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This study deals with the combined effects of Navier Slip, Convective cooling, variable viscosity, and suction/injection on the entropy generation rate in an unsteady flow of an incompressible viscous fluid flowing through a channel with permeable walls. The model equations for momentum and energy balance are solved numerically using semi-discretization finite difference techniques. Both the velocity and temperature profiles are obtained and utilized to compute the entropy generation number. The effects of key parameters on the fluid velocity, temperature, entropy generation rate and Bejan number are depicted graphically and analyzed in detail.

The cornerstone in the field of heat transfer and thermal design is the second law analysis and its design-related concept of entropy generation minimization. The foundation of knowledge of entropy production goes back to Clausius and Kelvin’s studies on the irreversible aspects of the Second Law of Thermodynamics [

The work of Chinyoka [

The pioneering works of Aziz [

The objective of the present work is to study the unsteady flow of a reactive variable viscosity fluid between two parallel porous plates acted upon by nonconstant pressure. Both the lower and upper walls of the channel are subjected to asymmetric convective heat exchange with the ambient and allow for uniform suction/injection in the transverse direction. The mathematical formulation of the problem is established in

We consider unsteady flow of an incompressible viscous fluid through a channel with permeable walls, see

Schematic diagram of the problem.

Under these assumptions, the governing equations for the momentum and energy balance in one dimension can be written as follows:

The appropriate initial and boundary conditions are:

Substituting Equation (

In the

Following Bejan [

The first term in Equation (

We then define the Bejan number as

Our numerical algorithm is based on the semi-implicit finite difference scheme [

The discretization of the governing equations is based on a linear Cartesian mesh and uniform grid on which finite-differences are taken. We approximate both the second and first spatial derivatives with second-order central differences.

The equations corresponding to the first and last grid points are modified to incorporate the boundary conditions. The semi-implicit scheme for the velocity component reads:

In Equation (

The solution procedure for

The equation for

Unless otherwise stated, we employ the parameter values:

These will be the default values in this work. In the succeeding graphics, if any of these parameter values is not explicitly mentioned, it will be understood that such parameters take on the default values.

We display the transient solutions in

Transient and steady state profiles.

The steady flow case with

with boundary conditions given as

The analytic solution reads:

A comparison plot of the exact solution and the corresponding numerical solution (obtained at

Analytic and numerical solution: (

The response of the velocity and temperature to varying values of the Prandtl number (

Larger values of the Prandtl number correspondingly increase the strength of the heat sources in the temperature equation and hence in turn increases the overall fluid temperature as clearly illustrated in

Effects of the Prandtl number,

The response of the velocity and temperature to varying values of the Reynolds number (

Effects of the Reynolds number,

Larger values of the Reynolds number correspond to higher suction/injection strength and hence clearly decreases the axial fluid velocity as illustrated in

The response of the velocity and temperature to varying values of the viscosity parameter (

Effects of the viscosity parameter,

As expected, both velocity and temperature increase with increasing viscosity parameter (

The effects of the wall Biot numbers on the velocity and temperature profiles is illustrated in

Effects of the lower wall Biot number

Effects of the upper wall Biot number

As seen from the temperature boundary conditions (12), higher Biot numbers mean correspondingly higher degrees of convective cooling at the channel walls, thus leading to lower temperatures at the channel walls and hence also in the bulk fluid. The overall temperature profiles thus decrease with increasing Biot number as the bulk fluid continually adjusts to the lower wall temperatures.

The response of the velocity and temperature to varying values of the Eckert number (

Effects of the Eckert number, (

The effects of the Eckert number are similar to those for the Prandtl number. The effects of the wall slip parameters on the velocity and temperature profiles is illustrated in

Effects of the lower wall slip parameter,

Effects of the upper wall slip parameter,

As expected, an increase in the slip parameters correspondingly increases the wall (and hence also the bulk) fluid velocity.

In this section, we plot the entropy generation rate

Variation of entropy generation rate with

Variation of Bejan number with

Variation of entropy generation rate with

Variation of entropy generation rate with

Variation of entropy generation rate with

Variation of entropy generation rate with

Variation of entropy generation rate with

Variation of entropy generation rate with

Variation of entropy generation rate with

Variation of entropy generation rate with

In this section, we plot the Bejan number

Away from the wall (

Variation of Bejan number with

Variation of Bejan number with

Variation of Bejan number with

Variation of Bejan number with

Variation of Bejan number with

Variation of Bejan number with

Variation of Bejan number with

Variation of Bejan number with

We computationally investigate the combined entropy generation rate in an unsteady porous channel flow with Navier slip subjected to asymmetrical convective boundary conditions. A major observation in the current work is the reduction in heat generation due to the presence of uniform suction/injection. We also notice that due to the nature of the coupling of the source terms, the fluid velocity and temperature either both increase or both decrease together. We have also demonstrated computationally that parameters that increase the entropy generation rate will correspondingly decrease the Bejan number and vice versa. In particular wall subjected to higher slip (