Effect of chemical reaction and radiation absorption on the unsteady MHD free convection Couette flow in a vertical channel filled with porous materials

Abstract

This paper deals with effect of chemical reaction and radiation absorption on the unsteady MHD natural convection flow in a vertical channel filled with porous materials. One of the plate moves with a constant velocity in the direction of fluid flow while the other plate is stationary. A uniform magnetic field acts perpendicular to the porous plate, which absorbs the fluid with a suction velocity varying with time. The dimensionless governing equations for this problem are solved analytically using two-term harmonic and non-harmonic perturbation series method. The results of the analytical solutions for velocity, temperature and concentration profiles of both phases are graphically presented and discussed. It is noteworthy to mention that growing permeability and channel width parameter lead to increase in the fluid velocity profile. Results in the present work have been compared with other works in literature and an excellent agreement is found.

Appendix

$$\begin{aligned} m_1= & {} \frac{-Sc+\sqrt{Sc^2+4Sc\gamma }}{2} \quad m_2=\frac{-Sc-\sqrt{Sc^2+4Sc\gamma }}{2} \\ m_3= & {} \frac{-Sc+\sqrt{Sc^2+4Sc(n+\gamma )}}{2} \quad m_4=\frac{-Sc-\sqrt{Sc^2+4Sc(n+\gamma )}}{2} \\ m_5= & {} \frac{-Pr+\sqrt{Pr^2+4Pr\lambda }}{2} \quad m_6=\frac{-Pr-\sqrt{Pr^2+4Pr\lambda }}{2}\\ m_7= & {} \frac{-Pr+\sqrt{Pr^2+4Pr(n+\lambda )}}{2} \quad m_8=\frac{-Pr-\sqrt{Pr^2+4Pr(n+\lambda )}}{2}\\ m_9= & {} \frac{-1+\sqrt{1+4\left( M+\frac{1}{K}\right) }}{2} \quad m_{10}=\frac{-1-\sqrt{1+4\left( M+\frac{1}{K}\right) }}{2}\\ m_{11}= & {} \frac{-1+\sqrt{1+4\left( M+\frac{1}{K}+n\right) }}{2} \quad m_{12}=\frac{-1-\sqrt{1+4\left( M+\frac{1}{K}+n\right) }}{2}\\ A_1= & {} \frac{e^{m_1H}}{e^{m_2H}-e^{m_1H}} \quad A_2=1-A_1 \quad A_3=\frac{(A_5+A_6-1)e^{m_3H}-A_5e^{m_1H}-A_6e^{m_2H}}{e^{m_4H}-e^{m_3H}}\\ A_4= & {} 1-(A_3+A_5+A_6) \quad A_5=\frac{AA_1Scm_1}{m_1^2+Scm_1-Sc(\gamma +n)} \quad A_5=\frac{AA_2Scm_2}{m_2^2+Scm_2-Sc(\gamma +n)}\\ A_7= & {} \frac{(1-A_9-A_{10})e^{m_6H}+A_9e^{m_1H}+A_{10}e^{m_2H}}{e^{m_6H}-e^{m_5H}} \quad A_8=1-(A_7+A_9+A_{10})\\ A_9= & {} \frac{-PrA_1Q_1}{m_1^2+Prm_1-Pr\lambda } \quad A_{10}=\frac{-PrA_2Q_1}{m_2^2+Prm_2-Pr\lambda } \\ A_{11}= & {} \frac{(A_{13}\!+\!A_{14}\!+\!A_{15}\!{+}\!A_{16}\!+\!A_{17}\!+\!A_{18}\!-\!1)e^{m_8H}\!\!-\!A_{13}e^{m_1H}\!\!-\!A_{14}e^{m_2H}\!\!-\!A_{15}e^{m_3H}\!\!-\!A_{16}e^{m_4H}\!\!-\!A_{17}e^{m_5H}\!\!-\!A_{18}e^{m_6H}}{e^{m_7H}\!\!-\!e^{m_8H}}\\ A_{12}= & {} 1-(A_{11}+A_{13}+A_{14}+A_{15}+A_{16}+A_{17}+A_{18}) \end{aligned}$$

$$\begin{aligned} A_{13}= & {} \frac{-Pr(AA_9m_1+Q_1A_5)}{m_1^2+Prm_1-Pr(\lambda +n)} \quad A_{14}=\frac{-Pr(AA_{10}m_2+Q_1A_6)}{m_2^2+Prm_2-Pr(\lambda +n)}\\ A_{15}= & {} \frac{-PrA_3Q_1}{m_3^2+Prm_3-Pr(\lambda +n)} \quad A_{16}=\frac{-PrA_4Q_1}{m_4^2+Prm_4-Pr(\lambda +n)}\\ A_{17}= & {} \frac{-PrAA_7m_5}{m_5^2+Prm_5-Pr(\lambda +n)} \quad A_{18}=\frac{-PrAA_8m_6}{m_6^2+Prm_6-Pr(\lambda +n)} \\ A_{19}= & {} \frac{(A_{21}+A_{22}+A_{23}+A_{24}-u_p)e^{m_{10}H}-A_{21}e^{m_1H}-A_{22}e^{m_2H}-A_{23}e^{m_5H}-A_{24}e^{m_6H}}{e^{m_9H}-e^{m_{10}H}}\\ A_{20}= & {} u_p-(A_{19}+A_{21}+A_{22}+A_{23}+A_{24}) \quad A_{21}=\frac{-(GrA_9+GmA_1)}{m_1^2+m_1-(M+\frac{1}{K})}\\ A_{22}= & {} \frac{-(GrA_{10}+GmA_2)}{m_2^2+m_2-(M+\frac{1}{K})} \quad A_{23}=\frac{-GrA_7}{m_5^2+m_5-(M+\frac{1}{K})}\\ A_{24}= & {} \frac{-GrA_8}{m_6^2+m_6-(M+\frac{1}{K})} \\ A_{25}= & {} \frac{X_1+X_2+X_3}{e^{m_{11}H}-e^{m_{12}H}}\\ X_1= & {} (A_{27}+A_{28}+A_{29}+A_{30}+A_{31}+A_{32}+A_{33}+A_{34}+A_{35}+A_{36})e^{m_{12}H}\\ X_2= & {} -(A_{27}e^{m_1H}+A_{28}e^{m_2H}+A_{29}e^{m_3H}+A_{30}e^{m_4H}+A_{31}e^{m_5H})\\ X_3= & {} -(A_{32}e^{m_6H}+A_{33}e^{m_7H}+A_{34}e^{m_8H}+A_{35}e^{m_9H}+A_{36}e^{m_{10}H})\\ A_{26}= & {} -(A_{25}+A_{27}+A_{28}+A_{29}+A_{30}+A_{31}+A_{32}+A_{33}+A_{34}+A_{35}+A_{36})\\ A_{27}= & {} \frac{-(GrA_{13}+GmA_5+AA_{21}m_1)}{m_1^2+m_1-(M+\frac{1}{K}+n)} \quad A_{28}=\frac{-(GrA_{14}+GmA_6+AA_{22}m_2)}{m_2^2+m_2-(M+\frac{1}{K}+n)}\\ A_{29}= & {} \frac{-(GrA_{15}+GmA_3)}{m_3^2+m_3-(M+\frac{1}{K}+n)} \quad A_{30}=\frac{-(GrA_{16}+GmA_4)}{m_4^2+m_4-(M+\frac{1}{K}+n)}\\ A_{31}= & {} \frac{-(GrA_{17}+AA_{23}m_5)}{m_5^2+m_5-(M+\frac{1}{K}+n)} \quad A_{32}=\frac{-(GrA_{18}+AA_{24}m_6)}{m_6^2+m_6-(M+\frac{1}{K}+n)}\\ A_{33}= & {} \frac{-GrA_{11}}{m_7^2+m_7-(M+\frac{1}{K}+n)} \quad A_{34}=\frac{-GrA_{12}}{m_8^2+m_8-(M+\frac{1}{K}+n)}\\ A_{35}= & {} \frac{-AA_{19}m_9}{m_9^2+m_9-(M+\frac{1}{K}+n)} \quad A_{36}=\frac{-AA_{20}m_{10}}{m_{10}^2+m_{10}-(M+\frac{1}{K}+n)} \end{aligned}$$