# Entropy Generation in a Couple Stress Fluid Flow Through a Vertical Channel Filled with Saturated Porous Media

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

_{0}while the right wall is placed at temperature T

_{w}such that T

_{0}< T

_{w}.

_{p}specific heat at constant pressure, k is the thermal conductivity of the fluid, K is the porous media permeability, c is the empirical constant in the second order (porous inertia) resistance such that c = 0 corresponds to the Darcy law, δ is the fluid particle size effect due to couple stresses. Introducing the following dimensionless variables:

_{f}and the local Nusselt number Nu, which are expressed as:

_{w}is the wall shear stress and q

_{w}is the heat flux at the channel walls given by:

## 3. Entropy Analysis

_{s}= N

_{1}+ N

_{2}, (Heat transfer Irreversibility), (Irreversibility due to viscous dissipation, couple stresses and porous media), (Irreversibility ratio).

## 4. Results and Discussion

#### 4.1. Variation of Parameters on Velocity Profile

#### 4.2. Variation of Parameters on Temperature Profile

#### 4.3. Variation of Parameters on Skin Friction

#### 4.4. Variation of Parameters on Nusselt Number

#### 4.4. Effect of Various Parameters on Entropy Generation Profiles

^{−1}. With an increase in BrΩ

^{−1}while all other parameters are kept constant, there exit an increase in the entropy generation profile at both walls and it is symmetric, but at η = 0.5, there is no change in the entropy generation. Figure 22 shows the effect of M on the entropy generation profile. An increase in M causes a decrease in the entropy generation at both walls, but between η = 0.3 and η = 0.75, an increase in entropy generation profile is noticed. Figure 23 shows the effect of A on the entropy generation profile. As A is increasing, large increases in the entropy generation at the lower wall are noticed while little increase is noticed on the upper wall. Increases in λ as shown in Figure 24 decrease the entropy generation profile at the lower wall and increase the entropy generation profile at the upper wall.

#### 4.5. Effect of Different Parameters on Bejan Number

^{−1}on the Bejan number. An increase in each of these parameters decreases the Bejan number across the flow uniformly. The influence of M on the Bejan number is shown in Figure 29. It is noticed that the Bejan number decrease between η = 0.3 and η = 0.7 but increases at the both walls as M increases. Figure 30 shows that increase in λ, increase the Bejan number across the flow.

## 5. Conclusions

- The velocity profiles in general are parabolic in nature. Increases in Gr, Pr and A increase the velocity profile, while increases in M, S and λ decrease the velocity profile.
- An increase in Gr, Pr, Ec and A increases the temperature profile, while increases in S, M and λ decrease the temperature profile.
- An increase in M verses S decreases the skin friction, while an increase in A versus Gr increases the skin friction.
- The Nusselt number increases with increasing A versus S, while it decreases with an increase in M versus S.
- Entropy generation increases with increases in Gr, BrΩ
^{−1}and A, while it decreases with an increase in S. As M increases, its decreases at both walls, but increases in the centre of the channel. - Increases in Gr, A and BrΩ
^{−1}decrease the Bejan number across the flow, while an increase in λ increases the Bejan number across the flow. Increases in M and S; increase the Bejan number at both walls, but at the centre of the channel, the Bejan number decreases.

## Conflicts of Interest

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**MDPI and ACS Style**

Makinde, O.D.; Eegunjobi, A.S.
Entropy Generation in a Couple Stress Fluid Flow Through a Vertical Channel Filled with Saturated Porous Media. *Entropy* **2013**, *15*, 4589-4606.
https://doi.org/10.3390/e15114589

**AMA Style**

Makinde OD, Eegunjobi AS.
Entropy Generation in a Couple Stress Fluid Flow Through a Vertical Channel Filled with Saturated Porous Media. *Entropy*. 2013; 15(11):4589-4606.
https://doi.org/10.3390/e15114589

**Chicago/Turabian Style**

Makinde, Oluwole Daniel, and Adetayo Samuel Eegunjobi.
2013. "Entropy Generation in a Couple Stress Fluid Flow Through a Vertical Channel Filled with Saturated Porous Media" *Entropy* 15, no. 11: 4589-4606.
https://doi.org/10.3390/e15114589