Onset of thermal convection in a porous layer with mixed boundary conditions

Abstract

Stability of a horizontal porous layer saturated by a fluid under conditions of heating from below is revisited. The aim is to extend the model for the boundary walls constraining the velocity and temperature by means of mixed, or third-kind, conditions. This model proved to be appropriate to describe departure from perfectly conducting or uniform heat flux conditions for the temperature, and impermeable or perfectly permeable conditions for the velocity. Linearised equations for general normal modes perturbing the basic rest state are obtained. The principle of exchange of stabilities is proven. The perturbation equations are solved analytically to deduce the dispersion relation at neutral stability, as well as to draw neutral stability curves and to yield the critical values of the Darcy–Rayleigh number and of the wave number for the onset of the instability. Specially interesting regimes are analysed in detail including the degenerate case, where the velocity and temperature boundary conditions feature the same mixing parameters, and the symmetric case, where identical conditions are prescribed on both boundaries.

Appendix: Exchange of stabilities

We multiply Eq. (12a) by the complex conjugate of $f$, namely $\bar{f}$, and integrate with respect to $z$ in the interval $[0,1]$. Likewise, we multiply Eq. (12b) by the complex conjugate of $h$, namely $\bar{h}$, and integrate with respect to $z$ in the interval $[0,1]$. Thus, we obtain

$$\begin{aligned}&\int _0^1 \bar{f}\!f^{{\prime }{\prime }}\ \mathrm{d}z - k^2 \int _0^1 |f|^2\ \mathrm{d}z + k^2 R \int _0^1 \bar{f} h\ \mathrm{d}z = 0 , \end{aligned}$$

(45a)

$$\begin{aligned}&\int _0^1 \bar{h} h^{{\prime }{\prime }}\ \mathrm{d}z - (k^2 + \gamma ) \int _0^1 |h|^2\ \mathrm{d}z + \int _0^1 \bar{h}f\ \mathrm{d}z = 0 . \end{aligned}$$

(45b)

Integration by parts, by taking into account Eq. (12c), yields

$$\begin{aligned}&\int _0^1 \bar{f}\!f^{{\prime }{\prime }}\ \mathrm{d}z = \left. \bar{f}\!f^{\prime } \right| _0^1 - \int _0^1 |f^{\prime }|^2 \mathrm{d}z = - \frac{1}{b_2} |f(1)|^2 - \frac{1}{b_1} |f(0)|^2 - \int _0^1 |f^{\prime }|^2 \mathrm{d}z , \end{aligned}$$

(46a)

$$\begin{aligned}&\int _0^1 \bar{h} h^{{\prime }{\prime }} \mathrm{d}z = \left. \bar{h} h^{\prime } \right| _0^1 - \int _0^1 |h^{\prime }|^2\ \mathrm{d}z = - \frac{1}{a_2} |h(1)|^2 - \frac{1}{a_1} |h(0)|^2 - \int _0^1 |h^{\prime }|^2 \mathrm{d}z . \end{aligned}$$

(46b)

This proves that both integrals

$$\begin{aligned} \int _0^1 \bar{f}\!f^{{\prime }{\prime }} \mathrm{d}z , \qquad \int _0^1 \bar{h} h^{{\prime }{\prime }} \mathrm{d}z , \end{aligned}$$

are real and non-positive.

We can now sum Eq. (45a) with $- k^2 R$ times the complex conjugate of Eq. (45b), so that we obtain

$$\begin{aligned} \int _0^1 \bar{f}\!f^{{\prime }{\prime }} \mathrm{d}z - k^2 \int _0^1 |f|^2 \mathrm{d}z - k^2 R \int _0^1 \bar{h} h^{{\prime }{\prime }} \mathrm{d}z + k^2 R \left( k^2 + \gamma \right) \int _0^1 |h|^2 \mathrm{d}z = 0 . \end{aligned}$$

(47)

Both the real part and the imaginary part of the right-hand side of Eq. (47) must vanish. A vanishing imaginary part means

$$\begin{aligned} k^2 R \hbox {Im}(\gamma ) \int _0^1 |h|^2 \mathrm{d}z = 0 . \end{aligned}$$

(48)

On account of Eq. (48), either $\hbox {Im}(\gamma )=0$ or $h$ is identically zero. The latter condition is ruled out, because Eq. (12b) would imply an identically vanishing $f$, so that the basic state is actually not perturbed. Therefore, the conclusion is that $\hbox {Im}(\gamma )=0$.