The flow in microchannel is an active area of study in the recent days due to its significance in the field of science and engineering. Fluid movement in microchannel

Makinde and Eegunjobi

A theoretical analysis of the Casson fluid flow along an inclined micro-annular channel is done by Idowu et al.

Alireza

We found good literature in the study of Williamson fluid flow along different channels with the influence of various parameters. Recently Gireesha et al.

But so far, there is no study is found to analyse the entropy generation in the mixed convection of Williamson fluid along the micro porous channel with the influence of chemical reaction, variable thermal conductivity. So in order to fill this research gap in this present article we aimed to analyse the entropy generation for the flow of Williamson fluid along the inclined micro porous channel with the influence of chemical reaction which includes activation energy in it and by taking thermal conductivity as a variable property.

Convection of Williamson fluid is along an inclined microchannel with the inclination ‘α’ is taken for the study and the two plates of the microchannel is separated by distance ‘a’ and the temperature of the plates are at distinct temperatures

Following are the assumptions we adopted for our present study,

The flow is taken as viscous, laminar and steady.

The thermal conductivity of the fluid taken as variable property whereas the density and viscosity of the fluid are taken as constant.

The flow is carried out through the porous medium with the influence of magnetic field.

Under the above mentioned conditions the momentum, temperature distribution and chemical reaction equations in dimensional form takes the form as below

Employed boundary conditions for the considered flow model in dimensional form,

Thermal conductivity of the Williamson fluid is dependent on temperature and varied throughout the flow is denoted as:

Where ε - variable thermal conductivity parameter.

Dimensionless variables used for the non-dimensionalization

Dimensionless form of the governing equations (1, 2) using (6) are

Boundary conditions in dimensionless form,

where Re =

Entropy is the evaluation of the irreversibility in the model. To obtain the efficient flow model it necessary to minimize the entropy production in the model. Entropy analysis can be done second law of thermodynamics. In present considered flow model magnetic field, permeability, viscous dissipation and heat transfer are affecting the entropy production in the flow.

The equation of irreversibility takes the form as:

On non-dimensionalising, equation of irreversibility of the system reduces to:

Above equation can be written as,

The ratio of entropy production due to heat transfer to the overall entropy production is denoted as Bejan number, it is defined as:

The reduced system of nonlinear equations (9), (10) and (11) and the corresponding equations of the employed boundary conditions for the considered flow are (12), (13) and (14) to solve these equations we used the Bvp4c Technique which involves the finite difference method. By this, we transform the ODE, by assigning new variables.

Assuming,

Transforming the equations (9), (10) and (11) using the above assumptions we can write,

and the boundary conditions can be written as,

The considered flow model is solved numerically and the solutions for velocity, temperature, entropy generation and Bejan number are obtained and the influence of chemical reaction parameter, variable thermal conductivity parameter, Biot number, mixed convection parameter, concentration buoyancy parameter, and inclination are discussed using the graphs. The results calculated by maintaining constant values as: P=1, Br=0.7, λ=2, Bi=1, We =0.5, M=1, α=π/4, σ =0.5, ϕ=1, ε=0.1,

The Velocity of the considered fluid shows the enhanced nature for the higher inclinations of the channel and similar nature is observed for the thermal field this is due to the increase in the applied force on the flowing fluid. In figure 9a we are noticing that entropy production at both the walls exaggerates with an increase in α. In figure 9b the Bejan number profile shows the diminishing nature this is due to the dominance of the total irreversibility with the irreversibility due to heat transfer. To minimize the entropy generation in the system inclination has to be maintained at a lower rate.

y Makinde and Eegunjobi Gireesha et al. Present study 0 0.0000000 0.0000000 0.000000 0.2 0.0711487 0.0711487 0.071149 0.4 0.0963903 0.1137694 0.113770 0.6 0.1215460 0.1215460 0.121550 0.8 0.0867637 0.0867637 0.086764 1 0.0000000 0.0000000 0.000000

the display of the numerical solution obtained for the velocity for the limiting case. The obtained results are in good agreement with theexisting studies Gireesha et al.

The entropy generated in the mixed convection of Williamson fluid along an inclined porous microchannel with the influence of variable thermal conductivity and chemical reaction in the presence of magnetic field is studied. The major outcomes of the present study are, porous

• ξ reduces the temperature and velocity of the fluid.

• Concentration buoyancy parameter, Br, ε, We, Bi and

• Magnified values of ε, λ, Bi and inclination of the channel enhance the entropy production.

• Increase in We shows the dual trend for the entropy production and Bejan number.

• Due to the dominant of the entropy production of the system to the entropy production due to heat transfer, Bejan number shows the declining trend for the λ. channel inclination.

• Bejan number shows upward nature for Bi and variable thermal conductivity parameter.

a - distance between the plates

Bi - Biot number;

Br - Brinkman number

g - acceleration due to gravity (m/

Gr - Grashof number;

K - thermal conductivity;

M - Hartman number;

P - pressure (kg

T - temperature(K);

u - velocity (m/s);

We - Wessenberg number;

λ - mixed convection parameter;

θ - dimensionless temperature;

α - channel inclination;

ψ - heat generation/absorption parameter;

σ - permeability parameter;