Entropy Generation Analysis in a Vertical Porous Channel with Navier Slip and Viscous Dissipation

Abstract

The intension of this paper is to investigate the effects of Navier slip and buoyancy force on the entropy in a vertical generation porous channel with suction/injection. This problem is solved analytically by perturbation technique. Closed form solutions are obtained for the fluid velocity and the temperature. The leads of slip parameter, injection/suction Reynolds number, Peclet number and Brinkmann number on the fluid velocity, temperature profiles, Bejan number, and rate of entropy generation are showed graphically and quantitatively discussed.

Appendix

$$\begin{aligned} K_{2} & = \frac{{K_{1} }}{Re},\quad K_{1} = Gr_{1} - K,\quad Gr_{1} = \frac{Gr}{{\left( {\exp (Pe) - 1} \right)}},\quad Gr_{2} = \frac{{Gr_{1} }}{{Pe\left( {Pe - Re} \right)}}, \\ J & = 1 - \left( {\beta_{2} Re} \right),\quad a_{1} = 1 - \beta_{1} Re,\quad a_{2} = \beta_{1} K_{2} + \beta_{1} PeGr_{2} - Gr_{2} , \\ b_{1} & = \exp (Re)\left( {1 - \left( {\beta_{2} \,Re} \right)} \right), \\ b_{2} & = \beta_{2} K_{2} + \beta_{2} PeGr_{2} \exp (Pe) - Gr_{2} \exp (Pe) - K_{2} , \\ d_{1} & = 2K_{2} PeC_{2} ,\quad d_{2} = - 2K_{2} PeGr_{2} ,\quad d_{3} = - Re^{2} C_{2}^{2} , \\ d_{4} & = - Pe^{2} Gr_{2}^{2} ,\quad d_{5} = 2RePeGr_{2} C_{2} ,\quad d_{6} = - K_{2}^{2} \\ \end{aligned}$$

$$\begin{aligned} E_{1} & = \frac{{d_{1} }}{{Re\left( {Re - Pe} \right)}},\quad E_{2} = \frac{{d_{2} }}{Pe},\quad E_{3} = \frac{{d_{3} }}{{2Re\left( {2Re - Pe} \right)}},\quad E_{4} = \frac{{d_{4} }}{{2Pe^{2} }}, \\ E_{5} & = \frac{{d_{5} }}{{Re\left( {Re + Pe} \right)}},\quad E_{6} = - \frac{{d_{6} }}{Pe},\quad F = E_{1} + E_{3} + E_{4} + E_{5} ,\quad H = 1 - \left( {\beta_{1} Re} \right) \\ I & = - \frac{{GrC_{3} \beta_{1} }}{Re} + \frac{{GrC_{4} }}{{Pe\left( {Pe - Re} \right)}}\left( {Pe\beta_{1} - 1} \right) + \frac{{GrE_{1} \beta_{1} }}{Re} + \frac{{GrE_{3} }}{Re}\left( {\beta_{1} - \frac{1}{{2Re}}} \right) \\&\quad+ \frac{{GrE_{2} }}{{Pe\left( {Pe - Re} \right)}}\left[ \begin{aligned} & \beta_{1} + \frac{2Pe - Re}{{Pe\left( {Pe - Re} \right)}} \\&- \frac{{\beta_{1} \left( {2Pe - Re} \right)}}{Pe - Re} \\ \end{aligned} \right] + \frac{{GrE_{4} }}{{2Pe\left( {2Pe - Re} \right)}}\left( {2Pe\beta_{1} - 1} \right) \\&\quad+ \frac{{GrE_{5} }}{{Pe\left( {Pe + Re} \right)}}\left[ {\beta_{1} \left( {Re + Pe} \right) - 1} \right] - \frac{{GrE_{6} \beta_{1} }}{{Re^{2} }} \\ \\ \end{aligned}$$

$$\begin{aligned} K & = \frac{{GrC_{3} }}{Re}\left( {1 - \beta_{2} Re} \right) + \frac{{GrC_{4} \,{ \exp }(Pe)}}{{Pe\left( {Pe - Re} \right)}}\left( {Pe\beta_{2} - 1} \right) + \frac{{GrE_{1} \,{ \exp }(Re)}}{Re}\left( {\beta_{2} + Re\beta_{2} - 1} \right) \\ & \quad + \frac{{GrE_{3} \, { \exp }\left( {2Re} \right)}}{{2Re^{2} }}\left( {2\beta_{2} Re - 1} \right) + \frac{{GrE_{4}\, { \exp }\left( {2Pe} \right)}}{{2Pe\left( {2Pe - Re} \right)}}\left( {2Pe\beta_{2} - 1} \right) \\ & \quad + \frac{{GrE_{2} \,{ \exp }(Pe)}}{{Pe\left( {Pe - Re} \right)}}\left[ {\beta_{2} + \beta_{2} Pe - 1 + \frac{2Pe - Re}{{Pe\left( {Pe - Re} \right)}} - \frac{{\beta_{2} \left( {2Pe - Re} \right)}}{Pe - Re}} \right] \\ & \quad + \frac{{GrE_{5}\, { \exp }\left( {Re + Pe} \right)}}{{Pe\left( {Pe + Re} \right)}}\left[ {\beta_{2} \left( {Re + Pe} \right) - 1} \right] \\ & \quad + \frac{{GrE_{6} }}{Re}\left( {\frac{1}{2} - \beta_{2} } \right) + \frac{{GrE_{6} }}{{Re^{2} }}\left( {1 - \beta_{2} } \right) \\ \end{aligned}$$

$$\begin{aligned} G & = E_{1} \exp (Re) + E_{2} \exp (Pe) + E_{3} \exp (2Re) \\ & \quad + E_{4} \exp (2Pe) + E_{5} \exp \left( {Re + Pe} \right) + E_{6} \\ C_{1} & = \frac{{a_{1} b_{2} - a_{2} b_{1} }}{{b_{1} - a_{1} }},\quad C_{2} = \frac{{a_{2} - b_{2} }}{{b_{1} - a_{1} }},\quad C_{3} = \frac{{G - F\left( {\exp (Pe)} \right)}}{\exp (Pe) - 1}, \\ C_{4} & = \frac{F - G}{\exp (Pe) - 1},\quad C_{5} = \frac{{HK - JI\left( {\exp (Re)} \right)}}{\exp (Re) - H},\quad C_{6} \frac{I - K}{\exp (Re) - H} \\ \end{aligned}$$