Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation

Abiodun A. Opanuga, Olasunmbo O. Agboola, Hilary I. Okagbue and Sheila A. Bishop

## Abstract

Entropy generation of fully developed steady, viscous, incompressible couple stress fluid in a vertical micro-porous-channel in the presence of horizontal magnetic field is analysed in this work. The governing equations for the flow are derived, and nondimensionalised and the resulting nonlinear ordinary differential equations are solved via a rapidly convergent technique developed by Zhou. The solution of the velocity and temperature profiles are utilised to obtain the flow irreversibility and Bejan number. The effects of couple stresses, fluid wall interaction parameter (FSIP), effective temperature ratio (ETR), rarefaction and magnetic parameter on the velocity profile, temperature profile, entropy generation and Bejan number are presented and discussed graphically.

Keywords: microchannel, entropy generation, MHD, natural convection, differential transform method (DTM)

## 1. Introduction

In the last decades, the study of microchannel flows has become an important subject for researchers because of the reduction in the size of such devices which increases the dissipated heat per unit area. The effective performance of these devices is dependent on the temperature; as a result a comprehensive knowledge of such flow behaviours is required for accurate prediction of performance during the design process. These microfluidics have characteristic lengths of 1 � 100 μm and are categorised by the dimensionless quantity called the Knudsen number ð Þ Kn . Researchers have shown that microchannel flows are influenced by several parameters of which velocity slip and temperature jump occurring at the solid-fluid interface in small-scale systems are the most important [2–4]. Velocity slip and temperature jump become more significant at higher Knudsen number. The latter boundary conditions are assumed at Knudsen number greater than 0.01 since below this value the classical Navier-Stokes equations is no longer valid.

The influence of velocity slip and temperature jump on microchannel flows has been extensively studied. Khadrawi and Al-Shyyab [5] obtained a close form

solution of the effect of velocity slip and temperature jump on heat and fluid flow for axially moving micro-concentric cylinders. Chen and Tian [6] applied lattice Boltzmann numerical technique with Langmuir model for the velocity slip and temperature jump to investigate fluid flow and heat transfer between two horizontal parallel plates. The study suggested the application of Langmuir-slip model as an alternative for the Maxwell-slip model. Earlier on, Larrode et al. [7] in his work, slip-flow heat transfer in circular tubes, has stated the significance of fluid-wall interaction. Moreover, Adesanya [8] has analytically studied the effects of velocity slip and temperature jump on the unsteady free convective flow of heatgenerating/heat-absorbing fluid with buoyancy force. The study concluded that an increase in the slip parameter enhanced fluid motion, while an increase in the temperature jump parameter increased fluid temperature. Aziz and Niedbalski [9] applied finite difference technique to investigate thermally developing microtube gas flow with respect to both radial and axial coordinates. The result indicated that increase in Knudsen number reduced local Nusselt number (Nu). Several investigations regarding microchannel flows have also been carried out by Jha and collaborators [10–12].

From the perspective of energy management, it has been established that thermal processes such as the microscale fluid flow and heat transfer modelling are irreversible. This implies that entropy generation which destroys the available energy leading to inefficiency of thermal designs exists. Therefore, the aim of this research endeavour is the minimisation of irreversibility associated with microchannel flow of couple stress fluid by applying the robust approach proposed by Bejan [13] and applied by numerous researchers such as Adesanya and collaborators [14–16], Adesanya and Makinde [17–18], Ajibade and Jha [19], Eegunjobi and Makinde [20–21], Das and Jana [22] and more recently Opanuga and his collaborators [23–26].

us ¼ � <sup>2</sup> � <sup>σ</sup><sup>v</sup> σv λ du dy0 � � � � y0 ¼h

DOI: http://dx.doi.org/10.5772/intechopen.81123

ρν<sup>∘</sup> du0 dy<sup>0</sup> ¼ � dp

Scheme of the vertical microchannel.

Figure 1.

ρcpv<sup>0</sup>

The boundary conditions are

<sup>u</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup> � <sup>f</sup> <sup>v</sup> f v λ du<sup>0</sup> dy<sup>0</sup> , d2 u0

<sup>u</sup><sup>0</sup> ¼ � <sup>2</sup> � <sup>f</sup> <sup>v</sup> f v λ du<sup>0</sup> dy<sup>0</sup> , d2 u0

> <sup>h</sup> , u <sup>¼</sup> <sup>u</sup><sup>0</sup> ν∘

<sup>y</sup> <sup>¼</sup> <sup>y</sup><sup>0</sup>

Pr <sup>¼</sup> νρcp

<sup>Ω</sup> <sup>¼</sup> <sup>T</sup><sup>1</sup> � <sup>T</sup><sup>0</sup> T<sup>0</sup>

<sup>β</sup><sup>t</sup> <sup>¼</sup> <sup>2</sup> � <sup>f</sup> <sup>v</sup> f v

77

dT0 dy<sup>0</sup> <sup>¼</sup> <sup>k</sup> <sup>d</sup><sup>2</sup>

The dimensionless parameters for this flow are

, <sup>θ</sup> <sup>¼</sup> <sup>T</sup><sup>0</sup> � <sup>T</sup><sup>0</sup> T<sup>1</sup> � T<sup>0</sup>

k Tð Þ <sup>1</sup> � T<sup>0</sup>

0h2

, H<sup>2</sup> <sup>¼</sup> <sup>σ</sup>B<sup>2</sup>

1 Pr, <sup>γ</sup><sup>s</sup> <sup>¼</sup> cp cv

<sup>k</sup> , Br <sup>¼</sup> μν<sup>0</sup>

2γs γ<sup>s</sup> þ 1 , Ts � Tw ¼ � <sup>2</sup> � <sup>σ</sup><sup>t</sup>

d4 u0 dy0<sup>4</sup> <sup>þ</sup> <sup>ρ</sup>gβ<sup>∗</sup>

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation

du0 dy0 � �<sup>2</sup>

þ η

2 � ft ft

ft

, G <sup>¼</sup> �h<sup>2</sup> μν<sup>0</sup>

<sup>k</sup> , Bi<sup>2</sup> <sup>¼</sup> <sup>γ</sup>2<sup>h</sup>

<sup>μ</sup> , <sup>ξ</sup>ð Þ¼ <sup>T</sup><sup>2</sup> � <sup>T</sup><sup>0</sup> <sup>T</sup><sup>1</sup> � <sup>T</sup>0, <sup>β</sup><sup>v</sup> <sup>¼</sup> <sup>2</sup> � <sup>f</sup> <sup>v</sup>

<sup>h</sup> , <sup>ψ</sup> <sup>¼</sup> <sup>β</sup><sup>t</sup> βv

While the governing equations for the flow are stated as [10]

d2 u0 dy0<sup>2</sup> � <sup>η</sup>

T0 dy0<sup>2</sup> <sup>þ</sup> <sup>μ</sup>

dy0<sup>2</sup> <sup>¼</sup> <sup>0</sup>, T<sup>0</sup> <sup>¼</sup> <sup>T</sup><sup>2</sup> <sup>þ</sup>

dy0<sup>2</sup> <sup>¼</sup> <sup>0</sup>, T<sup>0</sup> <sup>¼</sup> <sup>T</sup><sup>1</sup> � <sup>2</sup> � ft

, Re <sup>¼</sup> <sup>ρ</sup>v0<sup>h</sup> μ

, Bi<sup>1</sup> <sup>¼</sup> <sup>γ</sup>1<sup>h</sup>

, Kn <sup>¼</sup> <sup>λ</sup>

dx<sup>0</sup> <sup>þ</sup> <sup>μ</sup>

σt

d2 u0 dy0<sup>2</sup> !<sup>2</sup>

> 2γ γ þ 1

λ Pr dT<sup>0</sup> dy , y <sup>¼</sup> <sup>0</sup>

> λ Pr dT<sup>0</sup> dy , y <sup>¼</sup> <sup>h</sup>

dp dx , a<sup>2</sup> <sup>¼</sup> <sup>μ</sup>

<sup>k</sup> , <sup>ν</sup> <sup>¼</sup> <sup>μ</sup> ρ ,

, Ns <sup>¼</sup> SGT0h<sup>2</sup>

h2 η ,

f v ,

k Tð Þ <sup>1</sup> � T<sup>0</sup>

2γ γ þ 1

2γ γ þ 1

λ Pr dT dy0 � � � � y0 ¼h

ð Þ� <sup>T</sup> � <sup>T</sup><sup>0</sup> <sup>σ</sup>B<sup>2</sup>

<sup>þ</sup> <sup>σ</sup>B<sup>2</sup>

(1)

(4)

(5)

<sup>0</sup>u<sup>0</sup> (2)

<sup>0</sup>u0<sup>2</sup> (3)

In this work, an efficient technique introduced by Zhou has been employed to construct the solutions of the velocity and temperature profiles. Due to the accuracy of this technique in handling numerous linear and nonlinear models of both ordinary and partial equations, it has gained wide applications by investigators over the last decades. Arikoglu and Ozkol [27] applied it to obtain the solution of difference equations. Biazar and Eslami [28] solved quadratic Riccati differential equation using this method. Agboola et al. [29–30] applied it to third-order ordinary differential equations and natural frequencies of a cantilever beam. Solutions of Volterra integral equation via this technique was obtained by Odibat [31], while Opanuga et al. [32] compared the method with Adomian decomposition method to find the solution of multipoint boundary-value problem and more recently in couple stress fluid model [33].

## 2. Problem formulation

A fully developed laminar, viscous, incompressible and electrically conducting couple stress fluid in a vertical parallel microchannel of width h is considered. The x-axis is such that it is vertically upward along the plates while the y-axis is taken normal to it. There is an asymmetric heating of the plates such that the hotter plate ð Þ y ¼ 0 is maintained at temperature T1, while the cooler plate ð Þ y ¼ h is at temperature T2, ð Þ T1>T<sup>2</sup> and T<sup>0</sup> is the reference frame. Velocity slip and temperature jump are incorporated. Furthermore, induced magnetic field effect arising due to the motion of an electrically conducting fluid is taken into consideration (Figure 1). Following Cheng and Weng [34], the velocity slip and temperature jump are given as

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation DOI: http://dx.doi.org/10.5772/intechopen.81123

#### Figure 1. Scheme of the vertical microchannel.

solution of the effect of velocity slip and temperature jump on heat and fluid flow for axially moving micro-concentric cylinders. Chen and Tian [6] applied lattice Boltzmann numerical technique with Langmuir model for the velocity slip and temperature jump to investigate fluid flow and heat transfer between two horizontal parallel plates. The study suggested the application of Langmuir-slip model as an alternative for the Maxwell-slip model. Earlier on, Larrode et al. [7] in his work, slip-flow heat transfer in circular tubes, has stated the significance of fluid-wall interaction. Moreover, Adesanya [8] has analytically studied the effects of velocity

generating/heat-absorbing fluid with buoyancy force. The study concluded that an increase in the slip parameter enhanced fluid motion, while an increase in the temperature jump parameter increased fluid temperature. Aziz and Niedbalski [9] applied finite difference technique to investigate thermally developing microtube gas flow with respect to both radial and axial coordinates. The result indicated that increase in Knudsen number reduced local Nusselt number (Nu). Several investigations regarding microchannel flows have also been carried out by Jha and collab-

From the perspective of energy management, it has been established that thermal processes such as the microscale fluid flow and heat transfer modelling are irreversible. This implies that entropy generation which destroys the available energy leading to inefficiency of thermal designs exists. Therefore, the aim of this

microchannel flow of couple stress fluid by applying the robust approach proposed

In this work, an efficient technique introduced by Zhou has been employed to construct the solutions of the velocity and temperature profiles. Due to the accuracy

ordinary and partial equations, it has gained wide applications by investigators over the last decades. Arikoglu and Ozkol [27] applied it to obtain the solution of difference equations. Biazar and Eslami [28] solved quadratic Riccati differential equation using this method. Agboola et al. [29–30] applied it to third-order ordinary differential equations and natural frequencies of a cantilever beam. Solutions of Volterra integral equation via this technique was obtained by Odibat [31], while Opanuga et al. [32] compared the method with Adomian decomposition method to find the solution of multipoint boundary-value problem and more recently in couple stress

A fully developed laminar, viscous, incompressible and electrically conducting couple stress fluid in a vertical parallel microchannel of width h is considered. The x-axis is such that it is vertically upward along the plates while the y-axis is taken normal to it. There is an asymmetric heating of the plates such that the hotter plate ð Þ y ¼ 0 is maintained at temperature T1, while the cooler plate ð Þ y ¼ h is at temperature T2, ð Þ T1>T<sup>2</sup> and T<sup>0</sup> is the reference frame. Velocity slip and temperature jump are incorporated. Furthermore, induced magnetic field effect arising due to the motion of an electrically conducting fluid is taken into consideration (Figure 1). Following Cheng and Weng [34], the velocity slip and temperature jump are

research endeavour is the minimisation of irreversibility associated with

by Bejan [13] and applied by numerous researchers such as Adesanya and collaborators [14–16], Adesanya and Makinde [17–18], Ajibade and Jha [19], Eegunjobi and Makinde [20–21], Das and Jana [22] and more recently Opanuga and

of this technique in handling numerous linear and nonlinear models of both

slip and temperature jump on the unsteady free convective flow of heat-

Pattern Formation and Stability in Magnetic Colloids

orators [10–12].

his collaborators [23–26].

fluid model [33].

given as

76

2. Problem formulation

$$\mu\_{\rm s} = -\frac{2 - \sigma\_v}{\sigma\_v} \lambda \frac{du}{d\eta'}\Big|\_{\rm y'=h}, T\_s - T\_w = -\frac{2 - \sigma\_t}{\sigma\_t} \frac{2\gamma}{\gamma + 1} \frac{\lambda}{\Pr} \frac{dT}{d\eta'}\Big|\_{\rm y'=h} \tag{1}$$

While the governing equations for the flow are stated as [10]

$$
\rho \nu\_\* \frac{du'}{d\mathbf{y}'} = -\frac{dp}{d\mathbf{x}'} + \mu \frac{d^2 u'}{d\mathbf{y}'^2} - \eta \frac{d^4 u'}{d\mathbf{y}'^4} + \rho \mathbf{g} \boldsymbol{\beta}^\*(T - T\_0) - \sigma \mathbf{B}\_0^2 u' \tag{2}
$$

$$
\rho c\_p v\_0 \frac{dT'}{dy'} = k \frac{d^2 T'}{dy'^2} + \mu \left(\frac{du'}{dy'}\right)^2 + \eta \left(\frac{d^2 u'}{dy'^2}\right)^2 + \sigma B\_0^2 u'^2 \tag{3}
$$

The boundary conditions are

$$\begin{aligned} u' &= \frac{2 - f\_v}{f\_v} \lambda \frac{du'}{dy'}, \frac{d^2 u'}{dy'^2} = 0, \, T' = T\_2 + \frac{2 - f\_t}{f\_t} \frac{2\gamma}{\gamma + 1} \frac{2\gamma}{\text{Pr}} \frac{dT'}{dy'}, y = 0\\ u' &= -\frac{2 - f\_v}{f\_v} \lambda \frac{du'}{dy'}, \frac{d^2 u'}{dy'^2} = 0, \, T' = T\_1 - \frac{2 - f\_t}{f\_t} \frac{2\gamma}{\gamma + 1} \frac{2\gamma}{\text{Pr}} \frac{dT'}{dy'}, y = h \end{aligned} \tag{4}$$

The dimensionless parameters for this flow are

$$\begin{aligned} \mathbf{y} &= \frac{\mathbf{y}'}{h}, u = \frac{u'}{\nu\_\*}, \theta = \frac{T' - T\_0}{T\_1 - T\_0}, \text{Re} = \frac{\rho v\_0 h}{\mu}, \mathbf{G} = \frac{-h^2}{\mu \nu\_0} \frac{dp}{dx}, a^2 = \mu \frac{h^2}{\eta}, \\ \mathbf{Pr} &= \frac{\nu \rho c\_p}{k}, Br = \frac{\mu \nu\_0}{k(T\_1 - T\_0)}, Bi\_1 = \frac{\gamma\_1 h}{k}, Bi\_2 = \frac{\gamma\_2 h}{k}, \nu = \frac{\mu}{\rho}, \\ \boldsymbol{\Omega} &= \frac{T\_1 - T\_0}{T\_0}, H^2 = \frac{\sigma B\_0^2 h^2}{\mu}, \xi (T\_2 - T\_0) = T\_1 - T\_0, \beta\_v = \frac{2 - f\_v}{f\_v}, \\ \boldsymbol{\beta}\_t &= \frac{2 - f\_v}{f\_v} \frac{2 \gamma\_s}{\gamma\_s + 1} \frac{1}{\mathbf{Pr}}, \boldsymbol{\gamma}\_s = \frac{c\_p}{c\_v}, \mathbf{Kn} = \frac{\lambda}{h}, \boldsymbol{\Psi} = \frac{\beta\_t}{\beta\_v}, \mathbf{Ns} = \frac{\mathbf{S}\_G T\_0 h^2}{k(T\_1 - T\_0)} \end{aligned} \tag{5}$$

Using (5) in Eqs. (2)–(4) yields the boundary-value problems:

$$\operatorname{Re}\frac{du}{dy} = G + \frac{d^2u}{dy^2} - \frac{1}{a^2}\frac{d^4u}{dy^4} - H^2u + Gr\theta,\tag{6}$$

Original function Transformed function f yð Þ¼ u yð Þ� w yð Þ F kð Þ¼ U kð Þ� W kð Þ

dyn F kð Þ¼ ð Þ <sup>k</sup>þ<sup>n</sup> !

dy <sup>2</sup> F kð Þ¼ <sup>∑</sup><sup>k</sup>

<sup>2</sup> F kð Þ¼ <sup>∑</sup><sup>k</sup>

Operations and properties of differential transform method.

f yð Þ¼ <sup>u</sup><sup>2</sup> F kð Þ¼ <sup>∑</sup><sup>k</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81123

<sup>k</sup>! U kð Þ þ n

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation

<sup>r</sup>¼<sup>0</sup>U rð ÞU kð Þ � r

A. Couple stress parameter vs. fluid velocity. B. Couple stress parameter vs. fluid temperature. C. Couple stress

parameter vs. entropy generation. D. Couple stress parameter vs. Bejan number.

<sup>r</sup>¼<sup>0</sup>ð Þ <sup>r</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>k</sup> � <sup>r</sup> <sup>þ</sup> <sup>1</sup> U rð Þ <sup>þ</sup> <sup>1</sup> U kð Þ � <sup>r</sup> <sup>þ</sup> <sup>1</sup>

<sup>r</sup>¼<sup>0</sup>ð Þ <sup>r</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>r</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>k</sup> � <sup>r</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>k</sup> � <sup>r</sup> <sup>þ</sup> <sup>2</sup> U rð Þ <sup>þ</sup> <sup>2</sup> U kð Þ � <sup>r</sup> <sup>þ</sup> <sup>2</sup>

f yð Þ¼ <sup>d</sup><sup>n</sup> u yð Þ

f yð Þ¼ du yð Þ

u yð Þ dy<sup>2</sup>

f yð Þ¼ <sup>d</sup><sup>2</sup>

Table 1.

Figure 2.

79

$$\frac{d^2\theta}{dy^2} = \text{Re}\, p\_r \frac{d\theta}{dy} - Br \left\{ \left(\frac{du}{dy}\right)^2 + \frac{1}{a^2} \left(\frac{d^2u}{dy^2}\right)^2 + H^2u^2 \right\},\tag{7}$$

$$u = \beta\_v \text{Kn}\frac{du}{dy}, \frac{d^2 u}{dy^2} = 0, \theta = \xi + \beta\_v \text{Kn}\mu \frac{d\theta}{dy}, y = \mathbf{0} \tag{8}$$

$$u = -\beta\_v K n \frac{du}{dy}, \frac{d^2 u}{dy^2} = 0, \theta = 1 - \beta\_v K n \nu \frac{d\theta}{dy}, y = 1$$

## 3. Method of solution

DTM is applied in this work to obtain the solution of the velocity and temperature profiles. The results are used to calculate the entropy generation and irreversibility ratio.

### 3.1 Differential transformation method (DTM)

Consider a function, f xð Þ. The differential transformation of the function f xð Þ is defined as

$$F(k) = \frac{1}{k!} \left[ \frac{d^k f(\mathbf{x})}{d\mathbf{x}^k} \right]\_{\mathbf{x} = \mathbf{x}\_0} \tag{9}$$

where f xð Þ is the given function and F kð Þ is the transformed function which is also known as the spectrum of f xð Þ. The inverse differential transformation of F kð Þ is given by

$$f(\mathbf{x}) = \sum\_{k=0}^{\infty} \mathbf{x}^k F(k) \tag{10}$$

In practise, the function stated in Eq. (10) is usually represented by a finite series of the form:

$$f(\mathbf{x}) = \sum\_{k=0}^{n} \mathbf{x}^{k} F(k) \tag{11}$$

where n is the size of the series (Arikoglu and Ozkol [27], Biazar and Eslami [28]).

To apply DTM to the problem in view, the basic properties of DTM, which are outlined in Table 1, are invoked in Eqs. (6)–(8). Doing this, one obtains the following recurrence relations:

$$F(k+4) = \frac{1}{a^2(k+4)!} \left[ (k+1)(k+2)F(k+2) - \text{Re}(k+1)F(k+1) - H^2(F(k)) + G \right] \tag{12}$$

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation DOI: http://dx.doi.org/10.5772/intechopen.81123


Table 1.

Using (5) in Eqs. (2)–(4) yields the boundary-value problems:

dy � Br du

dy � �<sup>2</sup>

þ 1 a2

dy<sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>θ</sup> <sup>¼</sup> <sup>ξ</sup> <sup>þ</sup> <sup>β</sup>vKn<sup>ψ</sup>

DTM is applied in this work to obtain the solution of the velocity and temperature profiles. The results are used to calculate the entropy generation and irrevers-

Consider a function, f xð Þ. The differential transformation of the function f xð Þ is

dk f xð Þ dx<sup>k</sup> " #

where f xð Þ is the given function and F kð Þ is the transformed function which is also known as the spectrum of f xð Þ. The inverse differential transformation of F kð Þ

> f xð Þ¼ ∑ ∞ k¼0 xk

> f xð Þ¼ ∑ n k¼0 xk

In practise, the function stated in Eq. (10) is usually represented by a finite

where n is the size of the series (Arikoglu and Ozkol [27], Biazar and Eslami [28]). To apply DTM to the problem in view, the basic properties of DTM, which are outlined in Table 1, are invoked in Eqs. (6)–(8). Doing this, one obtains the fol-

<sup>a</sup><sup>2</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>4</sup> ! ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>k</sup> <sup>þ</sup> <sup>2</sup> F kð Þ� <sup>þ</sup> <sup>2</sup> Reð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> F kð Þ� <sup>þ</sup> <sup>1</sup> <sup>H</sup><sup>2</sup>

ð Þþ F kð Þ <sup>G</sup> � �

x¼x<sup>0</sup>

F kð Þ (10)

F kð Þ (11)

F kð Þ¼ <sup>1</sup> k!

dy<sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>θ</sup> <sup>¼</sup> <sup>1</sup> � <sup>β</sup>vKn<sup>ψ</sup>

d2 u dy<sup>2</sup> !<sup>2</sup>

<sup>þ</sup> <sup>H</sup><sup>2</sup> u2

dθ dy , y <sup>¼</sup> <sup>0</sup>

> dθ dy , y <sup>¼</sup> <sup>1</sup>

d2 u dy<sup>2</sup> � <sup>1</sup> a2 d4 u dy<sup>4</sup> � <sup>H</sup><sup>2</sup>

8 < : u þ Grθ, (6)

;, (7)

(8)

(9)

(12)

9 =

Re du

Pattern Formation and Stability in Magnetic Colloids

<sup>u</sup> <sup>¼</sup> <sup>β</sup>vKn du

<sup>u</sup> ¼ �βvKn du

3.1 Differential transformation method (DTM)

d2 θ dy<sup>2</sup> <sup>¼</sup> Re pr

3. Method of solution

ibility ratio.

defined as

is given by

series of the form:

F kð Þ¼ þ 4

78

lowing recurrence relations:

1

dy <sup>¼</sup> <sup>G</sup> <sup>þ</sup>

dθ

dy , d2 u

> dy , d2 u

Operations and properties of differential transform method.

#### Figure 2.

A. Couple stress parameter vs. fluid velocity. B. Couple stress parameter vs. fluid temperature. C. Couple stress parameter vs. entropy generation. D. Couple stress parameter vs. Bejan number.

Pattern Formation and Stability in Magnetic Colloids

r¼0

$$
\Theta(k+2) = \frac{1}{(k+2)!} \left[ \mathrm{RePr}(k+1)\Theta(k+1) - \mathrm{Br} \left( \sum\_{r=0}^{k} (r+1)F(r+1) \right) \right]
$$

$$
$$

$$
$$

where F kð Þ and Θð Þk are the transformed functions of f yð Þ and θð Þy , respectively. These are given by

$$f(\mathbf{y}) = \sum\_{k=0}^{\infty} \mathbf{y}^k F(k), \theta(\mathbf{y}) = \sum\_{k=0}^{\infty} \mathbf{y}^k \Theta(k). \tag{14}$$

Using the recursive relations of Eqs. (12) and (13), we can obtain the differential coefficients, Fð Þ 4 , Fð Þ5 , …, F nð Þ and Θð Þ 4 , Θð Þ5 , Θð Þ n by setting k ¼ 0, 1, 2, …. The values of F jð Þ for j ¼ 4, 5, … and Θð Þj for j ¼ 2, 3, … can now be evaluated in terms of Fð Þ 0 , Fð Þ1 , Fð Þ2 , Fð Þ3 , Θð Þ 0 and Θð Þ1 . For convenience, the values of Fð Þ 0 , Fð Þ1 ,

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation

The values of F kð Þ and Θð Þk for k ¼ 0, 1, … are now substituted back into

F kð Þ, θð Þ¼ y ∑

We next invoke the transformed form of boundary condition (8) on (16) to

A. Effective temperature ratio vs. fluid velocity. B. Effective temperature ratio vs. fluid temperature. C. Effective

temperature ratio vs. entropy generation. D. Effective temperature ratio vs. Bejan number.

determine the values of all the unknown coefficients stated in (15). Coding

<sup>F</sup>ð Þ¼ <sup>2</sup> <sup>a</sup>3, <sup>F</sup>ð Þ¼ <sup>3</sup> <sup>a</sup>4,

n k¼0 (15)

<sup>y</sup>kΘð Þ<sup>k</sup> : (16)

Fð Þ2 , Fð Þ3 , Θð Þ 0 and Θð Þ1 are set as unknowns such as

DOI: http://dx.doi.org/10.5772/intechopen.81123

Eq. (14) to obtain the series solutions in the form:

Figure 4.

81

f yð Þ¼ ∑ n k¼0 yk

Fð Þ¼ 0 a1, Fð Þ¼ 1 a2, Θð Þ¼ 0 b1, Θð Þ¼ 1 b2:

#### Figure 3.

A. Fluid wall interaction parameter vs. fluid velocity. B. Fluid wall interaction parameter vs. fluid temperature. C. Fluid wall interaction parameter vs. entropy generation. D. Fluid wall interaction vs. Bejan number.

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation DOI: http://dx.doi.org/10.5772/intechopen.81123

Using the recursive relations of Eqs. (12) and (13), we can obtain the differential coefficients, Fð Þ 4 , Fð Þ5 , …, F nð Þ and Θð Þ 4 , Θð Þ5 , Θð Þ n by setting k ¼ 0, 1, 2, …. The values of F jð Þ for j ¼ 4, 5, … and Θð Þj for j ¼ 2, 3, … can now be evaluated in terms of Fð Þ 0 , Fð Þ1 , Fð Þ2 , Fð Þ3 , Θð Þ 0 and Θð Þ1 . For convenience, the values of Fð Þ 0 , Fð Þ1 , Fð Þ2 , Fð Þ3 , Θð Þ 0 and Θð Þ1 are set as unknowns such as

$$\begin{aligned} F(\mathbf{0}) &= a\_{\mathbf{1}}, \quad F(\mathbf{1}) = a\_{\mathbf{2}}, \quad F(\mathbf{2}) = a\_{\mathbf{3}}, \; F(\mathbf{3}) = a\_{\mathbf{4}}, \\ \Theta(\mathbf{0}) &= b\_{\mathbf{1}}, \quad \Theta(\mathbf{1}) = b\_{\mathbf{2}}.\end{aligned} \tag{15}$$

The values of F kð Þ and Θð Þk for k ¼ 0, 1, … are now substituted back into Eq. (14) to obtain the series solutions in the form:

$$f(\boldsymbol{\jmath}) = \sum\_{k=0}^{n} \boldsymbol{\jmath}^{k} F(k), \ \boldsymbol{\Theta}(\boldsymbol{\jmath}) = \sum\_{k=0}^{n} \boldsymbol{\jmath}^{k} \boldsymbol{\Theta}(k). \tag{16}$$

We next invoke the transformed form of boundary condition (8) on (16) to determine the values of all the unknown coefficients stated in (15). Coding

Figure 4.

A. Effective temperature ratio vs. fluid velocity. B. Effective temperature ratio vs. fluid temperature. C. Effective temperature ratio vs. entropy generation. D. Effective temperature ratio vs. Bejan number.

<sup>Θ</sup>ð Þ¼ <sup>k</sup> <sup>þ</sup> <sup>2</sup> <sup>1</sup>

These are given by

Figure 3.

number.

80

ð Þ k þ 2 !

Pattern Formation and Stability in Magnetic Colloids

�BrH<sup>2</sup> <sup>∑</sup>

� Br <sup>a</sup><sup>2</sup> <sup>∑</sup> k r¼0

�

k r¼0

f yð Þ¼ ∑ ∞ k¼0 yk

F rð ÞF kð Þ � r � ��

RePrð Þ k þ 1 Θð Þ� k þ 1 Br

where F kð Þ and Θð Þk are the transformed functions of f yð Þ and θð Þy , respectively.

F kð Þ, θð Þ¼ y ∑

A. Fluid wall interaction parameter vs. fluid velocity. B. Fluid wall interaction parameter vs. fluid temperature. C. Fluid wall interaction parameter vs. entropy generation. D. Fluid wall interaction vs. Bejan

∞ k¼0 yk

∑ k r¼0

0

BB@

ð Þ r þ 1 ð Þ r þ 2 F rð Þ þ 2 ð Þ k � r þ 1 ð Þ k � r þ 2 F kð Þ � r þ 2 � �

ð Þ r þ 1 F rð Þ þ 1

ð Þ k � r þ 1 F kð Þ � r þ 1

Θð Þk : (14)

(13)

Figure 5.

A. Rarefaction vs. fluid velocity. B. Rarefaction vs. fluid temperature. C. Rarefaction vs. entropy generation. D. Rarefaction vs. Bejan number.

Eqs. (12)–(15) in symbolic Maple software yields the approximate solution. The results are presented in Figures 2–6.

To verify the accuracy of the results, the exact solution of the velocity profile (6) subject to the boundary conditions (8) at βvKn ¼ 0:05, Gr ¼ 0, a ¼ 1, Re ¼ 0:1, H ¼ 1 is obtained as.

$$\begin{aligned} u(y) &= (-3.063639e^{(0.866146y)} + 2.057901^{\circ}\text{Cos}\left[0.470472\,\text{y}\right] + \\ \mathbf{1.1}\,^{(1.73291^{\circ}y)}\text{Cos}\left[0.528272\,\text{y}\right] + \mathbf{1.808594}\,\text{Sin}\left[0.4704729\,\text{y}\right] - \\ \mathbf{0.093418}\,\text{e}\left(1.732291^{\circ}y\right)\text{Sin}\left[0.528272\,\text{y}\right]) \end{aligned} \tag{17}$$

SG <sup>¼</sup> <sup>k</sup> T2 0

T<sup>0</sup>

Ns <sup>¼</sup> <sup>d</sup><sup>θ</sup> dy � �<sup>2</sup>

þ Br Ω

next term <sup>μ</sup>

and <sup>σ</sup>B<sup>2</sup> 0 <sup>T</sup><sup>0</sup> <sup>u</sup><sup>∗</sup><sup>2</sup>

83

Figure 6.

dT<sup>∗</sup> dy<sup>∗</sup> � �<sup>2</sup>

The first term in Eq. (18) <sup>k</sup>

du<sup>∗</sup> dy<sup>∗</sup> þ μ T<sup>0</sup>

vs. entropy generation. D. Hartmann Number vs. Bejan number.

T2 0 dT<sup>∗</sup> dy<sup>∗</sup> � �<sup>2</sup>

du dy � �<sup>2</sup> ! <sup>þ</sup>

du<sup>∗</sup> dy<sup>∗</sup> � �<sup>2</sup> !

are couple stress and magnetic entropy generation.

þ 1 a2

A. Hartmann Number vs. fluid velocity. B. Hartmann Number vs. fluid temperature. C. Hartmann Number

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation

DOI: http://dx.doi.org/10.5772/intechopen.81123

� �<sup>2</sup> � � is entropy generation due to viscous dissipation, <sup>1</sup>

Using (4) in Eq. (18), the dimensionless form of entropy generation is written as

@

Br Ωa<sup>2</sup> d2 u dy<sup>2</sup>

@

d2 u dy<sup>2</sup>

!<sup>2</sup> 0

1 A þ

!<sup>2</sup> 0

1 <sup>A</sup> <sup>þ</sup> <sup>σ</sup>B<sup>2</sup> 0 T<sup>0</sup> u∗2

is entropy generation due to heat transfer; the

BrM<sup>2</sup> Ω u2 : (18)

d2 u dy<sup>2</sup> � �<sup>2</sup> � �

a2

, (19)

The above solution is compared with DTM solution as displayed in Table 2.

### 3.2 Analysis of entropy generation

The local entropy generation for the flow is given as Bejan [13]:

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation DOI: http://dx.doi.org/10.5772/intechopen.81123

Figure 6.

Eqs. (12)–(15) in symbolic Maple software yields the approximate solution. The

A. Rarefaction vs. fluid velocity. B. Rarefaction vs. fluid temperature. C. Rarefaction vs. entropy generation. D.

<sup>y</sup> <sup>Þ</sup>

The local entropy generation for the flow is given as Bejan [13]:

The above solution is compared with DTM solution as displayed in Table 2.

To verify the accuracy of the results, the exact solution of the velocity profile (6) subject to the boundary conditions (8) at βvKn ¼ 0:05, Gr ¼ 0, a ¼ 1, Re ¼ 0:1,

<sup>e</sup>ð Þ <sup>0</sup>:866146'<sup>y</sup> <sup>þ</sup> <sup>2</sup>:057901' Cos <sup>0</sup>:470472'

<sup>y</sup> <sup>þ</sup> <sup>1</sup>:808594' Sin <sup>0</sup>:4704729'

<sup>y</sup> <sup>þ</sup>

(17)

<sup>y</sup> �

results are presented in Figures 2–6.

u yð Þ ¼ ð�3:063639'

3.2 Analysis of entropy generation

eð Þ <sup>1</sup>:732291'<sup>y</sup> Cos 0:528272'

Pattern Formation and Stability in Magnetic Colloids

eð Þ <sup>1</sup>:732291'<sup>y</sup> Sin 0:528272'

H ¼ 1 is obtained as.

Rarefaction vs. Bejan number.

Figure 5.

1: '

82

0:093418'

A. Hartmann Number vs. fluid velocity. B. Hartmann Number vs. fluid temperature. C. Hartmann Number vs. entropy generation. D. Hartmann Number vs. Bejan number.

$$S\_G = \frac{k}{T\_0^2} \left(\frac{dT^\*}{d\mathbf{y}^\*}\right)^2 + \frac{\mu}{T\_0} \left(\left(\frac{du^\*}{d\mathbf{y}^\*}\right)^2\right) + \frac{1}{a^2} \left(\left(\frac{d^2u}{d\mathbf{y}^2}\right)^2\right) + \frac{\sigma B\_0^2}{T\_0} u^{\*^2}.\tag{18}$$

The first term in Eq. (18) <sup>k</sup> T2 0 dT<sup>∗</sup> dy<sup>∗</sup> � �<sup>2</sup> is entropy generation due to heat transfer; the next term <sup>μ</sup> T<sup>0</sup> du<sup>∗</sup> dy<sup>∗</sup> � �<sup>2</sup> � � is entropy generation due to viscous dissipation, <sup>1</sup> a2 d2 u dy<sup>2</sup> � �<sup>2</sup> � � and <sup>σ</sup>B<sup>2</sup> 0 <sup>T</sup><sup>0</sup> <sup>u</sup><sup>∗</sup><sup>2</sup> are couple stress and magnetic entropy generation.

Using (4) in Eq. (18), the dimensionless form of entropy generation is written as

$$\mathcal{N}s = \left(\frac{d\theta}{dy}\right)^2 + \frac{Br}{\Omega}\left(\left(\frac{du}{dy}\right)^2\right) + \frac{Br}{\Omega a^2}\left(\left(\frac{d^2u}{dy^2}\right)^2\right) + \frac{Br\mathcal{M}^2}{\Omega}u^2,\tag{19}$$


#### Table 2.

Comparison of the exact solution with the values of velocity (u).

where ð Þ SG; Ns are the dimensional and dimensionless entropy generation rates. The ratio of heat transfer entropy generation ð Þ N<sup>1</sup> to fluid friction entropy generation ð Þ N<sup>2</sup> is represented as

$$
\Phi = \frac{N\_2}{N\_1} \tag{20}
$$

4.1 Influence of couple stress parameter

DOI: http://dx.doi.org/10.5772/intechopen.81123

increases.

Figure 2A illustrates the influence of couple stress parameter on fluid velocity. The plot shows a significant reduction in fluid velocity as the values of couple stress parameter increase. This observation is due to the increase in the dynamic viscosity of the fluid. In Figure 2B, it is illustrated that fluid temperature drops as couple

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation

generation decreases at the microchannel walls as couple stress parameter increases. However, the effect is opposite near the middle of the microchannel. The same scenario is observed in Figure 2D; it is shown that Bejan number reduces in value at microchannel right wall which is an indication that fluid friction irreversibility is the major contributor to entropy generation as couple stress parameter

Figure 3A presents the effect of fluid-structure interaction parameter on fluid velocity. It is noted that FSIP does not have any significant effect on the slip velocity at the walls as well as the microchannel. However, Figure 3B reveals that fluid temperature is enhanced as FSIP increases. Response to the enhancement in fluid temperature in Figure 3B is the rise in fluid entropy generation at the hotter wall of the microchannel, while entropy generation is reduced at the cooler wall as depicted in Figure 3C. In Figure 3D, it is noticed that Bejan number increases at the hotter wall of the microchannel, while it reduces at the cooler wall. The implication of the latter is that heat transfer irreversibility is the cause of entropy generation at the hotter wall, while on the other hand, fluid friction irreversibility is the major

Next is the response of fluid velocity, fluid temperature and entropy generation to variation in effective temperature ratio. In Figure 4, it is observed that velocity is accelerated slightly except towards the microchannel centre and plate y ¼ 0. In Figure 4B, fluid temperature is significantly enhanced at increasing values of effective temperature ratio. The effect of this is noticed in Figure 4C with an increase in entropy production approaching the microchannel plate y ¼ 1. Furthermore, Bejan number rises approaching the plate y ¼ 1 but reduces towards y ¼ 0 in Figure 4D. It is then concluded that fluid friction irreversibility is dominant at plate

In Figure 5A, the effect of rarefaction parameter on fluid velocity is presented. The plot indicates that rarefaction parameter increases and fluid velocity at plate y ¼ 0 is not significant; however, it is decelerated towards the microchannel plate y ¼ 1. Fluid temperature is considerably enhanced at plate y ¼ 1 for different values of rarefaction as displayed in Figure 5B. Figures 5C and 5D presents similar results at plate y ¼ 1. Both entropy generation and Bejan number reduce as the values of rarefaction parameter increase. The implication of this observation is that entropy generation at microchannel wall y ¼ 1 is a consequence of viscous

y ¼ 0, while heat transfer irreversibility is dominant at plate y ¼ 1.

stress parameter increases. It is depicted in Figure 2C that fluid entropy

4.2 Influence of fluid-structure interaction parameter

contributor at the cooler wall of the microchannel.

4.3 Influence of effective temperature ratio

4.4 Influence of rarefaction

dissipation.

85

Alternatively, Bejan number gives the entropy generation distribution ratio parameter; it represents the ratio of heat transfer entropy generation ð Þ N<sup>1</sup> to the total entropy generation ð Þ Ns due to heat transfer and fluid friction; it is defined as

$$Be = \frac{N\_1}{N\_\sharp} = \frac{1}{1 + \Phi},\tag{21}$$

$$Be = \begin{cases} 0, N\_2 \gg N\_1 \\ 0.5, N\_1 = N\_2 \\ 1, N\_2 \ll N\_1 \end{cases} \tag{22}$$

Note that N<sup>1</sup> represents heat transfer irreversibility, while N<sup>2</sup> denotes irreversibility due to viscous dissipation, couple stresses and magnetic field.

## 4. Results and discussion

In this work, investigation has been conducted on fully developed, steady, viscous and incompressible flow of couple stress fluid in a vertical micro-porouschannel in the presence of magnetic field. Effects of couple stress parameter ð Þ a , fluid wall interaction parameter ð Þ ψ , effective temperature ratio (ETR) ð Þξ , rarefaction ð Þ βvKn and Hartmann number ð Þ H are presented in this section. Reasonable intervals for the above parameters as used by Chen and Weng [32] are adopted in this investigation: 0≤H≤10, 0≤vKn≤0:1, 0≤ψ≤10 and the selected reference values of βvKn ¼ 0:05, ln ¼ 1:667. Furthermore, 0≤ξ≤5 and 0≤a≤2 with reference values of a ¼ 1, ξ ¼ 0:5.

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation DOI: http://dx.doi.org/10.5772/intechopen.81123

## 4.1 Influence of couple stress parameter

Figure 2A illustrates the influence of couple stress parameter on fluid velocity. The plot shows a significant reduction in fluid velocity as the values of couple stress parameter increase. This observation is due to the increase in the dynamic viscosity of the fluid. In Figure 2B, it is illustrated that fluid temperature drops as couple stress parameter increases. It is depicted in Figure 2C that fluid entropy generation decreases at the microchannel walls as couple stress parameter increases. However, the effect is opposite near the middle of the microchannel. The same scenario is observed in Figure 2D; it is shown that Bejan number reduces in value at microchannel right wall which is an indication that fluid friction irreversibility is the major contributor to entropy generation as couple stress parameter increases.

## 4.2 Influence of fluid-structure interaction parameter

Figure 3A presents the effect of fluid-structure interaction parameter on fluid velocity. It is noted that FSIP does not have any significant effect on the slip velocity at the walls as well as the microchannel. However, Figure 3B reveals that fluid temperature is enhanced as FSIP increases. Response to the enhancement in fluid temperature in Figure 3B is the rise in fluid entropy generation at the hotter wall of the microchannel, while entropy generation is reduced at the cooler wall as depicted in Figure 3C. In Figure 3D, it is noticed that Bejan number increases at the hotter wall of the microchannel, while it reduces at the cooler wall. The implication of the latter is that heat transfer irreversibility is the cause of entropy generation at the hotter wall, while on the other hand, fluid friction irreversibility is the major contributor at the cooler wall of the microchannel.

### 4.3 Influence of effective temperature ratio

Next is the response of fluid velocity, fluid temperature and entropy generation to variation in effective temperature ratio. In Figure 4, it is observed that velocity is accelerated slightly except towards the microchannel centre and plate y ¼ 0. In Figure 4B, fluid temperature is significantly enhanced at increasing values of effective temperature ratio. The effect of this is noticed in Figure 4C with an increase in entropy production approaching the microchannel plate y ¼ 1. Furthermore, Bejan number rises approaching the plate y ¼ 1 but reduces towards y ¼ 0 in Figure 4D. It is then concluded that fluid friction irreversibility is dominant at plate y ¼ 0, while heat transfer irreversibility is dominant at plate y ¼ 1.

## 4.4 Influence of rarefaction

In Figure 5A, the effect of rarefaction parameter on fluid velocity is presented. The plot indicates that rarefaction parameter increases and fluid velocity at plate y ¼ 0 is not significant; however, it is decelerated towards the microchannel plate y ¼ 1. Fluid temperature is considerably enhanced at plate y ¼ 1 for different values of rarefaction as displayed in Figure 5B. Figures 5C and 5D presents similar results at plate y ¼ 1. Both entropy generation and Bejan number reduce as the values of rarefaction parameter increase. The implication of this observation is that entropy generation at microchannel wall y ¼ 1 is a consequence of viscous dissipation.

where ð Þ SG; Ns are the dimensional and dimensionless entropy generation rates. The ratio of heat transfer entropy generation ð Þ N<sup>1</sup> to fluid friction entropy

βvKn ¼ 0:05, Gr ¼ 0, a ¼ 1, Re ¼ 0:1, H ¼ 1

y Exact DTM 0 0.001872811813791 0.001872811814000 0.1 0.005546352540252 0.005546352541866 0.2 0.008819633339440 0.008819633342380 0.3 0.011379286267087 0.011379286271194 0.4 0.013003721837660 0.013003721842694 0.5 0.013560741309568 0.013560741315200 0.6 0.013006153436252 0.013006153442060 0.7 0.011383386215798 0.011383386221257 0.8 0.008824085393964 0.008824085398432 0.9 0.005549692555064 0.005549692557769 1 0.001873996484802 0.001873996484819

> <sup>Φ</sup> <sup>¼</sup> <sup>N</sup><sup>2</sup> N<sup>1</sup>

Alternatively, Bejan number gives the entropy generation distribution ratio parameter; it represents the ratio of heat transfer entropy generation ð Þ N<sup>1</sup> to the total entropy generation ð Þ Ns due to heat transfer and fluid friction; it is defined as

<sup>¼</sup> <sup>1</sup>

0, N2≫N<sup>1</sup> 0:5, N<sup>1</sup> ¼ N<sup>2</sup> 1, N2≪N<sup>1</sup>

Note that N<sup>1</sup> represents heat transfer irreversibility, while N<sup>2</sup> denotes irrevers-

In this work, investigation has been conducted on fully developed, steady, viscous and incompressible flow of couple stress fluid in a vertical micro-porouschannel in the presence of magnetic field. Effects of couple stress parameter ð Þ a , fluid wall interaction parameter ð Þ ψ , effective temperature ratio (ETR) ð Þξ , rarefaction ð Þ βvKn and Hartmann number ð Þ H are presented in this section. Reasonable intervals for the above parameters as used by Chen and Weng [32] are adopted in this investigation: 0≤H≤10, 0≤vKn≤0:1, 0≤ψ≤10 and the selected reference values of βvKn ¼ 0:05, ln ¼ 1:667. Furthermore, 0≤ξ≤5 and 0≤a≤2 with reference values

<sup>1</sup> <sup>þ</sup> <sup>Φ</sup> , (21)

Be <sup>¼</sup> <sup>N</sup><sup>1</sup> Ns

> 8 ><

> >:

Be ¼

ibility due to viscous dissipation, couple stresses and magnetic field.

(20)

(22)

generation ð Þ N<sup>2</sup> is represented as

Comparison of the exact solution with the values of velocity (u).

Pattern Formation and Stability in Magnetic Colloids

Table 2.

4. Results and discussion

of a ¼ 1, ξ ¼ 0:5.

84

## 4.5 Influence of Hartmann number

Finally, the response of fluid velocity, temperature, entropy generation and Bejan number to variation in Hartmann number is presented in Figure 6. In Figure 6A, fluid velocity is found to have decelerated within the microchannel region. The observed reduction in the motion of fluid is attributed to the presence of applied magnetic field which usually induces a resistive type of force known as Lorentz force. Also the presence of Ohmic heating in the flow significantly enhanced fluid temperature as depicted in Figure 6B. Figures 6C and 6D displays increase in fluid entropy generation and Bejan number at the microchannel walls as Hartmann number increases from 1 to 5. It is deduced that fluid entropy generation is induced by heat transfer irreversibility.

Nomenclature

ft

B<sup>0</sup> uniform magnetic field

DOI: http://dx.doi.org/10.5772/intechopen.81123

Cp specific heat capacity Re Reynolds number a couple stress parameter k thermal conductivity Kn Knudsen number H Hartmann number Pr Prandtl number T temperature of fluid T<sup>0</sup> reference temperature Br Brinkman number

respectively

EG local volumetric entropy generation rate

Ns dimensionless entropy generation parameter

η fluid particle size effect due to couple stresses

Abiodun A. Opanuga\*, Olasunmbo O. Agboola, Hilary I. Okagbue and

\*Address all correspondence to: abiodun.opanuga@covenantuiversity.edu.ng

© 2018 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,

Department of Mathematics, Covenant University, Ota, Nigeria

Cv specific heats at constant volume

, f <sup>v</sup> thermal and tangential momentum accommodation coefficients,

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation

u fluid velocity h channel width

Be Bejan number

ρ fluid density

βt, β<sup>v</sup> dimensionless variables γ<sup>s</sup> ratio of specific heat μ dynamic viscosity

provided the original work is properly cited.

ξ effective temperature ratio σ electrical conductivity Ω temperature difference

ψ fluid wall interaction parameter

Greek letters

Author details

Sheila A. Bishop

87

## 5. Conclusions

Entropy generation of fully developed steady, viscous, incompressible couple stress fluid in a vertical micro-porous-channel in the presence of magnetic field is analysed in this work. The equations governing the fluid flow are solved via an efficient technique proposed by Zhou. Then fluid entropy generation and Bejan number are calculated by the results obtained. This work reduces to Chen and Weng [34] when Hartmann number, couple stress parameter, entropy generation and Bejan number are neglected ð Þ H ! 0; a ! 0; Ns ! 0; Be ! 0 . Furthermore, it agrees with Jha and Aina [10] in the absence of couple stress parameter, entropy generation and Bejan number ð Þ a ! 0; Ns ! 0; Be ! 0 . The present study is significant in the cooling of microchannel devices and conservation of useful energy. The following conclusions are made based on the results above:


## Acknowledgements

Authors appreciate the funding provided by Covenant University, Ota.

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation DOI: http://dx.doi.org/10.5772/intechopen.81123

## Nomenclature

4.5 Influence of Hartmann number

Pattern Formation and Stability in Magnetic Colloids

is induced by heat transfer irreversibility.

5. Conclusions

entropy generation.

generation.

Acknowledgements

86

Finally, the response of fluid velocity, temperature, entropy generation and Bejan number to variation in Hartmann number is presented in Figure 6. In Figure 6A, fluid velocity is found to have decelerated within the microchannel region. The observed reduction in the motion of fluid is attributed to the presence of applied magnetic field which usually induces a resistive type of force known as Lorentz force. Also the presence of Ohmic heating in the flow significantly

enhanced fluid temperature as depicted in Figure 6B. Figures 6C and 6D displays increase in fluid entropy generation and Bejan number at the microchannel walls as Hartmann number increases from 1 to 5. It is deduced that fluid entropy generation

Entropy generation of fully developed steady, viscous, incompressible couple stress fluid in a vertical micro-porous-channel in the presence of magnetic field is analysed in this work. The equations governing the fluid flow are solved via an efficient technique proposed by Zhou. Then fluid entropy generation and Bejan number are calculated by the results obtained. This work reduces to Chen and Weng [34] when Hartmann number, couple stress parameter, entropy generation and Bejan number are neglected ð Þ H ! 0; a ! 0; Ns ! 0; Be ! 0 . Furthermore, it agrees with Jha and Aina [10] in the absence of couple stress parameter, entropy generation and Bejan number ð Þ a ! 0; Ns ! 0; Be ! 0 . The present study is significant in the cooling of microchannel devices and conservation of useful energy.

1. Couple stress parameter reduces fluid velocity, velocity slip, temperature and

2. Increase in fluid-structure interaction parameter increases fluid temperature. However, entropy generation enhances at the hotter wall and reduces at the cooler region of the microchannel. Furthermore, fluid irreversibility is

3. Effective temperature ratio slightly enhances fluid velocity and velocity slip,

4.An increase in the values of rarefaction mainly reduces fluid velocity and velocity slip, increases fluid temperature and reduces entropy

5. Hartmann number decreases fluid velocity and velocity slip, while the

Authors appreciate the funding provided by Covenant University, Ota.

temperature is enhanced considerably. Entropy generation and Bejan number

The following conclusions are made based on the results above:

enhanced at the hot wall and increases at the cold wall.

raising fluid temperature significantly.

are enhanced at microchannel walls.


## Greek letters


## Author details

Abiodun A. Opanuga\*, Olasunmbo O. Agboola, Hilary I. Okagbue and Sheila A. Bishop Department of Mathematics, Covenant University, Ota, Nigeria

\*Address all correspondence to: abiodun.opanuga@covenantuiversity.edu.ng

© 2018 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] Agboola OO, Opanuga AA, Okagbue HI, Bishop SA, Ogunniyi PO. Analysis of hall effects on the entropy generation of natural convection flow through a vertical microchannel. International Journal of Mechanical Engineering and Technology. 2008;9(8):712-721

[2] Arkilic EB, Schmidt MA, Breuer KS. Gaseous slip flow in long microchannels. Journal of Microelectromechanical Systems. 1997;6:167-178

[3] Sofonea V, Sekerka RF. Diffusereflection boundary conditions for a thermal lattice Boltzmann model in two dimensions: Evidence of temperature jump and slip velocity in microchannels. Physical Review E – Statistical Nonlinear and Soft Matter Physics. 2005;71:1-10 http://dx.doi.org/10.1103/ PhysRevE.71.066709

[4] Lv Q, Liu X, Wang E, Wang S. Analytical solution to predicting gaseous mass flow rates of microchannels in a wide range of Knudsen numbers. Physical Review E. 2013;88:013007 http://dx.doi.org/10.1103/ PhysRevE.88.013007

[5] Khadrawi AF, Al-Shyyab A. Slip flow and heat transfer in axially moving micro-concentric cylinders. International Communications in Heat and Mass Transfer. 2010;37:1149-1152 http://dx.doi.org/10.1016/j. icheatmasstransfer.2010.06.006

[6] Chen S, Tian Z. Simulation of thermal micro-flow using lattice Boltzmann method with Langmuir slip model. International Journal of Heat and Fluid Flow. 2010;31:227-235 http://dx. doi.org/10.1016/j.ijheatfluidflow. 2009.12.006

[7] Larrode FE, Housiadas C, Drossinos Y. Slip-flow heat transfer in circular tubes. International Journal of Heat and Mass Transfer. 2000;43:2669-26680

[8] Adesanya SO. Free convective flow of heat generating fluid through a porous vertical channel with velocity slip and temperature jump. Ain Shams Engineering Journal. 2015;6:1045-1052. DOI: http://dx.doi.org/10.1016/j. asej.2014.12.008

flow through a channel saturated with porous material. Energy. 2015;93:

DOI: http://dx.doi.org/10.5772/intechopen.81123

[23] Opanuga AA, Okagbue HI, Agboola OO, Imaga OF. Entropy generation analysis of buoyancy effect on hydromagnetic poiseuille flow with internal heat generation. Defect and Diffusion Forum. 2017;378:102-112

[24] Opanuga AA, Gbadeyan JA, Iyase

[25] Opanuga AA, Okagbue HI, Agboola OO. Irreversibility analysis of a radiative MHD Poiseuille flow through porous

Proceedings of The World Congress on Engineering 2017; July 5–7; London, U.K

[26] Opanuga AA, Bishop SA, Okagbue HI, Agboola OO. Hall current and joule heating effects on flow of couple stress

SA. Second law analysis of hydromagnetic couple stress fluid embedded in a non-Darcian porous medium. IAENG International Journal of Applied Mathematics. 2017;47(3):

medium with slip condition.

fluid with entropy generation. Engineering, Technology & Applied Science Research. 2008;8(3):2923-2930

[27] Arikoglu A. Ozkol solution of difference equations by using

[28] Biazar J, Eslami M. Differential transform method for quadratic Riccati differential equation. International Journal of Nonlinear Science. 2010;9(4):

[29] Agboola OO, Gbadeyan JA,

[30] Agboola OO, Opanuga AA, Gbadeyan JA. Solution of third order

July 5–7 London; UK

Opanuga AA, Agarana MC, Bishop SA, Oghonyon JG. Variational iteration method for natural frequencies of a cantilever beam with special attention to the higher modes. Proceedings of The World Congress on Engineering 2017;

differential transform method. Applied Mathematics and Computation. 2006; 173(1):126-136. http://dx.doi.org/ 10.1016/j.amc.2005.06.013

287-294

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation

444-447

[16] Jangili S, Adesanya SO, Falade JA, Gajjela N. Entropy generation analysis for a radiative micropolar fluid flow through a vertical channel saturated with non-Darcian porous medium. International Journal of Applied and Computational Mathematics. 3(4): 3759-3782. DOI: 10.1007/s40819-017-

[17] Adesanya SO, Makinde OD.

Statistical Mechanics and its Applications. 2015;432:222-229

of couple stresses on entropy generation rate in a porous channel

with convective heating. Computational & Applied

10.1007/s40314-014-0117-z

Modelling. 2011;35:4630-4646

522-535

e14061028

89

[20] Eegunjobi AS, Makinde OD. Effects of Navier slip on entropy generation in a porous channel with suction/injection. Journal of Thermal Science and Technology. 2002;7(4):

[21] Eegunjobi AS, Makinde OD.

14:1028-1044. DOI: 10.3390/

Journal. 2014;5:575-584

Combined effect of buoyancy force and Navier slip on entropy generation in a vertical porous channel. Entropy. 2012;

[22] Das S, Jana RN. Entropy generation due to MHD flow in a porous channel with Navier slip. Ain Shams Engineering

Irreversibility analysis in a couple stress film flow along an inclined heated plate with adiabatic free surface. Physica A:

[18] Adesanya SO, Makinde OD. Effects

Mathematics. 2015;34:293-307. DOI:

[19] Ajibade AO, Jha BK, Omame A. Entropy generation under the effect of suction/injection. Applied Mathematical

1239-1245

0322-8

[9] Aziz A, Niedbalski N. Thermally developing microtube gas flow with axial conduction and viscous dissipation. International Journal of Thermal Sciences. 2011;50:332-340 http://dx.doi.org/10.1016/j. ijthermalsci.2010.08.003

[10] Jha BK. Aina B. MHD natural convection flow in a vertical porous microchannel formed by nonconducting and conducting plates in the presence of induced magnetic field. Heat Transfer Research. 2017;48(15):1-24

[11] Jha BK, Aina B, Joseph SB. Natural convection flow in a vertical microchannel with suction/injection. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering. 2014; 228(3):171-180. DOI: 10.1177/ 0954408913492719

[12] Jha BK, Aina B, Ajiya AT. MHD natural convection flow in a vertical parallel plate microchannel. Ain Shams Engineering Journal. 2015;6:289-295

[13] Bejan A. Entropy Generation through Heat and Fluid Flow. New York: Wiley; 1982

[14] Adesanya SO, Falade JA, Jangili S, Beg OA. Irreversibility analysis for reactive third-grade fluid flow and heat transfer with convective wall cooling. Alexandria Engineering Journal. 2017; 56:153-160

[15] Adesanya SO, Kareem SO, Falade JA, Arekete SA. Entropy generation analysis for a reactive couple stress fluid Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation DOI: http://dx.doi.org/10.5772/intechopen.81123

flow through a channel saturated with porous material. Energy. 2015;93: 1239-1245

References

[1] Agboola OO, Opanuga AA, Okagbue HI, Bishop SA, Ogunniyi PO. Analysis of hall effects on the entropy generation of natural convection flow through a vertical microchannel. International Journal of Mechanical Engineering and

Pattern Formation and Stability in Magnetic Colloids

[8] Adesanya SO. Free convective flow of heat generating fluid through a porous vertical channel with velocity slip and temperature jump. Ain Shams Engineering Journal. 2015;6:1045-1052. DOI: http://dx.doi.org/10.1016/j.

[9] Aziz A, Niedbalski N. Thermally developing microtube gas flow with axial conduction and viscous dissipation. International Journal of Thermal Sciences. 2011;50:332-340

http://dx.doi.org/10.1016/j. ijthermalsci.2010.08.003

Research. 2017;48(15):1-24

convection flow in a vertical

228(3):171-180. DOI: 10.1177/

0954408913492719

York: Wiley; 1982

56:153-160

[10] Jha BK. Aina B. MHD natural convection flow in a vertical porous microchannel formed by nonconducting and conducting plates in the presence of induced magnetic field. Heat Transfer

[11] Jha BK, Aina B, Joseph SB. Natural

microchannel with suction/injection. Proceedings of the Institution of

[12] Jha BK, Aina B, Ajiya AT. MHD natural convection flow in a vertical parallel plate microchannel. Ain Shams Engineering Journal. 2015;6:289-295

[13] Bejan A. Entropy Generation through Heat and Fluid Flow. New

[14] Adesanya SO, Falade JA, Jangili S, Beg OA. Irreversibility analysis for reactive third-grade fluid flow and heat transfer with convective wall cooling. Alexandria Engineering Journal. 2017;

[15] Adesanya SO, Kareem SO, Falade JA, Arekete SA. Entropy generation analysis for a reactive couple stress fluid

Mechanical Engineers, Part E: Journal of Process Mechanical Engineering. 2014;

asej.2014.12.008

[2] Arkilic EB, Schmidt MA, Breuer KS. Gaseous slip flow in long microchannels. Journal of Microelectromechanical

[3] Sofonea V, Sekerka RF. Diffusereflection boundary conditions for a thermal lattice Boltzmann model in two dimensions: Evidence of temperature jump and slip velocity in microchannels.

Physical Review E – Statistical Nonlinear and Soft Matter Physics. 2005;71:1-10 http://dx.doi.org/10.1103/

[4] Lv Q, Liu X, Wang E, Wang S. Analytical solution to predicting gaseous mass flow rates of microchannels in a wide range of Knudsen numbers. Physical Review E. 2013;88:013007

[5] Khadrawi AF, Al-Shyyab A. Slip flow and heat transfer in axially moving

International Communications in Heat and Mass Transfer. 2010;37:1149-1152

[7] Larrode FE, Housiadas C, Drossinos Y. Slip-flow heat transfer in circular tubes. International Journal of Heat and Mass Transfer. 2000;43:2669-26680

PhysRevE.71.066709

http://dx.doi.org/10.1103/ PhysRevE.88.013007

micro-concentric cylinders.

http://dx.doi.org/10.1016/j. icheatmasstransfer.2010.06.006

2009.12.006

88

[6] Chen S, Tian Z. Simulation of thermal micro-flow using lattice Boltzmann method with Langmuir slip model. International Journal of Heat and Fluid Flow. 2010;31:227-235 http://dx. doi.org/10.1016/j.ijheatfluidflow.

Technology. 2008;9(8):712-721

Systems. 1997;6:167-178

[16] Jangili S, Adesanya SO, Falade JA, Gajjela N. Entropy generation analysis for a radiative micropolar fluid flow through a vertical channel saturated with non-Darcian porous medium. International Journal of Applied and Computational Mathematics. 3(4): 3759-3782. DOI: 10.1007/s40819-017- 0322-8

[17] Adesanya SO, Makinde OD. Irreversibility analysis in a couple stress film flow along an inclined heated plate with adiabatic free surface. Physica A: Statistical Mechanics and its Applications. 2015;432:222-229

[18] Adesanya SO, Makinde OD. Effects of couple stresses on entropy generation rate in a porous channel with convective heating. Computational & Applied Mathematics. 2015;34:293-307. DOI: 10.1007/s40314-014-0117-z

[19] Ajibade AO, Jha BK, Omame A. Entropy generation under the effect of suction/injection. Applied Mathematical Modelling. 2011;35:4630-4646

[20] Eegunjobi AS, Makinde OD. Effects of Navier slip on entropy generation in a porous channel with suction/injection. Journal of Thermal Science and Technology. 2002;7(4): 522-535

[21] Eegunjobi AS, Makinde OD. Combined effect of buoyancy force and Navier slip on entropy generation in a vertical porous channel. Entropy. 2012; 14:1028-1044. DOI: 10.3390/ e14061028

[22] Das S, Jana RN. Entropy generation due to MHD flow in a porous channel with Navier slip. Ain Shams Engineering Journal. 2014;5:575-584

[23] Opanuga AA, Okagbue HI, Agboola OO, Imaga OF. Entropy generation analysis of buoyancy effect on hydromagnetic poiseuille flow with internal heat generation. Defect and Diffusion Forum. 2017;378:102-112

[24] Opanuga AA, Gbadeyan JA, Iyase SA. Second law analysis of hydromagnetic couple stress fluid embedded in a non-Darcian porous medium. IAENG International Journal of Applied Mathematics. 2017;47(3): 287-294

[25] Opanuga AA, Okagbue HI, Agboola OO. Irreversibility analysis of a radiative MHD Poiseuille flow through porous medium with slip condition. Proceedings of The World Congress on Engineering 2017; July 5–7; London, U.K

[26] Opanuga AA, Bishop SA, Okagbue HI, Agboola OO. Hall current and joule heating effects on flow of couple stress fluid with entropy generation. Engineering, Technology & Applied Science Research. 2008;8(3):2923-2930

[27] Arikoglu A. Ozkol solution of difference equations by using differential transform method. Applied Mathematics and Computation. 2006; 173(1):126-136. http://dx.doi.org/ 10.1016/j.amc.2005.06.013

[28] Biazar J, Eslami M. Differential transform method for quadratic Riccati differential equation. International Journal of Nonlinear Science. 2010;9(4): 444-447

[29] Agboola OO, Gbadeyan JA, Opanuga AA, Agarana MC, Bishop SA, Oghonyon JG. Variational iteration method for natural frequencies of a cantilever beam with special attention to the higher modes. Proceedings of The World Congress on Engineering 2017; July 5–7 London; UK

[30] Agboola OO, Opanuga AA, Gbadeyan JA. Solution of third order

Chapter 6

Abstract

1. Introduction

91

Free Convection in a MHD Open

This chapter examines the magneto-hydrodynamic (MHD) free convection in a square enclosure filled with liquid gallium subjected to a [transverse magnetic field] in-plane magnetic field. First, the side vertical walls of the cavity have spatially varying linearly temperature distributions. The bottom wall is uniformly heated and the upper wall is adiabatic. The second configuration is an open enclosure heated linearly from the left wall. Lattice Boltzmann method (LBM) is applied in order to solve the coupled equations of flow and temperature fields.

transfer rate decreases with an increase of the Ha number which is a widely solicited result in different engineering applications. Streamlines, isotherm counters and Nu numbers are displayed and discussed. A good stability is observed for all studied cases employing an in-house code. The obtained results show that the free-convection heat transfer in the open MHD enclosure is

Free convection in open cavities is an important phenomenon in engineering systems because of this it have received a considerable attention due to their applications in various industries of high-performance insulation for buildings, injection molding chemical catalytic reactors, packed sphere beds, grain storage, float glass production, air-conditioning in rooms, cooling of electronic devices, and geophysical problems. Extensive research studies using various numerical simulations were conducted into free convection of open enclosures [1–5]. The latter works have acquired a basic understanding of free convection flows and heat transfer characteristics in an open enclosure with non-conducting fluid. However, in most studies, one vertical wall of the enclosure is cooled and another one heated while the remaining top and bottom walls are well insulated. Recently, MHD free convection flows have attracted attention since can be are encountered in numerous problems with industrial and technological interest, covering a wide range of basic sciences such as nuclear engineering, fire research, crystal growth, astrophysics and metallurgy [6]. Besides, the process of manufacturing materials in industrial problems and microelectronic heat transfer devices involves an electrically conducting fluid

enhanced and it is greater to that of a uniformly heated wall.

Keywords: MHD, open cavity, free convection, LBM, heat transfer

. The results show that the heat

Cavity with a Linearly Heated

Wall Using LBM

This study has been carried out for Ra of 10<sup>5</sup>

Raoudha Chaabane

ordinary differential equations using differential transform method. Global Journal of Pure and Applied Mathematics. 2015;11(4):2511-2517

[31] Odibat Z. Differential transform method for solving Volterra integral equation with separable kernels. Mathematical and Computer Modelling. 2008;48:144-149. http://dx.doi.org/ 10.1016/j.mcm.2007.12.022

[32] Opanuga AA, Agboola OO, Okagbue HI. Approximate solution of multipoint boundary value problems. Journal of Engineering and Applied Sciences. 2015; 10(4):85-89

[33] Opanuga AA, Okagbue HI, Agboola OO, Bishop SA. Second law analysis of ion slip effect on MHD couple stress fluid. International Journal of Mechanics. 2018;12:96-101

[34] Chen CK, Weng HC. Natural convection in a vertical microchannel. Journal of Heat Transfer. 2005;127: 1053-1056

## Chapter 6

ordinary differential equations using differential transform method. Global

Pattern Formation and Stability in Magnetic Colloids

[31] Odibat Z. Differential transform method for solving Volterra integral equation with separable kernels.

Mathematical and Computer Modelling. 2008;48:144-149. http://dx.doi.org/

[32] Opanuga AA, Agboola OO, Okagbue HI. Approximate solution of multipoint boundary value problems. Journal of Engineering and Applied Sciences. 2015;

[33] Opanuga AA, Okagbue HI, Agboola OO, Bishop SA. Second law analysis of ion slip effect on MHD couple stress fluid. International Journal of Mechanics. 2018;12:96-101

[34] Chen CK, Weng HC. Natural convection in a vertical microchannel. Journal of Heat Transfer. 2005;127:

Journal of Pure and Applied Mathematics. 2015;11(4):2511-2517

10.1016/j.mcm.2007.12.022

10(4):85-89

1053-1056

90
