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

Studies related to channel flow and heat transfer characteristics of couple-stress fluids not only present theoretical problem challenges, but also find several applications in many industrial processes such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of metallic plates in a bath, colloidal solutions, etc. These fluids are capable of describing various types of lubricants, blood and suspension fluids. Stokes [1] introduced the theory of couple stress fluids. The main feature of couple stresses is to introduce a size dependent effect. Classical continuum mechanics neglects the size effect of material particles within the continua. However, in some important cases such as fluid flow with suspended particles, this cannot be true and a size dependent couple-stress theory is needed. The spin field due to micro-rotation of freely suspended particles sets up an anti-symmetric stress, known as couple-stress, and thus forms a couple-stress fluid. A review of couple stress fluid dynamics was reported by Stokes [2]. Srinivasacharya and Srikanth [3] studied the effect of couple stresses on the flow in a constricted annulus. Moreover, the analysis of convective transport in a porous medium with the inclusion of non-Darcian effects has also been a matter of study in recent years. The inertia effect is expected to be important at a higher flow rate and it can be accounted for through the addition of a velocity squared term in the momentum equation, which is known as the Forchheimer’s extension of Darcy’s law. A detailed review of convective heat transfer in Darcy and non-Darcy porous medium can be found in the book by Nield and Bejan [4]. Postelnicu [5] investigated the combined effects of magnetic fields, thermal-diffusion and diffusion-thermo on heat and mass transfer by natural convection from vertical surfaces in porous media. Other studies on flow and heat transfer in porous media can be found in [6,7,8]. Meanwhile, the current trend of analysing flow and heat transfer is to apply a Second Law (of Thermodynamics) analysis, and its design-related concept of entropy generation minimization (EGM). Entropy generation is associated with thermodynamic irreversibility, which presents in all flow and heat transfer processes. Bejan [9] pioneered theoretical work on entropy generation in flow systems. He showed that by minimizing the entropy, the efficiency of a thermal system could be improved. Thereafter, several authors [10,11,12] have analyzed different problems to study the entropy effects on thermal systems and to find ways to minimize them. Mahmud and Fraser [13] applied the Second Law analysis to fundamental convective heat transfer problems. Tasnim et al. [14] examined entropy generation rate in an isothermal porous two dimensional channel with magnetic field effects. Chauhan and Kumar [15] numerically analysed the heat transfer and inherent irreversibility in a compressible fluid flow through a channel partially filled with porous medium. Eegunjobi and Makinde [16] studied the combined effect of buoyancy force and Navier slip on entropy generation in a vertical porous channel. Sheng et al. [17] investigated systematically the effects of Rayleigh number, curvature of annulus and Prandtl number on flow pattern, temperature distribution and entropy generation for natural convection inside vertically concentric annuli with the aid of the lattice Boltzmann method. Sheng [18] investigated for the first time, the entropy generation of double-diffusive convection in the presence of rotation.

To the best knowledge of the authors, the buoyancy effect on the inherent irreversibility in a couple stress fluid flows through a vertical channel filled with saturated porous media has not been reported yet in the literature. The main objective of this present study is to theoretically analyze the entropy generation rate in a couple stress fluid flows through a vertical channel filled with saturated porous media in the presence of buoyancy force. The mathematical formulation of the problem is established in sections two and three. Using a shooting method coupled with a Runge-Kutta Fehlberg integration scheme the boundary value problem is tackled numerically. Both numerical and graphical results for velocity, temperature, entropy generation rate and Bejan number are presented and discussed quantitatively with respect to various parameters embedded in the system in section four. Finally, in Section 5, we give the concluding remarks.

Consider the steady flow of an incompressible viscous couple stress fluid through a vertical channel filled with porous media as depicted in Figure 1, where the left channel wall is placed at temperature T₀ while the right wall is placed at temperature T_w such that T₀ < T_w.

Assuming Brinkman–Forchheimer flow model, the governing equations of momentum and energy under Boussinesq’s approximation are given as [1,8,12,16]: with: where u is the axial velocity, h is the channel width, μ is the dynamic viscosity, ρ is the fluid density, T is the fluid temperature, c_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:

Substituting Equation (5) into Equations (1)–(4), we obtain the dimensionless governing equations as: with: where Gr is the Grashof number due to buoyancy effect, Ec is the Eckert number, Pr is the Prandtl number, λ is the couple stress parameter, A is the axial pressure gradient parameter, S is the porous media shape factor parameter and M the second order porous media resistance parameter. Other physical quantities of practical significance in this work are the skin friction coefficient C_f and the local Nusselt number Nu, which are expressed as: where τ_w is the wall shear stress and q_w is the heat flux at the channel walls given by:

The set of Equations (6)–(7) together with the boundary conditions in Equations (8)–(9) form a two point boundary value problem and are solved numerically using a shooting algorithm together with a Runge-Kutta Fehlberg integration scheme [19]. This method involves transforming Equations (6)–(9) into a set of initial value problems which contain some unknown initial values that need to be determined by guessing, after which a Runge-Kutta Fehlberg integration scheme is employed to solve the set of initial valued problems until the given boundary conditions are satisfied. The entire computation procedure is implemented using a program written and carried out using the MAPLE computer language. From the process of numerical computation, the fluid velocity and temperature, the skin friction coefficient and the Nusselt number, are determined for a given set of parameter values.

The inherent irreversibility in channel flow of a couple stress fluids arise due to the exchange of energy and momentum, within the fluid and at solid boundaries. Consequently, entropy production may occur as a result of fluid friction and heat transfer in the direction of finite temperature gradients. Following, Bejan [9,10], the volumetric rate of entropy generation can be expressed as:

The first term in Equation (12) describes the heat transfer irreversibility, the second and third terms represent irreversibility due to fluid friction and couple stress effect respectively, the fourth and fifth account for the irreversibility due to the presence for porous media. Using Equation (5), the dimensionless form of local entropy generation rate in Equation (12) is given as: where is the temperature difference parameter and Br = EcPr is the Brinkmann number. The Bejan number Be is defined as: where N_s = N₁ + N₂, (Heat transfer Irreversibility), (Irreversibility due to viscous dissipation, couple stresses and porous media), (Irreversibility ratio).

From Equation (14), it is obvious that the Bejan number (Be) ranges from . While Be = 0 represents the limit case of fluid friction, couple stress and porous media dominated irreversibility, Be = 1 corresponds to the limit case of heat transfer dominated irreversibility. The contribution of both heat transfer and fluid friction to entropy production in the flow system is the same when Be = 0.5.

It is difficult to study the influence of all parameters involved in the present problem on the flow and thermal field along with entropy generation characteristics. Therefore, a selected set of graphical results is presented in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29 and Figure 30 that will give a good understanding of the influence of different parameters on the velocity, temperature, entropy generation profiles, skin friction, Nusselt number and Bejan number.

We analyze the entropy generation in couple stress fluid flows through a vertical channel filled with saturated porous media. The nonlinear governing equations are solved numerically using shooting methods together with a Runge-Kutta Fehlberg integration scheme. We discussed the behaviours of the velocity profile, temperature profile, skin friction, Nusselt number, entropy generation profile and Bejan number for different value of parameters in the governing equations. From the discussion, we may conclude the following:

• 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.