The investigation of Darcy-Rayleigh convection with uniform heat flux Model A

Abstract

The paper investigates the effect of uniform heat flux Model A on Darcy-Rayleigh convection with a free surface on the top. The assumption of Model A as thermal boundary conditions is used at the lower surface, while the upper one is only fixed to isothermal conditions. Both approaches of energy equations have been taken into consideration to account for the local thermal non-equilibrium. Linear stability analysis has been performed in the condition where the perturbation terms are evaluated in the form of plane waves. Otherwise, the eigenvalue problem is either dealt with the analytical solution by a dispersion relation or numerically by employing the Range–Kutta solver with the shooting method. According to the results, the transition of the critical values from isothermal to isoflux boundary conditions may emerge in one of the local thermal non-equilibrium assumptions. On the other hand, infinite values may appear in the critical Rayleigh number at the local thermal equilibrium regime. Overall, the thermal instability becomes extremely dominant when the configuration approximates the model of gas saturated with metal foam which is one of the cases of the local thermal non-equilibrium.

Appendix

A: The principle of exchange of stabilities: the limiting case of $H\rightarrow \infty$

The principle of exchange of stabilities deals with the case of LTE model whose governing equations are defined as

$$\begin{gathered} \hat{\psi }^{\prime\prime} - a^{2} \hat{\psi } + a\frac{\gamma }{{1 + \gamma }}R_{\gamma } \theta _{{\text{m}}} = 0, \hfill \\ \theta _{{{\text{m}}^{{\prime \prime }} }} - a^{2} \theta _{{\text{m}}} + a\hat{\psi } + i\omega _{{\text{R}}} \frac{{\lambda + \gamma }}{{1 + \gamma }}\theta _{{\text{m}}} = 0. \hfill \\ \end{gathered}$$

(36)

The boundary conditions related to LTE model are

$$\begin{aligned} z=0: \, \hat{\psi }=0, \,{\theta }'_{\mathrm{{m}}}=0, \nonumber \\ z=1: \, \hat{\psi }'=0, \, {\theta }_{\mathrm{{m}}}=0. \end{aligned}$$

(37)

Using the boundary conditions Eqs. (37) in the integration by part of Eqs. (36) after multiplying them by their complex conjugated quantities $\bar{\hat{\psi }}$ and $\bar{\theta }_\mathrm{{m}}$ gives rise to

$$\begin{aligned}&-\int _0^1 | \hat{\psi }' |^2\ \mathrm{{d}}z - a^2 \int _0^1 |\hat{\psi }|^2\ \mathrm{{d}}z \nonumber \\&+ a \, \dfrac{\gamma }{1+\gamma }{R_\gamma } \, \int _0^1 \theta _\mathrm{{m}} \, \overline{\hat{\psi }} \ \mathrm{{d}}z=0, \end{aligned}$$

(38a)

$$\begin{aligned}&- \int _0^1 |\theta '_\text{m}|^2 \ \mathrm{{d}}z - a^2 \int _0^1 |\theta _\mathrm{{m}}|^2 \ \mathrm{{d}}z + a \ \int _0^1 \hat{\psi }\, \overline{\theta }_\mathrm{{m}} \ \mathrm{{d}}z \nonumber \\&+i{\omega _\mathrm{{R}}} \dfrac{\lambda +\gamma }{1+\gamma }\int _0^1 |\theta _\mathrm{{m}}|^2 \ \mathrm{{d}}z= 0. \end{aligned}$$

(38b)

We divide Eq. (38a) by the parameter $-\dfrac{\gamma }{1+\gamma }{R_\gamma }$, and then, we summed up with Eq. (38b) to have

$$\begin{aligned}&\dfrac{1+\gamma }{ \gamma {R_\gamma }} \left( \int _0^1 | \hat{\psi }' |^2\ \mathrm{{d}}z + a^2 \int _0^1 | \hat{\psi }|^2\ \mathrm{{d}}z \right) \nonumber \\&-\int _0^1 |\theta '_\mathrm{{m}}|^2 \ \mathrm{{d}}z -a^2\int _0^1 |\theta _\mathrm{{m}}|^2 \ \mathrm{{d}}z + i{\omega _\mathrm{{R}}} \dfrac{\lambda +\gamma }{1+\gamma } \int _0^1 |\theta _\mathrm{{m}}|^2 \ \mathrm{{d}}z\nonumber \\&= 0 . \end{aligned}$$

(39)

The imaginary part of Eq. (39) is equal to zero only if the condition of Eq. (40) is satisfied,

$$\begin{aligned} \omega _\mathrm{{R}} \dfrac{\lambda +\gamma }{1+\gamma } \int _0^1 |\theta _\mathrm{{m}}|^2 \ \mathrm{{d}}z= 0 . \end{aligned}$$

(40)

The two dimensionless parameters $\lambda$ and $\gamma$ are defined as positive and trivial solutions. The assumption of $\theta _\mathrm{{m}}=0$ is not acceptable with what we are dealing with. Therefore, the case of $\omega _\mathrm{{R}}=0$ is valid in Eq. (40), as consequence, the principle of exchange of instabilities is fulfilled for a special case of $H\rightarrow \infty$ .