E-ISSN:2583-553X

Research Article

Couple Stress

Applied Science and Biotechnology Journal for Advanced Research

2025 Volume 4 Number 3 MAY
Publisherwww.vandanapublications.com

Flow Analysis of a Couple Stress Fluid Through Porous Media in the Absence of a Pressure Gradient

Joshua O Oladele1*
DOI:10.5281/zenodo.15545269

1* Joshua O Oladele, Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ, USA.

This study investigates the steady flow of a chemically reacting couple-stress fluid through a porous medium without an imposed pressure gradient. Using the second law of thermodynamics, we analyze entropy generation and thermal irreversibility in the system. The higher-order differential equations that govern the flow, incorporating couple stresses and porous permeability effects, are non-dimensionalized and simplified to obtain approximate analytical solutions. Key parameters such as the stress parameter of the couple and the permeability of the porous medium are examined to determine their influence on flow behavior and the rates of entropy generation. The results provide insights relevant to the optimization of heat and mass transfer in complex fluid systems with applications in chemical and thermal engineering.

Keywords: fluid, pressure, entropy, thermodynamics

Corresponding Author How to Cite this Article To Browse
Joshua O Oladele, Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ, USA.
Email: 		Joshua O Oladele, Flow Analysis of a Couple Stress Fluid Through Porous Media in the Absence of a Pressure Gradient. Appl Sci Biotechnol J Adv Res. 2025;4(3):1-6.
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https://abjar.vandanapublications.com/index.php/ojs/article/view/91

Manuscript Received Review Round 1 Review Round 2 Review Round 3 Accepted
2025-04-02 2025-04-22 2025-05-21
Conflict of Interest Funding Ethical Approval Plagiarism X-checker Note
None Nil Yes 3.55

© 2025 by Joshua O Oladele and Published by Vandana Publications. This is an Open Access article licensed under a Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by/4.0/ unported [CC BY 4.0].

Appl Sci Biotechnol J Adv Res 2025;4(3)1
Download PDFBack To Article1. Introduction2. Statement of the
Problem3. Nondimensionalization
and Reformulation4. Solution of the Problem5. Results and AnalysisReferences
Joshua OO. Flow Analysis of a Couple Stress Fluid Through Porous Media

1. Introduction

The second law of thermodynamics plays a crucial role in design and optimization of thermal systems by quantifying entropy generation, which reflects irreversibility and energy losses. Minimizing entropy generation improves system efficiency and exergy, as established by Bejan (1982) and subsequent research [3, 4]. Entropy generation analysis, based on second law, offers a more accurate evaluation of system performance than first-law methods, especially in complex fluid flows [1, 5]. Non-Newtonian fluids, including suspensions, polymer solutions, and biological fluids, exhibit behaviors that deviate from Newtonian models, necessitating advanced constitutive models. Couple stress fluid model, introduced by Stokes, accounts for microstructural effects such as couple stresses and asymmetric stress tensors, making it suitable for fluids with size-dependent properties [7].

Previous studies have explored generation of entropy in couple stress fluids flowing through porous media under various conditions [8, 9]. However, limited attention has been given to combined effects of chemical reactions and cross-diffusion phenomena (e.g., Dufour effect) on entropy generation in such flows. This work addresses that gap by analyzing thermodynamic irreversibility in chemically reacting couple stress fluids flowing steadily through porous channels.

2. Statement of the Problem

The governing equation guiding the thermodynamic analysis of couple stress fluid flow through a porous medium is a fourth-order differential equation coupled with boundary conditions:

abjar_91_01

The corresponding boundary conditions are given by:

abjar_91_02

We introduce the following dimensionless variables and parameters:
abjar_91_03

where h is the channel half-width, u and ū are dimensionless and dimensional axial velocities, respectively, μ is the dynamic viscosity, η is the couple stress viscosity, ρ is the fluid density, and K is the permeability of porous medium [11, 12, 13].

The study of couple stress fluid flow in porous media has been extensively discussed by Adesanya and Makinde [11, 14], who analyzed entropy generation and convective heating effects in such flows.

The modeling approach aligns with the formulations provided by Anderson [15, 2] and Hayat et al. [17], where higher-order derivatives account for viscoelastic effects. Similar boundary conditions have been applied in related works on flow through porous channels [12].

3. Nondimensionalization and Reformulation

To nondimensionalize the governing equation, we substitute:

abjar_91_04

Hence, the derivatives transform as follows:

abjar_91_05

abjar_91_06

abjar_91_07

Substituting abjar_91_08 into abjar_91_09 we obtain:

abjar_91_10

Dividing through by abjar_91_11 we get:

abjar_91_12


Appl Sci Biotechnol J Adv Res 2025;4(3)2
Joshua OO. Flow Analysis of a Couple Stress Fluid Through Porous Media

Let,

abjar_91_13

(3.5) becomes:

abjar_91_14

Rewriting in a standard form, we obtain the nondimensional momentum equation:

abjar_91_15

where a is the couple stress parameter and β is the porous permeability constant.

4.  Solution of the Problem

The differential equation is

abjar_91_16

Assume a solution of the form

abjar_91_17

Differentiating (4.1) we obtain:

abjar_91_18

Then, the second, third, and fourth derivatives of u(y) are given by:

abjar_91_19

abjar_91_20

abjar_91_21

Substitute into (3.8) :

abjar_91_22

abjar_91_24

Since abjar_91_23 we have

Let abjar_91_25 Then (4.7) becomes

abjar_91_26

Using the quadratic formula:

abjar_91_27

Recall that abjar_91_28 so

abjar_91_29

Therefore, the general solution is

abjar_91_30

where

abjar_91_31


Appl Sci Biotechnol J Adv Res 2025;4(3)3
Joshua OO. Flow Analysis of a Couple Stress Fluid Through Porous Media

Using the boundary conditions:

abjar_91_32

abjar_91_33

abjar_91_34

abjar_91_35

abjar_91_36

abjar_91_37

abjar_91_38

abjar_91_39

Bringing the boundary conditions together, we solve the system:

abjar_91_40

Solving the system, the constants A, B, C, D are given by,

abjar_91_41

Therefore,

abjar_91_42


Appl Sci Biotechnol J Adv Res 2025;4(3)4
Joshua OO. Flow Analysis of a Couple Stress Fluid Through Porous Media

5. Results and Analysis

This section presents numerical results for the velocity distribution u(y) of a steady, chemically reacting couple-stress fluid flowing through a porous channel. The analysis focuses on the effects of the couple stress parameter a and the porous permeability parameter β. The boundary conditions abjar_91_43 are enforced throughout.

5.1. Effect of Couple Stress Parameter

Table 1 and Figure 1 illustrate the variation of the velocity profile u(y) for different values of the couple stress parameter a, at a fixed porous permeability parameter abjar_91_44

ya = 0.01a = 0.05a = 0.5
0.00.000000.000000.00000
0.19.45344 x 10 - 29.21226 x 10 - 28.58026 x 10 - 2
0.21.89354 x 10 - 11.84603 x 10 - 11.72411 x 10 - 1
0.32.84746 x 10 - 12.77814 x 10 - 12.60642 x 10 - 1
0.43.81003 x 10 - 13.72166 x 10 - 13.51363 x 10 - 1
0.54.78425 x 10 - 14.68131 x 10 - 14.45363 x 10 - 1
0.65.77338 x 10 - 15.66282 x 10 - 15.43644 x 10 - 1
0.76.78129 x 10 - 16.67352 x 10 - 16.47154 x 10 - 1
0.87.81341 x 10 - 17.72318 x 10 - 17.56943 x 10 - 1
0.98.87936 x 10 - 18.82547 x 10 - 18.74145 x 10 - 1
1.01.000001.000001.00000

Table 1: Velocity profile u(y) for various values of couple stress parameter a withabjar_91_44

Figure 1: Velocity distribution u(y) for varying couple stress parameter a at abjar_91_44 Increasing a reduces peak velocity near channel centerline due to enhanced internal microstructural resistance.

The results indicate that as the couple stress parameter increases, the velocity decreases throughout the domain, particularly near the centerline where the flow is most active. This is attributed to the augmented viscous resistance induced by couple stresses, which inhibit shear deformation due to micro-rotational effects.

5.2.  Effect of Porous Permeability Parameter

Table 2 and Figure 2 show the influence of the porous permeability parameter β on the velocity distribution for a fixed couple stress parameter β at a = 0.01.

yβ = 1β = 2β = 3
0.00.000000.000000.00000
0.18.5108 × 10−27.36722 × 10−26.4144 × 10−2
0.21.71076 × 10−11.4885 × 10−11.30276 × 10−1
0.32.58775 × 10−12.27068 × 10−12.00444 × 10−1
0.43.49089 × 10−13.09918 × 10−12.76809 × 10−1
0.54.42933 × 10−13.99077 × 10−13.61705 × 10−1
0.65.41254 × 10−14.96318 × 10−14.57661 × 10−1
0.76.45045 × 10−16.03493 × 10−15.67378 × 10−1
0.87.55353 × 10−17.2243 × 10−16.93519 × 10−1
0.98.73282 × 10−18.54579 × 10−18.38007 × 10−1
1.01.000001.000001.00000

Table 2: Velocity profile u(y) for different values of permeability parameter β at a = 0.01.

Figure 2: Variations in porous permeability parameter β significantly affect velocity profile u(y) when couple stress parameter is fixed at a = 0.01 Specifically, as β increases, velocity decreases due to increased flow resistance induced by porous medium. It is observed that increasing β results in a suppressed velocity profile across domain.


Appl Sci Biotechnol J Adv Res 2025;4(3)5
Joshua OO. Flow Analysis of a Couple Stress Fluid Through Porous Media

This behavior reflects enhanced resistance imposed by porous structure, which retards fluid motion & reduces kinetic energy, consistent with physical expectations.

5.3. Interpretation and Physical Implications

Interplay between couple stress and porous permeability effects significantly alters the flow dynamics. Higher values of intensify internal rotational resistance, increasing viscous dissipation & thus reducing fluid velocity. Similarly, increased β reflects a denser porous medium, leading to stronger drag forces & diminished flow rates. These mechanisms are critical in applications involving suspensions, polymeric fluids, and industrial slurries, where both microstructure & porous environments govern performance. Observed trends conform with thermodynamic principles, particularly in terms of entropy generation & energy dissipation, highlighting non-negligible influence of both parameters on system irreversibility [16, 10, 6].

References

1. M. A. Hassan, & M. N. K. Khan. (2017). Entropy generation analysis of couple stress fluid in a porous medium. International Journal of Heat and Mass Transfer, 108, 1823–1830.

2. S. Chen, & H. Hsin. (2010). Thermodynamic irreversibility and entropy generation in couple stress fluid flow through porous media. Applied Mathematical Modelling, 34, 1637–1645.

3. A. Ariel, & T. Aranda. (2006). On the flow of couple stress fluids in porous channels. Journal of Non-Newtonian Fluid Mechanics, 136, 95–101.

4. J. Kwanza, & P. Sharma. (2003). Entropy generation minimization in thermal systems: An overview. Energy Conversion and Management, 44, 1585–1603.

5. T. R. Mahapatra, & S. Roy. (2003). Entropy generation in non-Newtonian fluid flows: A review. International Journal of Heat and Fluid Flow, 24, 185–194.

6. P. Choudhury, & R. Gupta. (2006). Couple stress effects in porous flows. International Journal of Engineering Science, 44(7), 567–578.

7. J. G. Oldroyd. (1950). On the formulation of rheological equations of state. Proceedings of the Royal Society, A, 200, 523–541.

8. S. Shateyi, & F. Mabood. (2003). Entropy generation in couple stress fluid flows through porous media. Chaos, Solitons and Fractals, 15, 665–673.

9. R. Singh. (2002). Effects of chemical reaction and magnetic field on couple stress fluid flow in porous channels. , 40, 895–905.

10. J. L. Andrieko. (2000). Fluid Dynamics in Porous Media. (2nd ed.). New York: Springer.

11. S. O. Adesanya, & O. D. Makinde. (2013). Entropy generation and convective heating effects in couple stress fluid flow through porous channels. Applied Mathematical Modelling, 37, 7383–7395.

12. A. O. Eegunjobi, S. O. Adesanya, & O. D. Makinde. (2013). Thermal and entropy generation analysis of couple stress fluid flow through a porous medium. Thermal Science, 17, 1203–1214.

13. M. I. Ahmed, & H. A. Shehzad. (2008). Flow of couple stress fluids through porous media with thermal radiation and chemical reaction. Communications in Nonlinear Science and Numerical Simulation, 13, 2601–2612.

14. S. O. Adesanya, & O. D. Makinde. (2015). Convective heat transfer and entropy generation in couple stress fluid flows in porous channels. International Journal of Numerical Methods for Heat & Fluid Flow, 25, 376–393.

15. J. D. Anderson Jr. (1995). Computational fluid dynamics: The basics with applications. New York: McGraw-Hill.

16. F. Alcocer, & J. Smith. (2002). Effects of porous media on fluid flow. Journal of Fluid Mechanics, 456, 123–145.

17. T. Hayat, M. Sajid, & S. Asghar. (2002). Exact solutions for couple stress fluid flows through porous media. International Journal of Engineering Science, 40, 1059–1074.


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