**3.3 One- and two-way interaction study of nanofluid characteristics in a finned tube with twisted tape**

This section presents an experimental and numerical study to investigate the improvement of the heat transfer and the interaction in a circular finned tube by utilizing one metal oxide [γ-Al2O3 (20 nm)]/distilled water nanofluid as a coolant with a typical twisted tape having a twist ratio (TR) of 1.85 [5]. The studied concentrations of nanofluids are φ = 0, 3, and 5% by volume under laminar and turbulent flow conditions. The study includes constructing a test section that consists of aluminum tube of 1.5 m long, with internal and external diameters of 22 and 32 mm, respectively; see **Figure 7**. The coolant flows through the inner pipe under laminar

flow (678 ≥ Re ≥ 2033) and turbulent flow (3390 ≥ Re ≥ 10,172) regime with a constant inlet temperature of 60°C. Because of the complexity of twisted tape configurations and the one- and two-way fluid-structure interaction (FSI), it is impossible to determine an analytical solution of the governing equations for the practical configuration. The numerical simulations permit the intricate geometry analysis of the domain flow and the interaction by multiphysics systems coupling. Therefore, the commercial software of the finite volume numerical methods have been used to solve those equations and to study the interaction pattern of the fluid-heat-structure among fluid flow, typical twisted tape insert, and the finned tube having multidegrees of

The mathematical equations utilized for describing the fluid flow are continuity and momentum equations that characterize the conservation of mass and momentum. In addition, the momentum equations are recognized as the Navier-Stokes equations. For flows including heat transfer, another group of equations is needed

for describing the energy conservation. Continuity equation is derived via

For laminar flow, the continuity, momentum, and energy equations are:

*∂u ∂x* þ *∂v ∂y* þ *∂w*

*ρnf u*

*ρnf u*

*ρnf u*

*nf u*

*ρCp* � �

> *u ∂u ∂x* þ *v ∂u ∂y* þ *w*

¼ � <sup>1</sup> *ρnf*

> *u ∂v ∂x* þ *v ∂v ∂y* þ *w ∂v ∂z*

¼ � <sup>1</sup> *ρnf*

*u ∂w ∂x* þ *v ∂w ∂y* þ *w*

¼ � <sup>1</sup> *ρnf*

**101**

*∂u ∂x* þ *v ∂u ∂y* þ *w ∂u ∂z*

*∂v ∂x* þ *v ∂v ∂y* þ *w ∂v ∂z*

*∂w ∂x* þ *v ∂w ∂y* þ *w ∂w ∂z*

*∂Tnf ∂x* þ *v*

� �

<sup>þ</sup> *<sup>γ</sup>* <sup>∇</sup><sup>2</sup> *u*

<sup>þ</sup> *<sup>γ</sup>* <sup>∇</sup><sup>2</sup> *v*

� �

*∂p ∂x*

> *∂p ∂y*

� �

<sup>þ</sup> *<sup>γ</sup>* <sup>∇</sup><sup>2</sup> *w*

*∂p ∂z*

� �

� �

� �

*∂Tnf ∂y*

� �

þ *w*

turbulence kinetic energy (*k*) and its dissipation rate (*ε*) are:

*∂u ∂z*

*∂w ∂z*

*∂ u ∂x* þ *∂ v ∂y* þ *∂ w*

> *∂ ∂x*

> > *∂ ∂x u*0 *<sup>v</sup>*<sup>0</sup> � � <sup>þ</sup>

*∂ ∂x u*0 *<sup>w</sup>*<sup>0</sup> � � <sup>þ</sup>

þ

þ

þ

*∂Tnf ∂z*

employing the mass conservation principle to a small differential fluid volume. In the Cartesian coordinates, three equations with the following forms are determined.

> ¼ � *<sup>∂</sup><sup>p</sup> ∂x* þ *μnf*

> ¼ � *<sup>∂</sup><sup>p</sup> ∂y* þ *μnf*

> ¼ � *<sup>∂</sup><sup>p</sup> ∂z* þ *μnf*

For turbulent flow, the continuity, momentum, and energy equations and

*<sup>u</sup>*0<sup>2</sup> � � <sup>þ</sup>

*∂ ∂y u*0 *<sup>v</sup>*<sup>0</sup> � � <sup>þ</sup>

*∂ ∂y*

> *∂ ∂y v*0 *<sup>w</sup>*<sup>0</sup> � � <sup>þ</sup>

*<sup>w</sup>*<sup>0</sup> � � � �

*v*02 � � <sup>þ</sup>

� � � �

*vw*<sup>0</sup> � � � �

¼ *Knf*

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> <sup>0</sup> (5)

*∂*2 *u ∂z*<sup>2</sup> � � (6)

*∂*2 *v ∂z*<sup>2</sup> � � (7)

*∂*2 *w ∂z*<sup>2</sup> � � (8)

> *∂*2 *Tnf ∂z*<sup>2</sup>

(9)

(11)

(12)

(13)

*∂*2 *u ∂y*<sup>2</sup> þ

*∂*2 *v ∂y*<sup>2</sup> þ

*∂*2 *w ∂y*<sup>2</sup> þ

*∂*2 *Tnf ∂y*<sup>2</sup> þ

!

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> <sup>0</sup> (10)

*∂ ∂z u*0

*∂ ∂z*

> *∂ ∂z*

*w*0<sup>2</sup>

*∂*2 *u ∂x*<sup>2</sup> þ

*∂*2 *v ∂x*<sup>2</sup> þ

*∂*2 *w ∂x*<sup>2</sup> þ

*∂*2 *Tnf ∂x*<sup>2</sup> þ

freedom flow-induced vibration for free and forced vibration.

*Nanofluids and Computational Applications in Medicine and Biology*

*DOI: http://dx.doi.org/10.5772/intechopen.88577*

#### **Figure 7.**

*Physical geometry of finned tube (all dimensions in mm), geometry of twisted tape inside a finned tube insert, and meshing geometry.*

#### *Nanofluids and Computational Applications in Medicine and Biology DOI: http://dx.doi.org/10.5772/intechopen.88577*

**3.3 One- and two-way interaction study of nanofluid characteristics**

This section presents an experimental and numerical study to investigate the improvement of the heat transfer and the interaction in a circular finned tube by utilizing one metal oxide [γ-Al2O3 (20 nm)]/distilled water nanofluid as a coolant with a typical twisted tape having a twist ratio (TR) of 1.85 [5]. The studied concentrations of nanofluids are φ = 0, 3, and 5% by volume under laminar and turbulent flow conditions. The study includes constructing a test section that consists of aluminum tube of 1.5 m long, with internal and external diameters of 22 and 32 mm, respectively; see **Figure 7**. The coolant flows through the inner pipe under laminar

*Physical geometry of finned tube (all dimensions in mm), geometry of twisted tape inside a finned tube insert,*

**in a finned tube with twisted tape**

*Applications of Nanobiotechnology*

**Figure 7.**

**100**

*and meshing geometry.*

flow (678 ≥ Re ≥ 2033) and turbulent flow (3390 ≥ Re ≥ 10,172) regime with a constant inlet temperature of 60°C. Because of the complexity of twisted tape configurations and the one- and two-way fluid-structure interaction (FSI), it is impossible to determine an analytical solution of the governing equations for the practical configuration. The numerical simulations permit the intricate geometry analysis of the domain flow and the interaction by multiphysics systems coupling. Therefore, the commercial software of the finite volume numerical methods have been used to solve those equations and to study the interaction pattern of the fluid-heat-structure among fluid flow, typical twisted tape insert, and the finned tube having multidegrees of freedom flow-induced vibration for free and forced vibration.

The mathematical equations utilized for describing the fluid flow are continuity and momentum equations that characterize the conservation of mass and momentum. In addition, the momentum equations are recognized as the Navier-Stokes equations. For flows including heat transfer, another group of equations is needed for describing the energy conservation. Continuity equation is derived via employing the mass conservation principle to a small differential fluid volume. In the Cartesian coordinates, three equations with the following forms are determined. For laminar flow, the continuity, momentum, and energy equations are:

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial \mathbf{z}} = \mathbf{0} \tag{5}$$

$$\rho\_{\eta f} \left( u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial \mathbf{y}} + w \frac{\partial u}{\partial \mathbf{z}} \right) = -\frac{\partial p}{\partial \mathbf{x}} + \mu\_{\eta f} \left( \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial \mathbf{y}^2} + \frac{\partial^2 u}{\partial \mathbf{z}^2} \right) \tag{6}$$

$$
\rho\_{\eta f} \left( \mu \frac{\partial v}{\partial \mathbf{x}} + \nu \frac{\partial v}{\partial \mathbf{y}} + \nu \frac{\partial v}{\partial \mathbf{z}} \right) = -\frac{\partial p}{\partial \mathbf{y}} + \mu\_{\eta f} \left( \frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial \mathbf{y}^2} + \frac{\partial^2 v}{\partial \mathbf{z}^2} \right) \tag{7}
$$

$$\rho\_{\eta\f} \left( \mu \frac{\partial w}{\partial \mathbf{x}} + \nu \frac{\partial w}{\partial \mathbf{y}} + \nu \frac{\partial w}{\partial \mathbf{z}} \right) = -\frac{\partial p}{\partial \mathbf{z}} + \mu\_{\eta\f} \left( \frac{\partial^2 w}{\partial \mathbf{x}^2} + \frac{\partial^2 w}{\partial \mathbf{y}^2} + \frac{\partial^2 w}{\partial \mathbf{z}^2} \right) \tag{8}$$

$$\left(\rho \mathbf{C}\_{\rm p}\right)\_{\rm nf} \left( u \frac{\partial T\_{\rm nf}}{\partial \mathbf{x}} + v \frac{\partial T\_{\rm nf}}{\partial \mathbf{y}} + w \frac{\partial T\_{\rm nf}}{\partial \mathbf{z}} \right) = K\_{\rm nf} \left( \frac{\partial^2 T\_{\rm nf}}{\partial \mathbf{x}^2} + \frac{\partial^2 T\_{\rm nf}}{\partial \mathbf{y}^2} + \frac{\partial^2 T\_{\rm nf}}{\partial \mathbf{z}^2} \right) \tag{9}$$

For turbulent flow, the continuity, momentum, and energy equations and turbulence kinetic energy (*k*) and its dissipation rate (*ε*) are:

$$\frac{\partial \overline{u}}{\partial \mathbf{x}} + \frac{\partial \overline{v}}{\partial \mathbf{y}} + \frac{\partial \overline{w}}{\partial \mathbf{z}} = \mathbf{0} \tag{10}$$

$$\begin{split} \left( \overline{u} \left( \frac{\partial \overline{u}}{\partial \mathbf{x}} + \overline{v} \, \frac{\partial \overline{u}}{\partial \mathbf{y}} + \overline{w} \, \frac{\partial \overline{u}}{\partial \mathbf{z}} \right) + \left( \frac{\partial}{\partial \mathbf{x}} \left( \overline{u'^2} \right) + \frac{\partial}{\partial \mathbf{y}} \left( \overline{u'v'} \right) + \frac{\partial}{\partial \mathbf{z}} \left( \overline{u'w'} \right) \right) \\ = -\frac{1}{\rho\_{\text{pf}}} \frac{\partial p}{\partial \mathbf{x}} + \chi \, \nabla^2 \overline{u} \end{split} \tag{11}$$

$$\begin{split} \left( \overline{u} \, \frac{\partial \overline{v}}{\partial \mathbf{x}} + \overline{v} \, \frac{\partial \overline{v}}{\partial \mathbf{y}} + \overline{w} \, \frac{\partial \overline{v}}{\partial \mathbf{z}} \right) &+ \left( \frac{\partial}{\partial \mathbf{x}} \left( \overline{u'v'} \right) + \frac{\partial}{\partial \mathbf{y}} \left( \overline{v'^2} \right) + \frac{\partial}{\partial \mathbf{z}} \left( \overline{\overline{v}w'} \right) \right) \\ = -\frac{1}{\rho\_{\text{pf}}} \frac{\partial p}{\partial \mathbf{y}} + \chi \, \nabla^2 \overline{v} \end{split} \tag{12}$$

$$\begin{split} \left( \overline{u} \left( \frac{\partial \overline{w}}{\partial \mathbf{x}} + \overline{v} \, \frac{\partial \overline{w}}{\partial \mathbf{y}} + \overline{w} \, \frac{\partial \overline{w}}{\partial \mathbf{z}} \right) + \left( \frac{\partial}{\partial \mathbf{x}} \left( \overline{u'w'} \right) + \frac{\partial}{\partial \mathbf{y}} \left( \overline{v'w'} \right) + \frac{\partial}{\partial \mathbf{z}} \left( \overline{w'^2} \right) \right) \\ = -\frac{1}{\rho\_{\eta f}} \frac{\partial p}{\partial \mathbf{z}} + \chi \, \nabla^2 \overline{w} \end{split} \tag{13}$$

$$\left(\overline{u}\,\frac{\partial\overline{T}\_{\eta\circ f}}{\partial\mathbf{x}}+\overline{v}\,\frac{\partial\overline{T}\_{\eta\circ f}}{\partial\mathbf{y}}+\overline{w}\,\frac{\partial\overline{T}\_{\eta\circ f}}{\partial\mathbf{z}}\right) = a\,\nabla^2\overline{T}\_{\eta\circ f}+\left(-\frac{\partial}{\partial\mathbf{x}}\left(\overline{u'T'\_{\eta\circ f}}\right)-\frac{\partial}{\partial\mathbf{y}}\left(\overline{v'T'\_{\eta\circ f}}\right)-\frac{\partial}{\partial\mathbf{z}}\left(\overline{w'T'\_{\eta\circ f}}\right)\right) \tag{14}$$

(the fluid-structure interface), where the outputs of one analysis are passed to the other one as a load. There are two different fluid-structure interaction approaches

multiphysics problems are too hard to solve via the analytical approaches. Accordingly, they must be solved either via employing experiments or numerical simulations. The progressed methods and the existence of the reckoned commercial software in both CFD and computational structural mechanics (CSM) have made such numerical simulation possible. There are two dissimilar methods to solve the problems of FSI utilizing such software: the monolithic method and the partitioned method. In one-way coupled FSI, the results (forces) from the fluid analysis at the fluid-structure interface are applied as a load to the structural analysis. The boundary displacement from the structure is not passed back to the fluid analysis. The assumption is that the deformation of the structure is small, having insignificant effect on the fluid flow prediction. This allows the fluid analysis and structure analysis to be run independently. This technique will be used for the theoretical analysis. In the two-way coupled FSI, the structural analysis results are conveyed to the fluid analysis as a load. In a similar way, the fluid analysis results are passed back to the structural analysis as a load. For instance, the fluid pressure at the boundary can be applied as a load on the structural analysis, and the resulted displacement, velocity, or acceleration determined in the structural simulation could be passed on as a load to the fluid analysis. The analysis will carry on till the whole equilibrium (convergence) is attained between the fluid flow solution and the structural solution. The simulated values of average Nusselt number are compared with the experimental results, as shown in **Figure 9**. The computed values agree with the experimental data within 13 and 12% for laminar and turbulent flow, respectively. The simulated friction factors along the tube against the Reynolds no. are also compared to those determined from the experiments. It's noticed that the simulated data are matching with the experimental data within 11% for the average Nusselt no. and 9% for the friction factor, respectively. However, both results have the same behavior, and the differences are with acceptable values. **Figure 10** shows the velocity vector at location of Z = 0.5 m along the test section. A secondary flow is created, and a rotational movement is noted along the tube, and this will enhance

**Figure 11** elucidates the 3D view for the distribution of static temperature along the test section at the midplane (environment and tube) for a nanofluid having a 3%

*Comparison of experimental and predicted Nu for turbulent flow and for Re and f for laminar flow.*

that can be used, depending on the physical nature of the interaction. The

*Nanofluids and Computational Applications in Medicine and Biology*

*DOI: http://dx.doi.org/10.5772/intechopen.88577*

the heat transfer within the tube.

**Figure 9.**

**103**

$$\frac{\partial}{\partial t} \left( \rho\_{\eta f} k \right) + \frac{\partial}{\partial \mathbf{x}\_i} \left( \rho\_{\eta f} k u\_i \right) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + G\_k + G\_b - \rho\_{\eta f} \varepsilon - Y\_M + \mathbb{S}\_k \tag{15}$$

$$\frac{\partial}{\partial t} \left( \rho\_{\eta f} e \right) + \frac{\partial}{\partial \mathbf{x}\_i} \left( \rho\_{\eta f} e u\_i \right) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + \mathbf{C}\_{1\varepsilon} \frac{\varepsilon}{k} (\mathbf{G}\_k + \mathbf{C}\_{\lambda} \mathbf{G}\_b) - \mathbf{C}\_{2\varepsilon} \frac{\varepsilon^2}{k} + \mathbf{S}\_{\varepsilon} \mathbf{C}\_{\lambda} \mathbf{G}\_b \tag{16}$$

Grid independence test is carried out to obtain the most suitable computational grid for which a finer grid provides the similar outputs with the initial one, and the outputs do not vary as grid becomes finer. The checking procedure, either the solution is grid independent or not, is to generate a grid with many cells for comparing the solutions of both models. The tests of grid refinement for Nusselt number explain that the grid size of almost 2 million cells provides a sufficient accuracy and resolution to be adopted as the standard for all cases. The grid independence test performed for typical twist tape with TR = 1.85 configuration is shown in **Figure 8**.

The analysis of fluid-structure interaction (FSI) is an instant of a multiphysics problem, where the interaction between two dissimilar analyses is taken into consideration. This analysis includes conducting a structural analysis taking into consideration the interaction with the corresponding fluid analysis. The interaction between both analyses distinctively occurs at the model solution boundary

**Figure 8.** *The grid-independent solution test for typical twisted tape.*

#### *Nanofluids and Computational Applications in Medicine and Biology DOI: http://dx.doi.org/10.5772/intechopen.88577*

*<sup>u</sup> <sup>∂</sup>Tnf*

*∂ <sup>∂</sup><sup>t</sup> <sup>ρ</sup>nf <sup>k</sup>* � �

*∂ <sup>∂</sup><sup>t</sup> <sup>ρ</sup>nf <sup>ε</sup>* � �

*<sup>∂</sup><sup>x</sup>* <sup>þ</sup> *<sup>v</sup> <sup>∂</sup>Tnf ∂y*

> þ *∂ ∂xi*

þ *∂ ∂xi*

shown in **Figure 8**.

**Figure 8.**

**102**

*The grid-independent solution test for typical twisted tape.*

!

*Applications of Nanobiotechnology*

<sup>þ</sup> *<sup>w</sup> <sup>∂</sup>Tnf ∂z*

*ρnf kui* � �

*ρnf εui* � � <sup>¼</sup> *<sup>α</sup>* <sup>∇</sup><sup>2</sup>

¼ *∂ ∂xj*

¼ *∂ ∂xj* *Tnf* þ �

� �

� �

*<sup>μ</sup>* <sup>þ</sup> *<sup>μ</sup><sup>t</sup> σk* � � *∂k*

*<sup>μ</sup>* <sup>þ</sup> *<sup>μ</sup><sup>t</sup> σk* � � *∂ε*

*∂ <sup>∂</sup><sup>x</sup> <sup>u</sup>*<sup>0</sup> *T*0 *nf* � �

*∂xj*

*∂xj*

Grid independence test is carried out to obtain the most suitable computational grid for which a finer grid provides the similar outputs with the initial one, and the outputs do not vary as grid becomes finer. The checking procedure, either the solution is grid independent or not, is to generate a grid with many cells for comparing the solutions of both models. The tests of grid refinement for Nusselt number explain that the grid size of almost 2 million cells provides a sufficient accuracy and resolution to be adopted as the standard for all cases. The grid independence test performed for typical twist tape with TR = 1.85 configuration is

The analysis of fluid-structure interaction (FSI) is an instant of a multiphysics

problem, where the interaction between two dissimilar analyses is taken into consideration. This analysis includes conducting a structural analysis taking into consideration the interaction with the corresponding fluid analysis. The interaction

between both analyses distinctively occurs at the model solution boundary

þ *C*1*<sup>ε</sup> ε k* � *∂ <sup>∂</sup><sup>y</sup> <sup>v</sup>*<sup>0</sup> *T*0 *nf* � �

� � � �

þ *Gk* þ *Gb* � *ρnf ε* � *YM* þ *Sk* (15)

ð Þ� *Gk* þ *C*3*<sup>ε</sup>Gb C*2*<sup>ε</sup>*

� *∂ <sup>∂</sup><sup>z</sup> <sup>w</sup>*<sup>0</sup> *T*0 *nf*

(14)

*ε*2 *<sup>k</sup>* <sup>þ</sup> *<sup>S</sup><sup>ε</sup>* (16) (the fluid-structure interface), where the outputs of one analysis are passed to the other one as a load. There are two different fluid-structure interaction approaches that can be used, depending on the physical nature of the interaction. The multiphysics problems are too hard to solve via the analytical approaches. Accordingly, they must be solved either via employing experiments or numerical simulations. The progressed methods and the existence of the reckoned commercial software in both CFD and computational structural mechanics (CSM) have made such numerical simulation possible. There are two dissimilar methods to solve the problems of FSI utilizing such software: the monolithic method and the partitioned method. In one-way coupled FSI, the results (forces) from the fluid analysis at the fluid-structure interface are applied as a load to the structural analysis. The boundary displacement from the structure is not passed back to the fluid analysis. The assumption is that the deformation of the structure is small, having insignificant effect on the fluid flow prediction. This allows the fluid analysis and structure analysis to be run independently. This technique will be used for the theoretical analysis. In the two-way coupled FSI, the structural analysis results are conveyed to the fluid analysis as a load. In a similar way, the fluid analysis results are passed back to the structural analysis as a load. For instance, the fluid pressure at the boundary can be applied as a load on the structural analysis, and the resulted displacement, velocity, or acceleration determined in the structural simulation could be passed on as a load to the fluid analysis. The analysis will carry on till the whole equilibrium (convergence) is attained between the fluid flow solution and the structural solution.

The simulated values of average Nusselt number are compared with the experimental results, as shown in **Figure 9**. The computed values agree with the experimental data within 13 and 12% for laminar and turbulent flow, respectively. The simulated friction factors along the tube against the Reynolds no. are also compared to those determined from the experiments. It's noticed that the simulated data are matching with the experimental data within 11% for the average Nusselt no. and 9% for the friction factor, respectively. However, both results have the same behavior, and the differences are with acceptable values. **Figure 10** shows the velocity vector at location of Z = 0.5 m along the test section. A secondary flow is created, and a rotational movement is noted along the tube, and this will enhance the heat transfer within the tube.

**Figure 11** elucidates the 3D view for the distribution of static temperature along the test section at the midplane (environment and tube) for a nanofluid having a 3%

**Figure 9.** *Comparison of experimental and predicted Nu for turbulent flow and for Re and f for laminar flow.*

#### **Figure 10.**

*Velocity vectors in m/s at Re =10,172 for nanofluid (φ = 3%) at Z = 0.5 m.*

**Figure 11.** *Temperature distribution along the test section for φ = 3%.*

volume concentration and a Reynolds no. equal to 5086 for a turbulent flow. **Figure 12** shows the test section deformation calculated using static structuralmechanical is a solution processing model, under the influence of enlarging its value (1.8 103 autoscale). The maximum deformation occurs at the beginning of the tube in all models because of the high temperature concentration at this region. From this figure, one can see that the total deformation decreases as the volume concentration and the mass flow rate increase due to the frequency effect, which is decreased with the increase in nanoparticle mass.

**Figure 13** highlights the 3D view for the twisted tape with the velocity vector along a focused distance of the test section for a nanofluid having a 3% volume concentration. In this figure, one can see that the vector magnitude and direction change at different periods of time due to effect of the twisted tape deformation, which is much higher than that of the tube, resulted from thermal expansion (elongation), fluid pressure effect, and reaction force on it.

**4. Selected topics in medicine and biology**

*Velocity vectors for two periods at Re = 678 and φ = 3% and T2 = 0.8 s.*

*Total deformation from one-way interaction for Re = 10,172 and φ = 5%.*

*Nanofluids and Computational Applications in Medicine and Biology*

*DOI: http://dx.doi.org/10.5772/intechopen.88577*

**Figure 12.**

**Figure 13.**

**105**

During the previous decades, there has been a significant raise in the use of quantitative techniques for studying the physiological regimes. Recent methods for *Nanofluids and Computational Applications in Medicine and Biology DOI: http://dx.doi.org/10.5772/intechopen.88577*

**Figure 13.** *Velocity vectors for two periods at Re = 678 and φ = 3% and T2 = 0.8 s.*
