**3. Particle-vortex interactions in turbulence**

In order to initiate meaningful interactions through the interfacial forces, particles must be brought in close proximity. This can be achieved through the mechanisms of Brownian motion, differential settling, or turbulent dispersion as shown in **Figure 4**. The probability of particle collisions and the frequency of such collisions also depend on the trajectory of the particle motion. Thomas et al. [22] identified two types of particle trajectories leading to particle collisions namely: curvilinear and rectilinear particle trajectories. The particles in suspension under the influence of turbulence will experience fluctuating fluid motion with the particle being transported by the fluid eddies that exists within the flow vortex [3].

Consequently, small particles suspended in fluid exist in an environment of small energy-dissipating eddies in most practical flow devices. Under such conditions, particle collisions are facilitated by eddy size similar to that of the colliding particles. Furthermore, studies have shown that in addition to the fluid properties,

#### **Figure 4.**

*Mechanisms of particle transport in fluid leading to collisions (a) Brownian motion (b) fluid turbulence (c) differential settling (reproduced from [19] with permissions © 2006 CRC press).*

**9**

*The Role of Micro Vortex in the Environmental and Biological Processes*

the particle properties such as size, density, and porosity also play important role in particle-vortex interactions [1, 23]. Smaller particles with lower density than the flow fluid have been shown to be fully entrained within the vortex in the case of a vortical flow produced by the interaction of two flow streams of unequal velocity. The particles will be in dynamic equilibrium with the carrier fluid and will faithfully follow the flow streamlines. On the other hand, large particles will be unaffected by the vortex due to their large inertia, while the intermediate particles will be driven from the center of the vortex to the periphery as shown in **Figure 5**. The determining factor in particle-vortex interactions is the ratio of the particle relaxation time to that of the fluid, which is given by the Stokes number (Eq. (8)). Depending on the flow scenario, the particles in suspension will experience additional forces such as the drag, lift, pressure gradient, virtual mass, basset, and viscous stress forces due to the fluid-particle interactions [4, 24]. Taken together, all these forces will ultimately determine the trajectory, dynamics, and fate of particles

*Vortex-Particle interactions in turbulent flow (reproduced from [1] with permissions © 1995 springer).*

18 <sup>τ</sup> = = τ µ

A p F

where is τA the particle relaxation time, τF is the time associated with fluid motion (fluid time), ϱp is the particle density, *d* is the particle diameter, L is the

Turbulence is the main driver of particle interactions in many practical applications. Consequently, the particle dynamics in terms of the particle collisions, coalescence/aggregation, growth, and breakage is primarily controlled by the fluid turbulence. The aggregate stability under the influence of hydrodynamic force has been suggested to be a function of the binding or cohesive force FB and

St

length scale associated with the vortex while U is the flow velocity.

**4. Particle dynamics and aggregate disruption**

→

2

mic

→→→ = ++ mol f p<sup>−</sup> d v m FFF

d U

L

(8)

dt (9)

in suspension. The trajectory equation is given in Eq. (9).

*DOI: http://dx.doi.org/10.5772/intechopen.93531*

**Figure 5.**

*The Role of Micro Vortex in the Environmental and Biological Processes DOI: http://dx.doi.org/10.5772/intechopen.93531*

**Figure 5.**

*Vortex Dynamics Theories and Applications*

two particles are in direct contact.

of discrete sub-elements as shown in **Figure 3**. All particle-particle interactions within this context are described by contact forces in the normal and tangential directions, while considering the elastic force-displacement, inelastic deformations or plastic dislocations, solid friction, and viscous damping [17]. Depending on the simplicity of these interactions, a soft or hard sphere description can be given. In the hard sphere model, only elastic force-displacement is allowed. Soft sphere model on the other hand allows for most of the interactions that are possible when

When all these contact forces are fully resolved, the behavior of the particles upon collisions or impact on a wall such as their translational and rotational velocities can be predicted with a high degree of accuracy. A detailed description of the micromechanical theory of particle collisions and its importance in the determination of particle trajectory in dispersed suspension is beyond the scope of this communication and is available elsewhere [20]. In addition to the contact forces, body forces such as gravity and buoyancy and surface forces due to the fluid are some of the other important forces acting on the particles and their quantification is highly indispensable in resolving the dynamics of particles in suspension [21]. Some

In order to initiate meaningful interactions through the interfacial forces, particles must be brought in close proximity. This can be achieved through the mechanisms of Brownian motion, differential settling, or turbulent dispersion as shown in **Figure 4**. The probability of particle collisions and the frequency of such collisions also depend on the trajectory of the particle motion. Thomas et al. [22] identified two types of particle trajectories leading to particle collisions namely: curvilinear and rectilinear particle trajectories. The particles in suspension under the influence of turbulence will experience fluctuating fluid motion with the particle being

Consequently, small particles suspended in fluid exist in an environment of small energy-dissipating eddies in most practical flow devices. Under such conditions, particle collisions are facilitated by eddy size similar to that of the colliding particles. Furthermore, studies have shown that in addition to the fluid properties,

*Mechanisms of particle transport in fluid leading to collisions (a) Brownian motion (b) fluid turbulence* 

*(c) differential settling (reproduced from [19] with permissions © 2006 CRC press).*

of these additional forces are discussed in the next section.

transported by the fluid eddies that exists within the flow vortex [3].

**3. Particle-vortex interactions in turbulence**

**8**

**Figure 4.**

*Vortex-Particle interactions in turbulent flow (reproduced from [1] with permissions © 1995 springer).*

the particle properties such as size, density, and porosity also play important role in particle-vortex interactions [1, 23]. Smaller particles with lower density than the flow fluid have been shown to be fully entrained within the vortex in the case of a vortical flow produced by the interaction of two flow streams of unequal velocity. The particles will be in dynamic equilibrium with the carrier fluid and will faithfully follow the flow streamlines. On the other hand, large particles will be unaffected by the vortex due to their large inertia, while the intermediate particles will be driven from the center of the vortex to the periphery as shown in **Figure 5**. The determining factor in particle-vortex interactions is the ratio of the particle relaxation time to that of the fluid, which is given by the Stokes number (Eq. (8)). Depending on the flow scenario, the particles in suspension will experience additional forces such as the drag, lift, pressure gradient, virtual mass, basset, and viscous stress forces due to the fluid-particle interactions [4, 24]. Taken together, all these forces will ultimately determine the trajectory, dynamics, and fate of particles in suspension. The trajectory equation is given in Eq. (9).

$$\mathbf{St} = \frac{\mathbf{\tau}\_{\text{A}}}{\mathbf{\tau}\_{\text{F}}} = \frac{\varrho\_{\text{p}} \,\mathrm{d}^{\text{a}} \mathbf{U}}{18 \mu \mathrm{L}} \tag{8}$$

where is τA the particle relaxation time, τF is the time associated with fluid motion (fluid time), ϱp is the particle density, *d* is the particle diameter, L is the length scale associated with the vortex while U is the flow velocity.

$$\mathbf{m}\frac{\stackrel{\rightarrow}{\mathbf{d}}\stackrel{\rightarrow}{\mathbf{v}}}{\mathbf{d}\mathbf{t}} = \stackrel{\rightarrow}{\mathbf{F}\_{\text{mol}}} + \stackrel{\rightarrow}{\mathbf{F}\_{\text{m}\text{s}}} + \stackrel{\rightarrow}{\mathbf{F}\_{\text{f}\text{--p}}} \tag{9}$$
