} \\ -\Delta \boldsymbol{\nu} &= \alpha, \quad \left. \boldsymbol{\nu} \right|\_{\partial \Omega} = \mathbf{0}. \end{aligned} \tag{34}$$

is called the Smoluchowski-Poisson equation. This model is concerned on the thermodynamics of self-gravitating Brownian particles [18] and has been studied in the context of chemotaxis [19–23]. We have a blowup threshold to (34) as a consequence of the quantized blowup mechanism [19, 23]. The results on the existence of the bounded global-in-time solution [24–26] and blowup of the solution in finite time [27] are valid even to the case that *β* is a function of *t* as in *β* ¼ *β*ð Þ*t* . provided with the vortex term <sup>∇</sup> � *<sup>ω</sup>*∇⊥*<sup>ψ</sup>* on the right-hand side. We thus obtain the following theorems.

**Theorem 3** *It holds that*

$$-\beta(t) \le \delta, \quad \|\alpha\_0\|\_1 < 8\pi\delta^{-1} \Rightarrow T = +\infty,\ \|\alpha(\cdot, t)\|\_\infty \le \mathbb{C} \tag{35}$$

*in* (31)*, where δ*> 0 *is arbitrary*. **Theorem 4** *It holds that*

$$-\beta(t) \ge \delta, \quad \|\alpha v\_0\|\_1 > 8\pi\delta^{-1} \Rightarrow \exists \alpha v\_0 > 0, \quad \|\alpha v\_0\|\_1 > 8\pi\delta^{-1} \quad \text{such that } T < +\infty \quad \text{(36)}$$

*in* (31)*, where δ*> 0 *is arbitrary*.

**Remark 4** *In the context of chemotaxis in biology, the boundary condition of ψ is required to be the form of Neumann zero. The Poisson equation in* (34) *is thus replaced by*

$$-\Delta \psi = \left. \alpha - \frac{1}{|\Omega|} \right|\_{\Omega} \alpha, \quad \left. \frac{\partial \psi}{\partial \nu} \right|\_{\partial \Omega} = 0 \tag{37}$$

*or*

$$-\Delta\Psi + \Psi = \alpha, \quad \left. \frac{\partial \Psi}{\partial \nu} \right|\_{\partial \Omega} = 0 \tag{38}$$

*by* [28] *and* [29]*, respectively. In this case there arises the boundary blowup, which reduces the value* 8*π in Theorems 3–4 to* 4*π. The value* 8*π in Theorems 3–4, therefore, is a consequence of the exclusion of the boundary blowup* [30]*. This property is valid even for* (37) *or* (38) *of the Poisson part, if* (22) *is assumed*.

**Remark 5** *The requirement to ω*<sup>0</sup> *in Theorem 4 is the concentration at an interior point, which is not necessary in the case of* (22)*. Hence Theorems 3 and 4 are refined as*

$$-\beta(t) \le \delta, \quad \|\alpha\_0\|\_1 < 8\pi\delta^{-1} \Rightarrow T = +\infty,\ \|\alpha(\cdot, t)\|\_\infty \le \mathbb{C} \tag{39}$$

*and*

$$-\beta(t) \ge \delta, \quad \|\alpha\_0\|\_1 > 8\pi\delta^{-1} \Rightarrow T < +\infty,\tag{40}$$

if (22) holds in (35). The main task for the proof of Theorems 1 and 2, therefore, is a control of *β* ¼ *β*ð Þ*t* in (1).

This paper is composed of four sections and an appendix. Section 2 is devoted to the study on the stationary solutions, and Theorems 1 and 2 are proven in Sections 3 and 4, respectively. Then Theorem 4 is confirmed in Appendix.

#### **2. Stationary states**

First, we take the canonical system (31) with *β* independent of *t*. By (32) and (33), its stationary state is defined by

$$
\log \omega + \beta \mu = \text{constant}, \quad \omega = \omega(\mathbf{x}) > \mathbf{0}, \quad \int\_{\Omega} \omega = M. \tag{41}
$$

Then it holds that

$$
\omega = \frac{M e^{-\beta \psi}}{\int\_{\Omega} e^{-\beta \psi}} \tag{42}
$$

and hence

$$-\Delta \psi = \frac{\mathcal{M}e^{-\beta \psi}}{\int\_{\Omega} e^{-\beta \psi}}, \quad \psi|\_{\partial \Omega} = 0. \tag{43}$$

There arises an oredered structure arises in *β* <0, as observed by [11], as a consequence of a quantized blowup mechanism [19, 20, 31]. In the microcanonical system (1) and (2), the value *β* in (43) has to be determined by *E* besides *M*.

Equality (21), however, still ensures (41) and hence (42) in the stationary state even for (1)–(3). Writing

$$
\nu = -\beta \nu, \quad \mu = \frac{-\beta \mathcal{M}}{\int\_{\Omega} e^{-\beta \nu}}, \tag{44}
$$

we obtain

$$
\Delta - \Delta v = \mu e^{\nu} \quad \text{in } \Omega, \quad v|\_{\partial \Omega} = 0, \quad \frac{E}{M^2} = \frac{\|\nabla v\|\_2^2}{\left(\int\_{\Omega} -\frac{\partial v}{\partial \nu}\right)^2} \tag{45}
$$

by (30) and (43).

This system is the stationary state of (1) and (2) introduced by [4]. The first two equalities

$$-\Delta v = \mu \epsilon^v, \quad v|\_{\partial \Omega} = 0 \tag{46}$$

comprise a nonlinear elliptic eigenvalue problem and the unknown eigenvalue *μ* is determined by the third equality,

*Relaxation Dynamics of Point Vortices DOI: http://dx.doi.org/10.5772/intechopen.100585*

$$\frac{E}{M^2} = \frac{\|\nabla v\|\_2^2}{\left(\int\_{\Omega} - \frac{\partial v}{\partial \nu}\right)^2}. \tag{47}$$

The elliptic theory ensures rather deailed features of the set of solutions to (46). Here we note the following facts [31].


We show the following theorem, consistent to Theorem 2. **Theorem 5** *If* <sup>Ω</sup> <sup>¼</sup> *<sup>B</sup>*ð Þ 0, 1 <sup>⊂</sup> **<sup>R</sup>**<sup>2</sup> *, there is δ*>0 *such that any solution v*ð Þ , *μ to* (45) *admits*

$$\frac{E}{M^2} \ge \delta.\tag{48}$$

*Proof:* If *μ* ¼ 0, it holds that *v* ¼ 0. We have �*v*> 0 exclusively in Ω, provided that �*μ*>0, respectively. By the elliptic theory [32], therefore, any solution *v* to (46) is radially symmetric as in *v* ¼ *v r*ð Þ, *r* ¼ ∣*x*∣. We have, furthermore, �*vr* <0 in 0<*r*≤1, if �*μ*>0, respectively.

Then it holds that *ψ* ¼ *ψ*ð Þ*r* , and hence

$$-\frac{1}{r}(r\wp\_r)\_r = \omega \quad \text{in} \ 0 < r \le \ 1, \quad \wp|\_{r=1} = 0 \tag{49}$$

by (42) and (43), which implies

$$-r\nu\_r(r) = \int\_0^r s\rho(s)ds > 0, \quad 0 < r \le 1. \tag{50}$$

We thus obtain *μ* 6¼ 0, in particular.

If *μ*<0 we have *β* >0 by (44), and therefore, *ψ<sup>r</sup>* >0 in 0< *r*≤ 1 by *vr* >0 there. It is a contradiction, and hence *μ*>0. In this case, the solution *v* ¼ *v r*ð Þ to (46) is explicit [31]. The numbers of the solution is 0, 1, and 2, according to *μ*>2, *μ* ¼ 2, and 0<*μ*< 2, respectively, and if 0 <*μ*≤2 the solutions *v* ¼ *v*� are given as

$$v\_{\pm}(r) = \log \frac{8\gamma\_{\pm}r}{\left(1 + \chi\_{\pm}r^2\right)^2}, \quad \chi\_{\pm} = \frac{4}{\mu} \left\{ 1 - \frac{\mu}{4} \pm \left(1 - \frac{\mu}{2}\right)^{1/2} \right\}.\tag{51}$$

In fact, we have *γ*<sup>þ</sup> ¼ *γ*� for *μ* ¼ 2. This solution is parametrized by

$$
\sigma = \int\_{\Omega} \mu e^{\upsilon} \in (0, 8\pi). \tag{52}
$$

Hence each 0 <*σ* < 8*π* admits a unique solution ð Þ *v*, *μ* to (46), and *v* ¼ *v*<sup>þ</sup> and *v* ¼ *v*� according as *σ* ≥ 4*π* and *σ* ≤ 4*π*, respectively. It holds also that *μ* ↓ 0 if either *σ* ↑ 8*π* or *σ* ↓ 0. Thus we have only to confirm that *E=M*<sup>2</sup> is bounded, both as *σ* ↑ 8*π* and *σ* ↓ 0.

As *σ* ↑ 8*π*, we have

$$v = v\_+(\mathbf{x}) \to \ \mathsf{4} \log \frac{\mathsf{1}}{|\mathsf{x}|} \quad \text{locally uniformly on } \overline{\Omega} \\ \backslash \{0\} \tag{53}$$

and hence

$$\|\nabla v\|\_{2}^{2} \to \ +\infty, \quad \int\_{\partial\Omega} -\frac{\partial v}{\partial \nu} \to \ 8\pi,\tag{54}$$

which implies

$$\lim\_{\sigma \uparrow 8\pi} \frac{E}{M^2} = +\infty.\tag{55}$$

As *σ* ↓ 0, on the other hand, we have

$$
v = v\_{-}(\mathbf{x}) \ \rightarrow \quad \mathbf{0} \quad \text{uniformly in } \overline{\Omega}.\tag{56}$$

Since *μ* ↓ 0, furthermore, there arises that

$$\gamma = \gamma\_- = \frac{4}{\mu} \left\{ \mathbf{1} - \frac{\mu}{4} - \left( \mathbf{1} - \frac{\mu}{2} \right)^{1/2} \right\} = \mu (\mathbf{1} + o(\mathbf{1})).\tag{57}$$

It holds also that

$$v(r) = \log \frac{8\gamma}{\mu} - 2\log\left(1 + \mu r^2\right) \tag{58}$$

and hence

$$v\_r(r) = -\frac{4\mu r}{\left(1 + \mu r^2\right)^2} = -4\mu r(1 + o(1)) \quad \text{uniformly on } \overline{\Omega}.\tag{59}$$

Then, (59) implies

$$\begin{split} \|\nabla v\|\_{2}^{2} &= 2\pi \int\_{0}^{1} v\_{r}^{2} r \ \, dr = 2\pi \cdot 16\mu^{2} \cdot \int\_{0}^{1} r^{3} \ \, dr \cdot (1 + o(1)) \\ &= 8\pi \mu^{2} (1 + o(1)) \end{split} \tag{60}$$

as well as

$$
\pi \left( \int\_{\partial \Omega} - \frac{\partial v}{\partial \nu} \right)^2 = 16\mu^2 \cdot 2\pi (1 + o(1)). \tag{61}
$$

It thus follows that

$$\lim\_{\sigma \downarrow 0} \frac{E}{M^2} = \frac{1}{4} \tag{62}$$

and hence the conclusion. □

**102**

#### **3. Proof of Theorem 1**

The first observation is the following lemma. **Lemma 1** *Under the assumption of* (22)*, it holds that*

$$
\beta = \beta(t) < 0, \quad \alpha\_r(r, t) < 0, \quad 0 < r \le 1, \ 0 \le t < T. \tag{63}
$$

*Proof:* We have (7) and hence

$$
\psi\_r(r,t) < 0, \quad 0 < r \le 1, \quad 0 \le t < T \tag{64}
$$

by (49), which implies, in particular,

$$\beta = -\frac{\left(\nabla \boldsymbol{\alpha}, \nabla \boldsymbol{\varphi}\right)}{\int\_{\Omega} \boldsymbol{\alpha} \left| \nabla \boldsymbol{\varphi} \right|^{2}} < \mathbf{0} \tag{65}$$

at *t* ¼ 0 by (22).

Since *<sup>ω</sup>* <sup>¼</sup> *<sup>ω</sup>*ð Þ *<sup>r</sup>*, *<sup>t</sup>* and *<sup>ψ</sup>* <sup>¼</sup> *<sup>ψ</sup>*ð Þ *<sup>r</sup>*, *<sup>t</sup>* , we obtain ∇⊥*<sup>ψ</sup>* <sup>¼</sup> 0, and hence

$$
\rho\_t = \rho\_{rr} + \frac{1}{r}\rho\_r + \beta\psi\_r\rho\_r - \beta\rho^2 \tag{66}
$$

by (1). Then *z* ¼ *ω<sup>r</sup>* satisfies

$$\begin{aligned} z\_t &= z\_{rr} - \frac{1}{r^2} z + \frac{1}{r} z\_r + \beta \nu\_{rr} z + \beta \nu\_r z\_r - 2\beta a x\_r, \quad 0 < r \le 1, \quad 0 \le t < T\\ z|\_{r=0} &= 0, \qquad z|\_{t=0} = a \nu\_{0r}(r) < 0, \quad 0 < r \le 1 \end{aligned} \tag{67}$$

and

$$z = -\beta \alpha \psi\_r, \quad r = \mathbf{1}, \; \mathbf{0} \le t < T. \tag{68}$$

Putting

$$m(t) = \min\_{\partial \Omega} z(\cdot, t) = a\_r(\cdot, t)|\_{r=1}, \tag{69}$$

we obtain *m*ð Þ 0 <0 from the assumption. If there is 0<*t*<sup>0</sup> < such that

$$m(t) < 0, \quad 0 \le t < t\_0 < T, \quad m(t\_0) = 0,\tag{70}$$

we obtain *z r*ð Þ , *t* >0 for 0 ≤*t*<*t*0, 0<*r*≤1, and *t* ¼ *t*0, 0< *r*<1 by the strong maximum principle. By (64), we have (65) for 0 ≤*t*≤ *t*0, that is,

$$\beta = -\frac{\int\_0^1 \wp\_r zr \,\, dr}{\int\_0^1 a \wp\_r^2 r \,\, dr} < 0, \quad 0 \le t \le t\_0,\tag{71}$$

and hence

$$z = -\beta o\nu\_r < 0 \quad r = 1, \ t = t\_0,\tag{72}$$

a contradiction. It holds that *z* ¼ *ω<sup>r</sup>* < 0 for 0 ≤*t* <*T*, *r* ¼ 1, and hence

$$\beta = -\frac{\int\_0^1 \mu\_r \alpha r dr}{\int\_0^1 a \psi\_r^2 r \ dr} < 0, \quad 0 \le t < T. \qquad \square \tag{73}$$

The proof of Theorem 3 relies on the fact

$$\beta \ge -\mathbb{C}, \ \int\_{\mathfrak{U}} \omega(\log \omega - 1) \le \mathbb{C} \Rightarrow T = +\infty, \ \|\!\!/ \omega(\cdot, t)\|\_{\infty} \le \mathbb{C}.\tag{74}$$

This property is known for the Smoluchoski-Poisson equation (34), but the proof is valid even to (31) with vortex term. Having (21), therefore, we have to provide the inequality *β* ≥ � *C*.

The inequality *β* <0, on the other hand, is sufficient for the following arguments.

**Lemma 2** *If β* ≤0*,* 0≤*t*< *T, it holds that*

$$
\underline{\alpha} \ge \underline{\alpha} \equiv \min\_{\overline{\Omega}} \; \; \alpha\_0 > 0 \; \; \; \; \; \; \; \; \; on \; \; \; \overline{\Omega} \times \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$$

*Proof:* Since (17) we obtain

$$\alpha \eta + \nabla^{\perp} \boldsymbol{\psi} \cdot \nabla \boldsymbol{\omega} = \Delta \boldsymbol{\alpha} + \beta \nabla \boldsymbol{\psi} \cdot \nabla \boldsymbol{\omega} + \beta \Delta \boldsymbol{\psi}$$

$$= \Delta \boldsymbol{\alpha} + \beta \nabla \boldsymbol{\psi} \cdot \nabla \boldsymbol{\omega} - \beta \boldsymbol{\alpha}^2 \tag{76}$$

$$\geq \Delta \boldsymbol{\omega} + \beta \nabla \boldsymbol{\psi} \cdot \nabla \boldsymbol{\omega} \quad \text{in} \ \mathfrak{Q} \times \ (0, T)$$

with

$$-\frac{\partial \rho}{\partial \nu} = \beta \rho \frac{\partial \rho}{\partial \nu} > 0 \quad \text{on} \quad \partial \Omega \times \quad [0, T) \tag{77}$$

$$\text{By (8). Then the result follows from the comparison theorem.}$$

by (8). Then the result follows from the comparison theorem. □ **Lemma 3** *Under the assumption of the previous lemma, there is C*<sup>0</sup> ¼ *C*0ð Þ Ω >0 *such that*

$$\|C\_0\|\|o\_0\|\|\_2^3 \le E\underline{\alpha} \Rightarrow \|\|o(\cdot, t)\|\|\_2 \le \|\|o\_0\|\|\_2, \quad -\beta(t) \le a \equiv \frac{\|\|o\_0\|\|\_2^2}{E\underline{\alpha}}, \quad 0 \le t < T. \tag{78}$$

*Proof:* Using (11) and (17), we obtain

$$\begin{split} \int\_{\Omega} \left[ \nabla \cdot \left( a \nabla^{\perp} \boldsymbol{\mu} \right) \right] \boldsymbol{\mu} &= \int\_{\Omega} a \nabla \boldsymbol{\omega} \cdot \nabla^{\perp} \boldsymbol{\mu} = \frac{1}{2} \int\_{\Omega} \nabla a^{2} \cdot \nabla^{\perp} \boldsymbol{\mu} \\ &= -\frac{1}{2} \int\_{\Omega} a^{2} \nabla \cdot \nabla^{\perp} \boldsymbol{\mu} = \mathbf{0}. \end{split} \tag{79}$$

Hence (1) with (2) implies

$$\begin{split} \frac{1}{2} \frac{d}{dt} \|\boldsymbol{\alpha}\|^2\_2 + \|\nabla \boldsymbol{\alpha}\|^2\_2 &= -\beta \int\_{\Omega} \boldsymbol{\alpha} \nabla \boldsymbol{\nu} \cdot \nabla \boldsymbol{\omega} = -\frac{\beta}{2} \left( \nabla \boldsymbol{\nu}, \nabla \boldsymbol{\omega}^2 \right) \\ &= -\frac{\beta}{2} \int\_{\partial \Omega} \boldsymbol{\alpha}^2 \frac{\partial \boldsymbol{\nu}}{\partial \boldsymbol{\nu}} + \frac{\beta}{2} \left( \Delta \boldsymbol{\nu}, \boldsymbol{\omega}^2 \right) \leq -\frac{\beta}{2} \|\boldsymbol{\omega}\|^3\_3 \end{split} \tag{80}$$

by *β* ≤0 and (88). Since

$$\int\_{\Omega} \nabla \boldsymbol{\omega} \cdot \nabla \boldsymbol{\varphi} = \int\_{\partial \Omega} \boldsymbol{\alpha} \frac{\partial \boldsymbol{\varphi}}{\partial \boldsymbol{\nu}} + \int\_{\Omega} \boldsymbol{\alpha} (-\Delta \boldsymbol{\varphi}) \leq \int\_{\Omega} \boldsymbol{\alpha}^2 \tag{81}$$

follows from (8), furthermore, it holds that

$$-\beta = \frac{\int\_{\Omega} \nabla \boldsymbol{\alpha} \cdot \nabla \boldsymbol{\mu}}{\int\_{\Omega} \boldsymbol{\alpha} |\nabla \boldsymbol{\mu}|^{2}} \leq \underline{\boldsymbol{\alpha}}^{-1} \cdot \frac{\|\boldsymbol{\alpha}\|\_{2}^{2}}{\|\nabla \boldsymbol{\mu}\|\_{2}^{2}} = \frac{1}{E\underline{\boldsymbol{\alpha}}} \|\boldsymbol{\alpha}\|\_{2}^{2}.\tag{82}$$

Then ineqality (80) induces

$$\frac{1}{2}\frac{d}{dt}\|\boldsymbol{o}\boldsymbol{o}\|\_{2}^{2} + \|\nabla\boldsymbol{o}\|\_{2}^{2} \leq \frac{1}{2E\underline{\alpha}}\|\boldsymbol{o}\boldsymbol{o}\|\_{2}^{2} \cdot \|\boldsymbol{o}\boldsymbol{o}\|\_{3}^{3}.\tag{83}$$

Here we use the Gagliardo-Nirenberg inequality (see (4.16) of [19]) in the form of

$$\|\|\boldsymbol{\alpha}\|\|\_{3}^{3} \leq \mathcal{C} \|\|\boldsymbol{\alpha}\|\|\_{H^{1}} \cdot \|\|\boldsymbol{\alpha}\|\|\_{2}^{2} = \mathcal{C} \|\|\boldsymbol{\alpha}\|\|\_{2}^{2} (\|\nabla \boldsymbol{\alpha}\|\|\_{2} + \|\|\boldsymbol{\alpha}\|\|\_{2}),\tag{84}$$

to obtain

$$\begin{split} \frac{1}{2} \frac{d}{dt} \|\boldsymbol{\omega}\|^2\_2 + \|\nabla \boldsymbol{\omega}\|^2\_2 &\leq \frac{C}{E\underline{\omega}} \|\boldsymbol{\omega}\|^4\_2 (\|\nabla \boldsymbol{\omega}\|\_2 + \|\boldsymbol{\omega}\|\_2) \\ &\leq \frac{1}{2} \|\nabla \boldsymbol{\omega}\|^2\_2 + \frac{C^2}{8(E\underline{\omega})^2} \|\boldsymbol{\omega}\|^8\_2 + \frac{C}{2E\underline{\omega}} \|\boldsymbol{\omega}\|^5\_2 \end{split} \tag{85}$$

and hence

$$\frac{d}{dt} \|\boldsymbol{\alpha}\|\|\_{2}^{2} + \|\nabla \boldsymbol{\alpha}\|\_{2}^{2} \leq \frac{C}{E\underline{\boldsymbol{\alpha}}} \|\boldsymbol{\alpha}\|\_{2}^{5} \left(\frac{C}{E\underline{\boldsymbol{\alpha}}} \|\boldsymbol{\alpha}\|\_{2}^{3} + \mathbf{1}\right). \tag{86}$$

Then, Poincaré-Wirtinger's inequality ensures

$$\frac{d}{dt} \|\boldsymbol{\alpha}\|\|\_{2}^{2} + \mu \|\boldsymbol{\alpha}\|\|\_{2}^{2} \leq \frac{C}{E\underline{\boldsymbol{\alpha}}} \left(\frac{C}{E\underline{\boldsymbol{\alpha}}} \|\boldsymbol{\alpha}\|\|\_{2}^{6} + \|\boldsymbol{\alpha}\|\|\_{2}^{3}\right) \|\boldsymbol{\alpha}\|\|\_{2}^{2},\tag{87}$$

where *μ* ¼ *μ*ð Þ Ω >0 is a constant. Writing

$$y(t) = \frac{C}{E\underline{\alpha}} \|\boldsymbol{\alpha}\|\_{2}^{3},\tag{88}$$

we obtain

$$\frac{d}{dt} \|\|\boldsymbol{\alpha}\|\|\_{2}^{2} + \mu \|\|\boldsymbol{\alpha}\|\|\_{2}^{2} \le \left(\boldsymbol{\mathcal{Y}}^{2} + \boldsymbol{\mathcal{Y}}\right) \|\|\boldsymbol{\alpha}\|\|\_{2}^{2},\tag{89}$$

and therefore, if

$$y^2 + y < \mu/2\tag{90}$$

holds at *t* ¼ 0, it keeps to hold that

$$\frac{d}{dt} \|\boldsymbol{\alpha}\|\_{2}^{2} \le 0 \tag{91}$$

and (90) for 0≤ *t*<*T*. Then, we obtain

*Vortex Dynamics - From Physical to Mathematical Aspects*

$$\|\|o(\cdot, t)\|\|\_{2} \le \|\|o\_0\|\|\_{2}, \quad 0 \le t < T,\tag{92}$$

and hence

$$-\beta(t) \le \frac{\|\alpha\_0\|\_2^2}{E\underline{\alpha}} = a, \quad 0 \le t < T \tag{93}$$

by (82). The condition *<sup>y</sup>*ð Þ <sup>0</sup> <sup>&</sup>lt; *<sup>μ</sup>* <sup>2</sup> means

$$C\_0 \| \| \rho\_0 \|\_2 \le E \underline{\varrho} \tag{94}$$

for *<sup>C</sup>*<sup>0</sup> <sup>&</sup>gt;0 sufficiently large, and hence we obtain the conclusion. □ *Proof of Theorem 1:* By the parabolic regularity, it suffices to show that

$$\|\|\boldsymbol{\rho}(\cdot,t)\|\|\_{\infty} \leq C, \quad 0 \leq t < T \tag{95}$$

under the assumption. We have readily shown

$$\|\|\rho(\cdot, t)\|\|\_{2} \le \mathcal{C}, \ \mathbf{0} \le -\beta(t) \le \mathcal{C}, \quad \mathbf{0} \le t < T \tag{96}$$

by Lemma 3. Then, the conclusion (95) is obtained similarly to (34). See [26] for more details.

In fact, we have

$$\int\_{\Omega} \left[ \nabla \cdot \left( a \nabla^{\perp} \boldsymbol{\mu} \right) \right] a^{p} = - \int\_{\Omega} a \nabla^{\perp} \boldsymbol{\mu} \cdot \nabla a^{p} = -p \int\_{\Omega} a^{p} \nabla^{\perp} \boldsymbol{\mu} \cdot \nabla a$$

$$= - \frac{p}{p+1} \int\_{\Omega} \nabla^{\perp} \boldsymbol{\mu} \cdot \nabla a^{p+1} = \frac{p}{p+1} \int\_{\Omega} a^{p+1} \nabla \cdot \left( \nabla^{\perp} \boldsymbol{\mu} \right) = 0 \tag{97}$$

for *p*> 0 by (11) and (34). Then it follows that

$$\begin{split} \frac{1}{p+1} \frac{d}{dt} \Big[ \int\_{\Omega} \alpha^{p+1} + \frac{4p}{\left(p+1\right)^{2}} \|\nabla \alpha^{\frac{p+1}{2}}\|\_{2}^{2} = -\beta \Big]\_{\Omega} \alpha \nabla \psi \cdot \nabla \alpha^{p} \\\\ = -\beta \cdot \frac{p}{p+1} \Big[ \int\_{\Omega} \nabla \psi \cdot \nabla \alpha^{p+1} \leq -\beta \frac{p}{p+1} \Big]\_{\Omega} \alpha^{p+1} (-\Delta \psi) \\\\ = -\beta \cdot \frac{p}{p+1} \Big[ \int\_{\Omega} \alpha^{p+1} \leq C \Big]\_{\Omega} \alpha^{p+2} \end{split} \tag{98}$$

by *β* <0 and (8). Then, Moser's iteration scheme ensures (95) as in [33].

#### **4. Proof of Theorem 2**

We begin with the following lemma. **Lemma 4** *Under the assumption of* (22)*, it holds that*

$$-\beta(t) \ge \delta, \quad 0 \le t < T, \ M = \|\alpha\_0\|\_1 > \frac{8\pi}{\delta} \Rightarrow T < +\infty \tag{99}$$

*in* (31)*, where δ*> 0 *is a constant.*

*Relaxation Dynamics of Point Vortices DOI: http://dx.doi.org/10.5772/intechopen.100585*

*Proof:* We have *ω* ¼ *ω*ð Þ *r*, *t* and *ψ* ¼ *ψ*ð Þ *r*, *t* for *r* ¼ ∣*x*∣ under the assumption, which implies <sup>∇</sup><sup>⊥</sup>*<sup>ψ</sup>* <sup>¼</sup> 0. Then we obtain

$$
\nabla \cdot \boldsymbol{\alpha} \nabla^{\perp} \boldsymbol{\varphi} = \nabla \boldsymbol{\alpha} \cdot \nabla^{\perp} \boldsymbol{\varphi} = \mathbf{0} \tag{100}
$$

by (17). It holds also that

$$\begin{split} \nabla \cdot (a \nabla \psi) &= \nabla \cdot \left( a \psi\_r \frac{\boldsymbol{\mathfrak{x}}}{r} \right) = \left( \nabla \cdot \frac{\boldsymbol{\mathfrak{x}}}{r} \right) a \psi\_r + \frac{\boldsymbol{\mathfrak{x}}}{r} \cdot \nabla (a \psi\_r) \\ &= \frac{\mathbf{1}}{r} a \psi\_r + (a \psi\_r)\_r = \frac{\mathbf{1}}{r} (r a \psi\_r)\_r, \end{split} \tag{101}$$

and therefore, there arises that

$$
\rho\_l \alpha\_l = \frac{1}{r} (r\alpha\_r + \beta r\alpha \psi\_r)\_r, \quad \alpha\_r + \beta a \psi\_r|\_{r=1} = 0. \tag{102}
$$

from (31). Then (102) implies

$$\frac{d}{dt}\int\_{0}^{1} ar^3 \ \ dr = \int\_{0}^{1} \alpha\_l r^3 \ \ dr = \int\_{0}^{1} (r\alpha\_r + \beta r a \psi\_r)\_r r^2 \ \ dr$$

$$= -\int\_{0}^{1} 2r^2 (\alpha\_r + \beta a \psi\_r) \ \ dr \tag{103}$$

$$= -2r^2 \alpha \vert\_{r=0}^{r=1} + \int\_{0}^{1} 4r\alpha - 2\beta a \psi\_r r^2 \ \ dr.$$

Here we use (50) derived from the Poisson part of (31), that is,

$$-r\wp\_r(r,t) = A(r,t) \equiv \int\_0^r s\wp(s,t)ds.\tag{104}$$

Putting

$$
\lambda = \int\_0^1 \rho r \,\, dr = \frac{M}{2\pi},
\tag{105}
$$

we obtain

$$\begin{split} \frac{d}{dt} \int\_{0}^{1} ar^3 \ \ dr &= -2a \vert\_{r=1} + 4\lambda + 2\beta \int\_{0}^{1} AA\_r \ \ dr \\ &= -2a \vert\_{r=1} + 4\lambda + \beta A^2 \vert\_{r=0}^{r=1} \\ &= -2a \vert\_{r=1} + 4\lambda + \beta \lambda^2 \end{split} \tag{106}$$

$$ < 4\lambda \left( \beta + \frac{M}{8\pi} \right) \le 4\lambda \left( -\delta + \frac{M}{8\pi} \right).$$

Since �*<sup>δ</sup>* <sup>þ</sup> *<sup>M</sup>* <sup>8</sup>*<sup>π</sup>* <sup>&</sup>lt;0, therefore, *<sup>T</sup>* ¼ þ<sup>∞</sup> is impossible, and we obtain *<sup>T</sup>* <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>. □ **Lemma 5** *Under the assumption* (22)*, there is δ*>0 *such that*

$$\frac{E}{M^2} < \delta, \quad \beta(t) \le 0, \quad 0 \le t < T \Rightarrow \beta(t) \le -\frac{1}{C E^{1/2}} \quad 0 \le t < T. \tag{107}$$

*Proof:* First, Lemma 1 implies

$$\left.w\right\|\left.w\right\|\_{\*} \equiv \left.w\right|\_{\left|\right|=1}.\tag{108}$$

Second, we have

$$\begin{split} \int\_{\Omega} \nabla \boldsymbol{\nu} \cdot \nabla \boldsymbol{\nu} &= \int\_{\partial \Omega} \frac{\partial \boldsymbol{\nu}}{\partial \boldsymbol{\nu}} \boldsymbol{\omega} + \int\_{\Omega} (-\Delta \boldsymbol{\mu}) \boldsymbol{\omega} = \boldsymbol{\alpha}\_{\*} \int\_{\partial \Omega} \frac{\partial \boldsymbol{\nu}}{\partial \boldsymbol{\nu}} + \|\boldsymbol{\omega}\|\_{2}^{2} \\ &= \boldsymbol{\alpha}\_{\*} \int\_{\Omega} \Delta \boldsymbol{\nu} + \|\boldsymbol{\omega}\|\_{2}^{2} = \|\boldsymbol{\omega}\|\_{2}^{2} - \boldsymbol{\alpha}\_{\*} \boldsymbol{M}, \end{split} \tag{109}$$

and hence

$$-\beta = \frac{\int\_{\Omega} \nabla \boldsymbol{\varphi} \cdot \nabla \boldsymbol{\alpha}}{\int\_{\Omega} \boldsymbol{\alpha} \left| \nabla \boldsymbol{\varphi} \right|^{2}} = \frac{\left\| \boldsymbol{\alpha} \right\|\_{2}^{2} - \boldsymbol{\alpha}\_{\*} \boldsymbol{M}}{\int\_{\Omega} \boldsymbol{\alpha} \left| \nabla \boldsymbol{\varphi} \right|^{2}}. \tag{110}$$

Here, we use the Gagliardo-Nirenberg inequality in the form of

$$\|\|\boldsymbol{w}\|\|\_{4}^{2} \leq C \|\|\boldsymbol{w}\|\|\_{2} \|\|\boldsymbol{w}\|\|\_{H^{1}},\tag{111}$$

which implies

$$\begin{split} \int\_{\Omega} \left| \alpha |\nabla \boldsymbol{\nu} |^{2} \leq \| \boldsymbol{\alpha} \| \_{2} \| \nabla \boldsymbol{\nu} \| \_{4}^{2} \leq C \| \boldsymbol{\alpha} \| \_{2} \| \nabla \boldsymbol{\nu} \| \_{2} \| \nabla \boldsymbol{\nu} \| \_{2} \\\\ \leq C E^{1/2} \| \boldsymbol{\alpha} \| \_{2}^{2} \end{split} \tag{112}$$

by the elliptic estimate of the Poisson equation in (2),

$$\|\|\varphi\|\|\_{H^2} \le C \|\|\alpha\|\|\_2. \tag{113}$$

We have, on the other hand,

$$
\rho\_\* M \le \frac{M}{E} \int\_{\Omega} \rho |\nabla \boldsymbol{\varphi}|^2 \tag{114}
$$

by (110), and therefore,

$$-\beta \ge \frac{1}{CE^{1/2}} - \frac{E}{M} \ge \frac{1}{2CE^{1/2}},\tag{115}$$

provided that

$$\left(\frac{E}{M^2} < \left(\frac{1}{2C}\right)^2\right. \tag{116}$$

Then the conclusion follows. □ *Proof of Theorem 2:* By Lemma 5, there is *δ*<sup>0</sup> > such that

$$\frac{E}{\text{M}^2} < \delta \Rightarrow -\beta \ge \frac{1}{\text{CE}^{1/2}} \equiv \delta\_1,\tag{117}$$

*Relaxation Dynamics of Point Vortices DOI: http://dx.doi.org/10.5772/intechopen.100585*

and then, Lemma 4 ensures

$$M > \frac{8\pi}{\delta\_1} \Rightarrow T < +\infty. \tag{118}$$

The assumption in (118) means

$$\frac{E}{M^2} < \left(\frac{1}{8\pi c}\right)^2,\tag{119}$$

and hence we obtain the conclusion. □

#### **Appendix Proof of Theorem 4**

This theorem is valid to the general case of Ω and *ω*<sup>0</sup> without (22). We assume *δ* ¼ 1 without loss of generation, so that

$$
\beta \le -1.\tag{120}
$$

We follow the argument [27] concerning (34) with the Poisson part replaced by (42) or (43). Thus we have to take case of the vortex term <sup>∇</sup> � *<sup>ω</sup>*∇⊥*ψ*, time varying *β* ¼ *β*ð Þ*t* , and the Dirichlet boundary condition in (31).

We recall the cut-off function used in [34] (see also Chapter 5 of [19]). Hence each *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> <sup>Ω</sup> and 0<sup>&</sup>lt; *<sup>R</sup>*≤1 admit *<sup>φ</sup>* <sup>¼</sup> *<sup>φ</sup><sup>x</sup>*0,*<sup>R</sup>* <sup>∈</sup>*C*<sup>2</sup> <sup>Ω</sup> � � with

$$\left. \frac{\partial \rho}{\partial \nu} \right|\_{\partial \Omega} = 0, \quad 0 \le \rho \le 1, \quad \rho = 1 \quad \text{in } \Omega \cap B(\mathbf{x}\_0, R/2), \quad \rho = 0 \quad \text{in } \Omega \backslash B(\mathbf{x}\_0, R), \tag{121}$$

and

$$|\nabla \boldsymbol{\rho}| \le \mathsf{CR}^{-1} \boldsymbol{\rho}^{1/2}, \quad |\nabla^2 \boldsymbol{\rho}| \le \mathsf{CR}^{-2} \boldsymbol{\rho}^{1/2}.\tag{122}$$

In more details, we take a cut-off function, denoted by *ψ*, satisfying (121), using a local conformal mapping, and then put *<sup>φ</sup>* <sup>¼</sup> *<sup>ψ</sup>*4.

Let

$$\left. \begin{array}{c} \rho \in \mathsf{C}^{2} \left( \overline{\Omega} \right), \quad \left. \frac{\partial \rho}{\partial \nu} \right|\_{\partial \Omega} = \mathbf{0}. \end{array} \tag{123}$$

be given. First, we have

$$\begin{aligned} \frac{d}{dt} \int\_{\Omega} \rho \rho &= \int\_{\Omega} \rho \nabla^{\perp} \boldsymbol{\mu} \cdot \nabla \boldsymbol{\rho} - (\nabla \boldsymbol{\alpha} + \beta \boldsymbol{\alpha} \nabla \boldsymbol{\mu}) \cdot \nabla \boldsymbol{\rho} \, d\mathbf{x} \\\\ &= \int\_{\Omega} \boldsymbol{\alpha} \nabla^{\perp} \boldsymbol{\mu} \cdot \nabla \boldsymbol{\rho} + \boldsymbol{\alpha} \Delta \boldsymbol{\rho} - \beta \boldsymbol{\alpha} \nabla \boldsymbol{\mu} \cdot \nabla \boldsymbol{\rho} \, d\mathbf{x} \end{aligned} \tag{124}$$

by (11). It holds that

$$\begin{aligned} \int\_{\Omega} \rho \nabla \boldsymbol{\varPsi} \cdot \nabla \boldsymbol{\varPhi} &= \iint\_{\Omega \times \Omega} \boldsymbol{\varPhi}(\mathbf{x}, t) [\nabla\_{\mathbf{x}} \mathbf{G}(\mathbf{x}, \mathbf{x}') \cdot \nabla \boldsymbol{\varPhi}(\mathbf{x})] \boldsymbol{\upalpha}(\mathbf{x}', t) \, \, d\mathbf{x} d\mathbf{x}' \\ &= \iint\_{\Omega \Omega} \boldsymbol{\upalpha}(\mathbf{x}, t) \boldsymbol{\upalpha}\_{\mathbf{x}, 2\mathbf{R}}(\mathbf{x}') [\nabla\_{\mathbf{x}} \mathbf{G}(\mathbf{x}, \mathbf{x}') \cdot \nabla \boldsymbol{\uprho}(\mathbf{x})] \boldsymbol{\upalpha}(\mathbf{x}', t) \, \, d\mathbf{x} d\mathbf{x}' \\ &+ \iint\_{\Omega \times \Omega} \boldsymbol{\upalpha}(\mathbf{x}, t) \big( 1 - \boldsymbol{\uprho}\_{\mathbf{x}, 2\mathbf{R}}(\mathbf{x}') \big) [\nabla\_{\mathbf{x}} \mathbf{G}(\mathbf{x}, \mathbf{x}') \cdot \nabla \boldsymbol{\uprho}(\mathbf{x})] \boldsymbol{\uprho}(\mathbf{x}', t) \, \, d\mathbf{x} d\mathbf{x}' \\ &= I + II. \end{aligned} \tag{125}$$

Let, furthermore, *x*<sup>0</sup> ∈ Ω and 0<*R* ≪ 1 in the above equality. Then,

$$\rho = \left| \boldsymbol{\kappa} - \boldsymbol{\kappa}\_0 \right|^2 \rho\_{\boldsymbol{\kappa}\_0, \boldsymbol{R}} \tag{126}$$

satisfies the requirement (123). It holds that

$$\nabla \rho = \mathcal{Z}(\mathbf{x} - \mathbf{x}\_0) \rho\_{\mathbf{x}\_0, \mathcal{R}} + |\mathbf{x} - \mathbf{x}\_0|^2 \nabla \rho\_{\mathbf{x}\_0, \mathcal{R}} \tag{127}$$

and hence

$$|\nabla \boldsymbol{\rho}| \le \mathbf{C} |\boldsymbol{\varkappa} - \boldsymbol{\varkappa}\_0| \left( \boldsymbol{\rho}\_{\boldsymbol{\varkappa}\_0, \boldsymbol{R}} + |\boldsymbol{\varkappa} - \boldsymbol{\varkappa}\_0| \boldsymbol{R}^{-1} \boldsymbol{\rho}\_{\boldsymbol{\varkappa}\_0, \boldsymbol{R}}^{1/2} \right) \le \mathbf{C} |\boldsymbol{\varkappa} - \boldsymbol{\varkappa}\_0| \boldsymbol{\rho}\_{\boldsymbol{\varkappa}\_0, \boldsymbol{R}}^{1/2}. \tag{128}$$

We obtain, furthermore,

$$|\boldsymbol{\pi}' - \boldsymbol{\pi}\_0| \ge 2R, \quad |\boldsymbol{\pi} - \boldsymbol{\pi}\_0| \le R \Rightarrow |\boldsymbol{\pi} - \boldsymbol{\pi}'| \ge R,\tag{129}$$

and hence

$$|\nabla\_{\mathbf{x}}G(\mathbf{x}, \mathbf{x}')| \le \mathbf{C} \mathbf{R}^{-1} \tag{130}$$

in this case. Then it follows that

$$|II| \le \mathcal{C} \mathcal{R}^{-1} M \int\_{\mathfrak{U}} |\mathbf{x} - \mathbf{x}\_0| \rho\_{\mathbf{x}\_0, \mathbf{R}}^{1/2} o(\mathbf{x}, t) \, \, d\mathbf{x} \le \mathcal{C} \mathcal{R}^{-1} M^{3/2} A^{1/2}, \tag{131}$$

where

$$A = \int\_{\Omega} |\mathfrak{x} - \mathfrak{x}\_0|^2 \rho\_{\mathfrak{x}\_0, \mathbb{R}} a \,. \tag{132}$$

We have, on the other hand,

$$\begin{split} II &= \iint\_{\Omega \times \Omega} \boldsymbol{\alpha}(\mathbf{x}, t) \boldsymbol{\varrho}\_{\mathbf{x}\_{0}, \mathbf{2R}}(\mathbf{x}') [\nabla\_{\mathbf{x}} \mathbf{G}(\mathbf{x}, \mathbf{x}') \cdot \nabla \boldsymbol{\varrho}(\mathbf{x})] \boldsymbol{\alpha}(\mathbf{x}', t) \, d\mathbf{x} d\mathbf{x}' \\ &= \frac{1}{2} \iint\_{\Omega \Omega} \left[ \boldsymbol{\varrho}\_{\mathbf{x}\_{0}, \mathbf{2R}}(\mathbf{x}') \nabla \boldsymbol{\varrho}(\mathbf{x}) \cdot \nabla\_{\mathbf{x}} \mathbf{G}(\mathbf{x}, \mathbf{x}') + \boldsymbol{\varrho}\_{\mathbf{x}\_{0}, \mathbf{2R}}(\mathbf{x}) \nabla \boldsymbol{\varrho}(\mathbf{x}') \cdot \nabla\_{\mathbf{x}'} \mathbf{G}(\mathbf{x}, \mathbf{x}') \right] \boldsymbol{\varrho} \otimes \boldsymbol{\varrho}, \end{split} \tag{133}$$

where *G* ¼ *G x*, *x*<sup>0</sup> ð Þ is the Green's function to

$$-\Delta \boldsymbol{\mu} = \boldsymbol{\alpha}, \quad \boldsymbol{\alpha}|\_{\partial \Omega} = \mathbf{0} \tag{134}$$

and

$$
\alpha \otimes \alpha = \alpha(\mathfrak{x}, t)\alpha(\mathfrak{x}', t) \,\, d\mathfrak{x}d\mathfrak{x}'.\tag{135}
$$

Here we use the local property of the Green's function

$$G(\mathbf{x}, \mathbf{x}') = \Gamma(\mathbf{x} - \mathbf{x}') + K(\mathbf{x}, \mathbf{x}'), \quad K \in \mathbb{C}^2(\overline{\Omega} \times \Omega) \cap \mathbb{C}^2(\Omega \times \overline{\Omega}), \tag{136}$$

where

$$\Gamma(\mathbf{x}) = \frac{1}{2\pi} \log \frac{1}{|\mathbf{x}|} \tag{137}$$

stands for the fundamental solution to �Δ. Let

$$
\rho\_{\mathbf{x}\_{0},\mathbf{R}}^{2}(\mathbf{x},\mathbf{x}() = \rho\_{\mathbf{x}\_{0},2\mathbf{R}}(\mathbf{x}')\nabla\rho(\mathbf{x}) \cdot \nabla\_{\mathbf{x}}K(\mathbf{x},\mathbf{x}') + \rho\_{\mathbf{x}\_{0},2\mathbf{R}}\nabla\rho(\mathbf{x}') \cdot \nabla\_{\mathbf{x}'}K(\mathbf{x},\mathbf{x}').\tag{138}
$$

Since (128) implies

$$\begin{split} |\varrho\_{\mathbf{x}\_{0},2\mathbb{R}}(\mathbf{x}')\nabla\rho(\mathbf{x})| &\leq \mathsf{C}\rho\_{\mathbf{x}\_{0},2\mathbb{R}}(\mathbf{x}')|\mathbf{x}-\mathbf{x}\_{0}|\rho\_{\mathbf{x}\_{0},\mathbb{R}}^{1/2}(\mathbf{x}) \\ &\leq \mathsf{C}|\mathbf{x}-\mathbf{x}\_{0}|\rho\_{\mathbf{x}\_{0},\mathbb{R}}^{1/2}(\mathbf{x}), \end{split} \tag{139}$$

it holds that

$$|\rho\_{\mathbf{x}\_{0},R}^{1}(\mathbf{x},\mathbf{x}')| \le \mathcal{C} \left( |\mathbf{x} - \mathbf{x}\_{0}| \rho\_{\mathbf{x}\_{0},R}^{1/2}(\mathbf{x}) + |\mathbf{x}' - \mathbf{x}\_{0}| \rho\_{\mathbf{x}\_{0},R}^{1/2}(\mathbf{x}') \right). \tag{140}$$

Then, we obtain

$$I = \frac{1}{2} \iint\_{\Omega \times \Omega} \rho^0\_{\chi\_0, R}(\mathfrak{x}, \mathfrak{x}') \omega \otimes \omega + III \tag{141}$$

with

$$|\mathrm{III}| \le \mathrm{CM}^{3/2} A^{1/2} \le \mathrm{CR}^{-1} \mathbf{M}^{3/2} A^{1/2},\tag{142}$$

where

$$\rho^{0}\_{\mathbf{x}\_{0},\mathbb{R}}(\mathbf{x},\mathbf{x}') = \nabla \Gamma(\mathbf{x} - \mathbf{x}') \cdot \left(\rho\_{\mathbf{x}\_{0},2\mathbb{R}}(\mathbf{x}')\nabla\rho(\mathbf{x}) - \rho\_{\mathbf{x}\_{0},2\mathbb{R}}(\mathbf{x})\nabla\rho(\mathbf{x}')\right). \tag{143}$$

Here, we have

$$\nabla \Gamma(\mathbf{x}) = -\frac{\mathbf{x}}{2\pi |\mathbf{x}|^2},\tag{144}$$

and therefore,

$$
\rho^0\_{\mathbf{x}\_0, \mathbb{R}}(\mathbf{x}, \mathbf{x}') = \rho^2\_{\mathbf{x}\_0, \mathbb{R}}(\mathbf{x}, \mathbf{x}') + \rho^3\_{\mathbf{x}\_0, \mathbb{R}}(\mathbf{x}, \mathbf{x}') \tag{145}
$$

fo

$$\rho\_{\mathbf{x}\_{0},\mathbf{R}}^{2}(\mathbf{x},\mathbf{x}') = -\frac{1}{2\pi} \frac{\mathbf{x} - \mathbf{x}'}{|\mathbf{x} - \mathbf{x}'|} \rho\_{\mathbf{x}\_{0},2\mathbf{R}}(\mathbf{x}') \cdot \left(\nabla\rho(\mathbf{x}) - \nabla\rho(\mathbf{x}')\right) \tag{146}$$

$$\rho^{3}\_{\mathbf{x}\_{0},\mathbb{R}}(\mathbf{x},\mathbf{x}') = -\frac{1}{2\pi} \frac{\mathbf{x} - \mathbf{x}'}{|\mathbf{x} - \mathbf{x}'|} \left(\rho\_{\mathbf{x}\_{0},2\mathbb{R}}(\mathbf{x}') - \rho\_{\mathbf{x}\_{0},2\mathbb{R}}(\mathbf{x})\right) \cdot \nabla \rho(\mathbf{x}).\tag{147}$$

Since (128) implies

$$|\rho\_{\mathbf{x}\_{0},\mathcal{R}}^{3}(\mathbf{x},\mathbf{x}')| \le \mathsf{CR}^{-1} |\nabla \rho(\mathbf{x})| \le \mathsf{CR}^{-1} |\mathbf{x} - \mathbf{x}\_{0}| \rho\_{\mathbf{x}\_{0},\mathcal{R}}^{1/2}(\mathbf{x}),\tag{148}$$

there arises that

$$I = \frac{1}{2} \iint\_{\Omega \times \Omega} \rho\_{\mathbf{x}\_{0}, \mathbb{R}}^{2}(\mathbf{x}, \mathbf{x}') \, \, \boldsymbol{\alpha} \otimes \boldsymbol{\alpha} + IV,\tag{149}$$

with

$$|IV| \le CR^{-1}M^{3/2}A^{1/2},\tag{150}$$

similarly. We have, furthermore,

$$\begin{split} \nabla \rho(\mathbf{x}) - \nabla \rho(\mathbf{x}') &= \mathbf{2}(\mathbf{x} - \mathbf{x}') \rho\_{\mathbf{x}\_0, \mathbb{R}}(\mathbf{x}) + \mathbf{2}(\mathbf{x}' - \mathbf{x}\_0) \left( \rho\_{\mathbf{x}\_0, \mathbb{R}}(\mathbf{x}) - \rho\_{\mathbf{x}\_0, \mathbb{R}}(\mathbf{x}') \right) \\ &+ \left| \mathbf{x}' - \mathbf{x}\_0 \right|^2 \left( \nabla \rho\_{\mathbf{x}\_0, \mathbb{R}}(\mathbf{x}) - \nabla \rho\_{\mathbf{x}\_0, \mathbb{R}}(\mathbf{x}') \right) + \left( \left| \mathbf{x} - \mathbf{x}\_0 \right|^2 - \left| \mathbf{x}' - \mathbf{x}\_0 \right|^2 \right) \nabla \rho\_{\mathbf{x}\_0, \mathbb{R}}(\mathbf{x}), \end{split} \tag{151}$$

and hence

$$\rho\_{\mathbf{x}\_{0},\mathbf{R}}^{2}(\mathbf{x},\mathbf{x}') = -\frac{1}{\pi}\rho\_{\mathbf{x}\_{0},2\mathbf{R}}(\mathbf{x}')\rho\_{\mathbf{x}\_{0},\mathbf{R}}(\mathbf{x}) + \rho\_{\mathbf{x}\_{0},\mathbf{R}}^{4}(\mathbf{x},\mathbf{x}') + \rho\_{\mathbf{x}\_{0},\mathbf{R}}^{5}(\mathbf{x},\mathbf{x}') + \rho\_{\mathbf{x}\_{0},\mathbf{R}}^{6}(\mathbf{x},\mathbf{x}') \tag{152}$$

with

$$\begin{split} |\rho^{\mathsf{A}}\_{\mathbf{x}\_{0},\mathsf{R}}(\mathbf{x},\mathsf{x}')| &\leq C|\mathsf{x}-\mathsf{x}'|^{-1}\rho\_{\mathbf{x}\_{0},\mathsf{2R}}(\mathsf{x}')|\mathsf{x}'-\mathsf{x}\_{0}||\rho\_{\mathbf{x}\_{0},\mathsf{R}}(\mathsf{x})-\rho\_{\mathbf{x}\_{0},\mathsf{R}}(\mathsf{x}')| \\ &\leq CR^{-1}|\mathsf{x}'-\mathsf{x}\_{0}|\rho\_{\mathbf{x}\_{0},\mathsf{2R}}(\mathsf{x}'), \end{split} \tag{153}$$

$$\begin{split} \left| \rho\_{\mathbf{x}\_{0},\mathbb{R}}^{\mathsf{S}}(\mathbf{x},\mathbf{x}') \right| &\leq C \left| \mathbf{x} - \mathbf{x}' \right|^{-1} \rho\_{\mathbf{x}\_{0},2\mathbb{R}}(\mathbf{x}') \left| \mathbf{x}' - \mathbf{x}\_{0} \right|^{2} \left| \nabla \rho\_{\mathbf{x}\_{0},\mathbb{R}}(\mathbf{x}) - \rho\_{\mathbf{x}\_{0},\mathbb{R}}(\mathbf{x}') \right| \\ &\leq CR^{-2} \left| \mathbf{x}' - \mathbf{x}\_{0} \right|^{2} \rho\_{\mathbf{x}\_{0},2\mathbb{R}}(\mathbf{x}') \\ &\leq CR^{-1} \left| \mathbf{x}' - \mathbf{x}\_{0} \right| \rho\_{\mathbf{x}\_{0},2\mathbb{R}}(\mathbf{x}'), \end{split} \tag{154}$$

and

$$\begin{split} \left| \rho\_{\mathbf{x}\_{0},\mathbf{R}}^{6} (\mathbf{x}, \mathbf{x}') \right| &\leq \mathbf{C} |\mathbf{x} - \mathbf{x}'| \left| \rho\_{\mathbf{x}\_{0},2\mathbf{R}} (\mathbf{x}') || \mathbf{x} - \mathbf{x}\_{0} \right|^{2} - \left| \mathbf{x}' - \mathbf{x}\_{0} \right|^{2} | \cdot | \nabla \rho\_{\mathbf{x}\_{0},\mathbf{R}} (\mathbf{x}) | \\ &\leq \mathbf{C} \mathbf{R}^{-1} (|\mathbf{x} - \mathbf{x}\_{0}| + |\mathbf{x}' - \mathbf{x}\_{0}|) \rho\_{\mathbf{x}\_{0},\mathbf{R}} (\mathbf{x}) \rho\_{\mathbf{x}\_{0},2\mathbf{R}} (\mathbf{x}') \\ &\leq \mathbf{C} \left( \mathbf{R}^{-1} |\mathbf{x} - \mathbf{x}\_{0}| \rho\_{\mathbf{x}\_{0},\mathbf{R}} (\mathbf{x}) + \mathbf{R}^{-1} | \mathbf{x}' - \mathbf{x}\_{0}| \rho\_{\mathbf{x}\_{0},2\mathbf{R}} (\mathbf{x}') \right) \end{split} \tag{155}$$

by

$$\left| \left| \left| \mathbf{x} - \mathbf{x}\_{0} \right|^{2} - \left| \mathbf{x}' - \mathbf{x}\_{0} \right|^{2} \right| = \left| \left( \mathbf{x} - \mathbf{x}', \mathbf{x} + \mathbf{x}' - 2\mathbf{x}\_{0} \right) \right| \le \left| \mathbf{x} - \mathbf{x}' \right| (\left| \mathbf{x} - \mathbf{x}\_{0} \right| + \left| \mathbf{x}' - \mathbf{x}\_{0} \right|). \tag{156}$$

The residual terms are thus treated similarly, and it follows that

$$\left| I + \frac{1}{2\pi} \int\_{\Omega} \alpha \rho\_{\mathbf{x}\_{0}, \mathbb{R}} \cdot \int\_{\Omega} \alpha \rho\_{\mathbf{x}\_{0}, 2\mathbb{R}} \right| \leq \mathsf{CR}^{-1} M^{3/2} A^{1/2}, \tag{157}$$

which results in

*Relaxation Dynamics of Point Vortices DOI: http://dx.doi.org/10.5772/intechopen.100585*

$$\left| \int\_{\Omega} a \nabla \boldsymbol{\varphi} \cdot \nabla \boldsymbol{\varphi} + \frac{1}{2\pi} \int\_{\Omega} a \boldsymbol{\varphi}\_{\mathbf{x}\_{0}, \mathbf{R}} \cdot \int\_{\Omega} a \boldsymbol{\varphi}\_{\mathbf{x}\_{0}, \mathbf{2R}} \right| \leq \mathbf{C} \mathbf{R}^{-1} \mathbf{M}^{3/2} A^{1/2}. \tag{158}$$

We can argue similarly to the vortex term in (124). This time, from

$$\nabla^{\perp} \Gamma(\mathfrak{x}) \cdot \mathfrak{x} = \mathbf{0} \tag{159}$$

it follows that

$$\left| \int\_{\Omega} \rho \nabla^{\perp} \boldsymbol{\psi} \cdot \nabla \boldsymbol{\varphi} \right| \leq \mathsf{CR}^{-1} \mathsf{M}^{3/2} \mathsf{A}^{1/2}. \tag{160}$$

Concerning the principal term of (124), we use

$$
\Delta \boldsymbol{\varrho} = 4 \boldsymbol{\varrho}\_{\mathbf{x}\_0, \mathbb{R}} + 4(\boldsymbol{\varkappa} - \boldsymbol{\varkappa}\_0) \cdot \nabla \boldsymbol{\varrho}\_{\mathbf{x}\_0, \mathbb{R}} + |\boldsymbol{\varkappa} - \boldsymbol{\varkappa}\_0|^2 \Delta \boldsymbol{\varrho}\_{\mathbf{x}\_0, \mathbb{R}}.\tag{161}
$$

From

$$|(\boldsymbol{\omega} - \boldsymbol{\omega}\_0) \cdot \nabla \boldsymbol{\rho}\_{\boldsymbol{x}\_0, \boldsymbol{R}}| \leq \boldsymbol{\mathcal{CR}}^{-1} |\boldsymbol{\omega} - \boldsymbol{\omega}\_0| \boldsymbol{\rho}\_{\boldsymbol{x}\_0, \boldsymbol{R}}^{1/2} \tag{162}$$

and

$$\begin{split} \left| \left| -\chi\_{0} \right|^{2} \Delta \boldsymbol{\rho}\_{\mathbf{x}\_{0}, \mathbf{R}} \right| &\leq \mathsf{CR}^{-2} |\boldsymbol{\varkappa} - \boldsymbol{\varkappa}\_{0}|^{2} \boldsymbol{\rho}\_{\mathbf{x}\_{0}, \mathbf{R}}^{1/2} \\ &\leq \mathsf{CR}^{-1} |\boldsymbol{\varkappa} - \boldsymbol{\varkappa}\_{0}| \boldsymbol{\varphi}\_{\mathbf{x}\_{0}, \mathbf{R}}^{1/2}, \end{split} \tag{163}$$

it follows that

$$\left| \int\_{\Omega} a \Delta \rho - 4 \int\_{\Omega} a \rho\_{\mathbf{x}\_0, \mathbf{R}} \right| \leq C \int\_{\Omega} R^{-1} |\mathbf{x} - \mathbf{x}\_0| \rho\_{\mathbf{x}\_0, \mathbf{R}} a \,\mathrm{d}\mathbf{x} \tag{164}$$

$$\leq C R^{-1} M^{1/2} A^{1/2}.$$

Let *M*<sup>1</sup> ¼ *Mx*0,*<sup>R</sup>* and *M*<sup>2</sup> ¼ *Mx*0,2*<sup>R</sup>* for

$$M\_{\mathbf{x}\_0, \mathbb{R}} = \int\_{\Omega} a \boldsymbol{q} \boldsymbol{\rho}\_{\mathbf{x}\_0, \mathbb{R}}.\tag{165}$$

Then, using (120), we end up with

$$\frac{dA}{dt} \le 4M\_1 - \frac{M\_1^2}{2\pi} + \text{CR}^{-1} \left( M^{3/2} + M^{1/2} \right) \text{A}^{1/2} + \text{C} (\text{M}\_2 - \text{M}\_1). \tag{166}$$

Inequalilty (166) implies *T* < þ ∞ if *A*ð Þ 0 ≪ 1, as is observed by [27] (see also Chapter 5 of [19]). Here we describe the proof for completeness.

The first observation is the monotoniity formula

$$\left| \frac{d}{dt} \int\_{\Omega} \alpha \rho \rho \right| \le C \left( M + M^2 \right) \| \nabla \rho \|\_{C^4}, \tag{167}$$

derived from (124) and the symmetry of the Green's function: *G x*, *x*<sup>0</sup> ð Þ¼ *G x*<sup>0</sup> ð Þ , *x* . The proof is the same as in (34) and is omitted.

$$I\_{\mathbf{x}\_{0},\mathbf{R}} = \int\_{\mathfrak{U}} \left| \mathbf{x} - \mathbf{x}\_{0} \right|^{2} a \rho \rho\_{\mathbf{x}\_{0},\mathbf{R}}.\tag{168}$$

$$\begin{split} \left| M\_2 - M\_1 \leq \right|\_{R < |\mathbf{x} - \mathbf{x}\_0| < 2R} \rho\_{\mathbf{x}\_0, 2R} \alpha \\ \leq 2R^{-1} \int\_{\Omega} |\mathbf{x} - \mathbf{x}\_0| \rho\_{\mathbf{x}\_0, 2R} \alpha \leq 2M^{1/2} R^{-1} I\_2^{1/2} \end{split} \tag{169}$$

$$\begin{split} A\_2 &= A\_1 + \int\_{\Omega} |\mathbf{x} - \mathbf{x}\_0|^2 \left(\rho\_{\mathbf{x}\_0, 2R} - \rho\_{\mathbf{x}\_0, R}\right) a \\ &\leq A\_1 + 4R^2 \int\_{\Omega} \left(\rho\_{\mathbf{x}\_0, 2R} - \rho\_{\mathbf{x}\_0, R}\right) a, \end{split} \tag{170}$$

$$\begin{split} \frac{dA\_1}{dt} &\le 4M\_1 - \frac{M\_1^2}{2\pi} + CR^{-1} \left( M^{3/2} + M^{1/2} \right) A\_1^{1/2} \\ &+ C \left( M^{3/2} + M^{1/2} \right) \left\{ \int\_{\Omega} \left( \rho\_{\mathbf{x}\_0, 2\mathbf{R}} - \rho\_{\mathbf{x}\_0, \mathbf{R}} \right) a \right\}^{1/2} . \end{split} \tag{171}$$

$$\left| \frac{d}{dt} \left( 4M\_1 - \frac{M^2}{2\pi} \right| \le C \left( M + M^2 \right) R^{-2} \right. \tag{172}$$

$$\left| \frac{d}{dt} \int\_{\Omega} (\rho\_{\mathbf{x}\_0, 2R} - \rho\_{\mathbf{x}\_0, R}) \, \omega \right| \le C (M + M^2) R^{-2}. \tag{173}$$

$$4M\_1 - \frac{M\_1^2}{2\pi} \le 4M\_1(\mathbf{0}) - \frac{M\_1(\mathbf{0})^2}{2\pi} + \text{CBa}\left(\mathbf{R}^{-1}t^{1/2}\right) \tag{174}$$

$$\begin{split} \int\_{\Omega} \left( \rho\_{\mathbf{x}\_{0}, 2R} - \rho\_{\mathbf{x}\_{0}, R} \right) a \leq \int\_{\Omega} \left( \rho\_{\mathbf{x}\_{0}, 2R} - \rho\_{\mathbf{x}\_{0}, R} \right) a \mathbf{o}\_{0} + \mathbf{C} \mathbf{B} a \left( R^{-1} t^{1/2} \right) \\ \leq 2 \mathbf{R}^{-2} A\_{2}(\mathbf{0}) + \mathbf{C} \mathbf{B} a \left( R^{-1} t^{1/2} \right) \end{split} \tag{175}$$

$$B = \mathbf{M}^{3/2} + \mathbf{M}^{1/2}, \quad \mathfrak{a}(\mathfrak{s}) = \mathfrak{s}^2 + \mathfrak{s}. \tag{176}$$

*Relaxation Dynamics of Point Vortices DOI: http://dx.doi.org/10.5772/intechopen.100585*

Thus we obtain

$$\begin{split} \frac{dA\_1}{dt} \leq & 4M\_1(\mathbf{0}) - \frac{M\_1(\mathbf{0})^2}{2\pi} + \mathcal{C}\mathcal{R}^{-1}\mathcal{B}A\_1^{1/2} + \mathcal{C}\mathcal{B}A\_2(\mathbf{0})^{1/2} + \mathcal{C}\mathcal{B}a\left(\mathcal{R}^{-1}t^{1/2}\right) \\ = & f(\mathbf{0}) + \mathcal{C}\mathcal{B}a\left(\mathcal{R}^{-1}t^{1/2}\right) + \mathcal{C}\mathcal{B}\mathcal{R}^{-1}A\_1^{1/2} \end{split} \tag{177}$$

for

$$J = 4M\_1 - \frac{M\_1^2}{4\pi} + \text{CBR}^{-1} A\_2^{1/2}. \tag{178}$$

Assume *M*1ð Þ 0 >8*π*, and put

$$-4\delta = 4M\_1(0) - \frac{M\_1(0)^2}{2\pi} < 0.\tag{179}$$

Let, furthemore,

$$\frac{1}{R^2} \int\_{\Omega} |\varkappa - \varkappa\_0|^2 \rho\_{\varkappa\_0, 2R} a\_0 \le \eta. \tag{180}$$

Now we define *s*<sup>0</sup> by

$$\text{CBa}(\mathfrak{s}\_0) = \delta \tag{181}$$

in (177), and take 0< *η* ≪ 1 such that

$$
\eta \le \delta \mathfrak{s}\_0^2. \tag{182}
$$

Then, if *R* and *T*<sup>0</sup> satisfy *R*�<sup>2</sup> *<sup>T</sup>*<sup>0</sup> <sup>¼</sup> *ηδ*�<sup>1</sup> , it holds that

$$A\_1(\mathbf{0}) \le \mathbb{R}^2 \eta < 2\delta T\_0. \tag{183}$$

Making 0 <*η* ≪ 1, furthermore, we may assume

$$\begin{split} J(\mathbf{0}) + \mathbf{CBR}^{-1} A\_1(\mathbf{0})^{1/2} &\leq -4\delta + \mathbf{CBR}^{-1} A\_2(\mathbf{0})^{1/2} \\ &\leq -4\delta + \mathbf{CB} \eta^{1/2} \leq -3\delta, \end{split} \tag{184}$$

which results in

$$\frac{dA\_1}{dt} \le f(\mathbf{0}) + CBa \left( R^{-1} T\_0^{1/2} \right) + BR^{-1} A\_1(t)^{1/2}$$

$$= f(\mathbf{0}) + \delta + \mathbf{CBR}^{-1} A\_1^{1/2}, \quad \mathbf{0} \le t < T\_0,\tag{185}$$

provided that *T* ≥*T*0.

A continuation argument to (184)–(185) guarantees

$$\frac{dA\_1}{dt} \le -2\delta, \quad 0 \le t < T\_0,\tag{186}$$

and then we obtain

*Vortex Dynamics - From Physical to Mathematical Aspects*

$$A\_1(T\_0) \le A\_1(\mathbf{0}) - 2\delta T\_0 < \mathbf{0} \tag{187}$$

by (183), a contradiction. □

#### **Acknowledgements**

This work was supported by JSPS Grand-in-Aid for Scientific Research 19H01799.

#### **Author details**

Ken Sawada<sup>1</sup> and Takashi Suzuki<sup>2</sup> \*

1 Meteorological College, Asahi-cho, Kashiwashi, Japan

2 Center for Mathematical Modeling and Data Science, Osaka University, Toyonakashi, Japan

\*Address all correspondence to: suzuki@sigmath.es.osaka-u.ac.jp

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Relaxation Dynamics of Point Vortices DOI: http://dx.doi.org/10.5772/intechopen.100585*

#### **References**

[1] P.H. Chavanis, *Generalized thermodynamics and Fokker-Planck equations: Applications to stellar dynamics and two-dimensional turbulence*, Phys. Rev. E68, (2003) 036108.

[2] P.-H. Chavanis,*Two-dimensional Brownian vortices*, Physica A 387 (2008) 6917-6942.

[3] P.H. Chavanis, J. Sommeria, R. Robert, *Statistical mechanics of twodimensional vortices and collisionless stellar systems*, Astrophys. J. 471 (1996) 385-399.

[4] K. Sawada, T, Suzuki, *Relaxation theory for point vortices*, In; Vortex Structures in Fluid Dynamic Problems (H. Perez-de-Tejada ed.), INTECH 2017, Chapter 11, 205-224.

[5] D. Lynden-Bell, *Statistical mechanics of violent relaxation in stellar systems*, Monthly Notices of Royal Astronmical Society 136 (1967) 101-121.

[6] E. Caglioti, P.L. Lions, C. Marchioro, M. Pulvirenti, *A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description* Comm. Math. Phys. 143 (1992) 501.

[7] G.L. Eyink, H. Spohn, *Negativetemperature states and large scale, longvisited vortices in two-dimensional turbulence*, Statistical Physics 70 (1993) 833.

[8] G. Joyce, D. Montgomery, *Negative temperature states for two-dimensional guiding-centre plasma*, J. Plasma Phys. 10 (1973) 107.

[9] M.K.H. Kiessling, *Statistical mechanics of classical particles with logarithmic interaction*, Comm. Pure Appl. Math. 46 (1993) 27-56.

[10] K. Nagasaki, T. Suzuki, *Asymptotic analysis for two-dimensional elliptic*

*eigenvalue problems with exponentially dominated nonlinearities*, Asymptotic Analysis 3 (1990) 173-188.

[11] L. Onsager, *Statistical hydrodynamics*, Suppl. Nuovo Cimento 6 (1949) 279-287.

[12] Y.B. Pointin, T.S. Lundgren, *Statistical mechanics of two-dimensional vortices in a bounded container*, Phys. Fluids 19 (1976) 1459-1470.

[13] R. Robert, J. Sommeria, *Statistical equilibrium states for two-dimensional flows*, J. Fluid Mech. 229 (1991) 291-310.

[14] R. Robert, J. Sommeria, *Relaxation towards a statistical equilibrium state in two-dimensional perfect fluid dynamics*, Phys. Rev. Lett. 69 (1992) 2776-2779.

[15] R. Robert, *A maximum-entropy principle for two-dimensional perfect fluid dynamics* J. Stat. Phys. **65** (1991) 531-553.

[16] R. Robert, C. Rosier,*The modeling of small scales in two-dimensional turbulent flows: A statistical Mechanics Approach*, J. Stat. Phys. 86 (1997) 481-515.

[17] K. Sawada, T. Suzuki, *Derivation of the equilibrium mean field equations of point vortex system and vortex filament system*, Theor. Appl. Mech. Japan 56 (2008) 285-290.

[18] C. Sire, P.-H. Chavanis, *Thermodynamics and collapse of selfgravitating Brownian particles in D dimensions*, Phys. Rev. E 66 (2002) 046133.

[19] T. Suzuki, *Free Energy and Self-Interacting Particles*, Birkhäuser, Boston, 2005.

[20] T. Suzuki, *Mean Field Theories and Dual Variation - Mathematical Structures of Mesoscopic Model*, 2nd edition, Atlantis Press, Paris, 2015.

[21] T. Suzuki, *Chemotaxis, Reaction, Network - Mathematics for Self-Organization*, World Scientific, Singapore, 2018.

[22] T. Suzuki, *Liouville's Theory in Linear and Nonlinear Partial Differential Equations - Interaction of Analysis, Geometry, Physics*, Springer, Berlin, (to appear).

[23] T. Suzuki, *Applied Analysis - Mathematics for Science, Engineering, Technology*, 3rd edition, Imperial College Press, London, (to appear).

[24] P. Biler, *Local and global solvability of some parabolic systems modelling chemotaxis*, Adv. Math. Sci. Appl. 8 (1998) 715-743.

[25] H. Gajewski, K. Zacharias, *Global behaviour of a reaction-diffusion system modelling chemotaxis*, Math. Nachr. 195 (1998) 77-114.

[26] T. Nagai, T. Senba, and K. Yoshida, *Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis*, Funkcial. Ekvac. 40 (1997) 411-433.

[27] T. Senba and T. Suzuki, *Parabolic system of chemotaxis; blowup in a finite and in the infinite time*, Meth. Appl. Anal. 8 (2001) 349-368.

[28] W. Jäger, S. Luckhaus, *On explosions of solutions to a system of partial differential equations modelling chemotaxis*, Trans. Amer. Math. Soc. 329 (1992) 819-824.

[29] T. Nagai, *Blow-up of radially symmetric solutions to a chemotaxis system*, Adv. Math. Sci. Appl. 5 (1995) 581-601.

[30] T. Suzuki, *Exclusion of boundary blowup for* 2*D chemotaxis system provided with Dirichlet condition for the Poisson part*, J. Math. Pure Appl. 100 (2013) 347-367.

[31] T. Suzuki, *Semilinear Elliptic Equations - Classical and Modern Theories*, De Gruyter, Berlin, 2020.

[32] B. Gidas, W.M. Ni, L. Nirenberg, *Symmetry and related properties via the maximum principle*, Comm. Math. Phys. 68 (1979) 209-243.

[33] N.D. Alikakos, *Lp bounds of solutions of reaction-diffusion equations*, Comm. Partial Differential Equations 4 (1979) 827-868.

[34] T. Senba and T. Suzuki, *Chemotactic collapse in a parabolic-elliptic system of mathematical biology*, Adv. Differential Equations 6 (2001) 21-50.

### Section 3
