Robin Boundary Effects in the Darcy–Rayleigh Problem with Local Thermal Non-equilibrium Model

Abstract

The contribution of Robin boundaries on the onset of convection in a horizontal saturated porous layer covered by a free surface on the top is investigated here. The saturated solid matrix is assumed in a regime where the temperature profile of the solid phase differs from a fluid one. Two energy equations are adopted as a consequence of the local thermal non-equilibrium model (LTNE), and four Biot numbers are arising out of the third kind of boundaries imposed on both surfaces. The dimensionless parameters H and $\gamma$ which rule the transition from local thermal equilibrium (LTE) to non-equilibrium one or vice versa are taken into account. The cases of equal and different Biot numbers have been considered beside the asymptotic limits of LTE and LTNE one. A linear stability analysis of the basic motionless state has been performed. The perturbation terms of the main steady flows are evaluated in the form of plane waves. The eigenvalue problem is solved either analytically or numerically depending on the temperature gradient of the fluid phase. The analytical solution is handled through a dispersion relation, while the numerical one is computed by the Runge–Kutta solver combined with the shooting method. The variation in Darcy–Rayleigh number and wave number is obtained with respect to Biot numbers for all resulting cases.

Appendix A : The Principle of Exchange of Instabilities for Limiting Case \(H\rightarrow 0\)

The eigenvalue problem obtained for the case characterized by no transfer of the heat is carried out between the two phases is

$$\begin{aligned}&{\psi }{''}-a^2 {\psi }+a \dfrac{1+\gamma }{\gamma } R{\theta } =0, \end{aligned}$$

(48a)

$$\begin{aligned}&{\theta }''-a^{2} {\theta }+a {\tilde{B}}_{f12}{\psi } +i {\omega _R} \dfrac{\chi +\gamma }{1+\gamma }\theta =0. \end{aligned}$$

(48b)

$$\begin{aligned}&z=0: \quad {\psi }(0)=0,\quad {\theta }'(0)+B_{f1} {\theta }(0)=0, \end{aligned}$$

(48c)

$$\begin{aligned}&z=1: \quad {\psi }'(1)=0, \quad {\theta }'(1)+B_{f2} {\theta }(1)=0. \end{aligned}$$

(48d)

Multiplying Eqs. (48a), (48b) by the complex conjugate quantities $\bar{\psi }$ and $\bar{\theta }$, respectively, gives rise to two complex resulting equations. These equations are integrated by part with the use of boundary conditions, namely

$$\begin{aligned}&-\int _0^1 | \psi ' |^2\ \hbox {d}z - a^2 \int _0^1 | \psi |^2\ \hbox {d}z + a \, \, R \, \dfrac{1+\gamma }{\gamma } \int _0^1 \theta \, {\overline{\psi }} \ \hbox {d}z=0, \end{aligned}$$

(49a)

$$\begin{aligned}&\quad B_{f1}|\theta (0)|^2 -B_{f2}|\theta (1)|^2 - \int _0^1 |\theta '|^2 \ \hbox {d}z - a^2 \int _0^1 |\theta |^2 \ \hbox {d}z \nonumber \\&\quad +a {\tilde{B}}_{f12} \int _0^1 \psi \, {\overline{\theta }} \ \hbox {d}z + i{\omega _R} \dfrac{\chi +\gamma }{1+\gamma } \int _0^1 |\theta |^2 \ \hbox {d}z= 0. \end{aligned}$$

(49b)

After multiplying Eq. (49a) by the parameter $\dfrac{-\gamma {{\tilde{B}}}_{f12}}{R(1+\gamma )}$, we can now add it to Eq. (49b) to obtain

$$\begin{aligned}&\dfrac{\gamma {{\tilde{B}}}_{f12}}{R(1+\gamma )} \left( \int _0^1 | \psi ' |^2\ \hbox {d}z + a^2 \int _0^1 | \psi |^2\ \hbox {d}z \right) \nonumber \\&\quad + B_{f1}|\theta (0)|^2 -B_{f2}|\theta (1)|^2 - \int _0^1 |\theta '|^2 \ \hbox {d}z - a^2 \int _0^1 |\theta |^2 \ \hbox {d}z + i{\omega _R} \dfrac{\chi +\gamma }{1+\gamma } \int _0^1 |\theta |^2 \ \hbox {d}z= 0 .\nonumber \\ \end{aligned}$$

(50)

The two parts of real and imaginary in Eq. (50) have to be independently equal to zero to satisfy the condition of

$$\begin{aligned} \omega _R \int _0^1 |\theta |^2 \ \hbox {d}z= 0 . \end{aligned}$$

(51)

Equation (51) defines two different assumptions. The first one is $\theta =0$ which means no secondary flow exists in the basic state. This condition cannot be acceptable because it would imply a contradiction with what we are looking for. Thus, this result supports the validity of the second assumption which is $\omega _R=0$. Consequently, we can assure that the eigenvalue problem of this limiting cases holds the principle of exchange of instabilities.