Study of Rayleigh-Bénard Convection of a Newtonian Nanoliquid in a Porous Medium Using General Boundary Conditions

Abstract

In the paper we make a linear stability analysis of Rayleigh-Bénard convection (RBC) in a Newtonian, nanoliquid-saturated porous medium. Single-phase model is used for nanoliquid description and values of thermophysical quantities concerning ethylene glycol-copper and the saturated porous medium it occupies are calculated using mixture theory or phenomenological relations. The study is carried out using general boundary conditions on the velocity and temperature. The Galerkin method is used to obtain the critical eigen value. The results of free-free, rigid-free and rigid-rigid isothermal/adiabatic boundaries are obtained from the present study by considering appropriate limits. The results of the limiting cases of the present study are in excellent agreement with those observed in earlier investigations. This problem is an integrated approach to studying Rayleigh-Bénard convection covering 34 different boundary combinations.

Appendix

The eigen functions F(Z) and G(Z) of the boundary value problem

$$\begin{aligned} \left[ a_{1} \Lambda \left( \dfrac{d^{2}}{dZ^{2}}-\nu ^{2}\right) ^{2}-a_{1} \sigma ^{2} \left( \dfrac{d^{2}}{dZ^{2}}-\nu ^{2}\right) \right] F(Z)+a_{1}^{2}\nu \text { Ra } G(Z)=0 \end{aligned}$$

(12)

$$\begin{aligned} -\nu F(Z)+a_{1}M\left( \dfrac{d^{2}}{dZ^{2}}-\nu ^{2}\right) G(Z)=0 \end{aligned}$$

(13)

subject to the boundary conditions

$$\begin{aligned} \dfrac{d^{2}F}{dZ^{2}}-Da_{sl}F= & {} \dfrac{dG}{dZ}-Bi_{l}G=0 \ \text {at} \ Z=\dfrac{-1}{2},\end{aligned}$$

(14)

$$\begin{aligned} \dfrac{d^{2}F}{dZ^{2}}+Da_{su}F= & {} \dfrac{dG}{dZ}+Bi_{u}G=0 \ \text {at } Z=\dfrac{1}{2} \end{aligned}$$

(15)

are chosen in the form

$$\begin{aligned} F(Z)= & {} Z^4+a Z^3+b_1Z^2+c_1Z+d_1,\end{aligned}$$

(16)

$$\begin{aligned} G(Z)= & {} Z^4+b_2Z^2+c_2Z+d_2. \end{aligned}$$

(17)

Fourth degree polynomials are chosen for F(Z) and G(Z) keeping in mind the order of the differential equations in Eqs. (12) and (13). The constants $a, b_i, c_i$ and $d_i, i=1,2$ are determined such that the eigen functions F(Z) and G(Z) are mutually orthogonal in the domain R=$\lbrace (X,Z)/ X\in [0,1]$ and $Z\in \left[ -\frac{1}{2},\frac{1}{2}\right] \rbrace$ , i.e., $\int _{-\frac{1}{2}}^{\frac{1}{2}}F(Z)\ G(Z) \ dZ=0$ and these functions satisfy the boundary conditions (14)–(15). The quantities $Da_{sl}$ and $Da_{su}$ are slip Darcy numbers at lower and upper plates respectively, $\text {Bi}_{l}$ and $\text {Bi}_{u}$ are Biot numbers at lower and upper plates respectively. From the above considerations the constants are found as follows:

$$\begin{aligned} a= & {} \dfrac{2 \text {Da}_{\text {su}}-2 \text {Da}_{\text {sl}}}{\text {Da}_{\text {sl}}\,(\text {Da}_{\text {su}}+4)+4 (\text {Da}_{su}+3)}, \; b_1=-\dfrac{(\text {Da}_{\text {sl}}+6) (\text {Da}_{\text {su}}+6)}{2 \text {Da}_{\text {sl}}\,(\text {Da}_{\text {su}}+4)+8 (\text {Da}_{su}+3)},\\ c_1= & {} \dfrac{\text {Da}_{\text {sl}}-\text {Da}_{\text {su}}}{2 \text {Da}_{\text {sl}}\,(\text {Da}_{\text {su}}+4)+8 (\text {Da}_{\text {su}}+3)},\; d_1=\dfrac{\text {Da}_{\text {sl}}\,(\text {Da}_{\text {su}}+8)+8 \text {Da}_{\text {su}}+60}{16 \text {Da}_{\text {sl}}\,(\text {Da}_{\text {su}}+4)+64 (\text {Da}_{\text {su}}+3)},\\ b_2= & {} -\frac{16 d_{2} ({\text {Bi}}_l {\text {Bi}}_u+ {\text {Bi}}_l+ {\text {Bi}}_u)+ {\text {Bi}}_l \text {Bi}_u+5 \text {Bi}_l+5 \text {Bi}_u+16}{4 \left( \text {Bi}_l \text {Bi}_u+3 \text {Bi}_l+3 \text {Bi}_u+8\right) },\\ c_2= & {} -\dfrac{\text {Bi}_l-\text {Bi}_u-16 d_{2} \text {Bi}_l+16 d_{2} \text {Bi}_u}{4 \left( \text {Bi}_l \text {Bi}_u+3 \text {Bi}_l+3 \text {Bi}_u+8\right) }.\\ \end{aligned}$$

The constant $d_{2}$ varies according to the boundary combination.