Structural Stability in Local Thermal Non-equilibrium Porous Media

Abstract

We consider a theory for local thermal non-equilibrium in a porous medium where the solid and fluid components may have different temperatures. A priori estimates are derived for a solution to the governing partial differential equations and these are employed in an analysis of continuous dependence and of convergence. It is shown that the solution depends continuously on changes in the coefficient governing the interaction between the fluid and solid temperatures. This coefficient is key to the theory since this is where the equations are coupled. We also prove a convergence result demonstrating that the solution converges appropriately as the coupling coefficient vanishes.

Appendix

We use Rellich identities, cf. Chirita et al. [17] and the references therein, to estimate ∥G _α ∥ and ∥∇G _α ∥ in (7).

Consider the identity, with G≡G ₁ or G ₂,

$$\int_{\varOmega} x_{k}G_{,k}\varDelta Gdx=0. $$

We integrate this by parts and write on Γ,

$$G_{,i}=n_{i}\dfrac{\partial G}{\partial n}+a^{\alpha\beta}x^i_{;\alpha}G_{;\beta}, $$

where a ^αβ is the fundamental form for the surface Γ, ξ ¹ and ξ ² are the surface coordinates, $x^{i}_{;\alpha}=\partial x^{i}/\partial \xi^{\alpha}$, n _i  represents the unit normal to Γ, and ∂/∂n is the normal derivative on Γ. In this manner we may show

$$\begin{aligned} \frac{1}{2}\Vert \nabla G \Vert ^{2} =&\frac{1}{2} \oint_{\varGamma} x_{i}n_{i} \vert \nabla G\vert ^{2}dA- \oint_{\varGamma} x_{i}n_{i} \biggl( \frac{\partial G}{\partial n} \biggr) ^{2}dA \\ &{}-\oint_{\varGamma} \frac{\partial G}{\partial n}\,x_ka^{\alpha\beta}x^k_{;\alpha}G_{;\beta} \,dA. \end{aligned}$$

(28)

Suppose now

$$ \max_{\varGamma}\bigl\vert x_{i} x^i_{;\alpha} \bigr\vert =\delta_{0}, $$

(29)

and given the properties of Ω suppose also that

$$x_{i}n_{i}\geq k_{0}>0\quad\text{on}\ \varGamma, $$

for some constant k ₀. We use the arithmetic-geometric mean inequality with a constant α>0, to show from (28) that

$$\begin{aligned} &\Vert \nabla G\Vert ^{2}+ \biggl( k_{0}-\frac{\delta_{0}}{\alpha} \biggr) \oint_{\varGamma} \biggl( \frac{\partial G}{\partial n} \biggr)^{2}dA\\ &\quad{}\leq\delta_{0}\alpha \oint_{\varGamma} \vert \nabla_{s}G\vert ^{2}dA, \end{aligned}$$

where |∇ _s G|²=a ^αβ G _;α a ^μβ G _;μ . Select now α=δ ₀/k ₀ to find

$$ \Vert \nabla G\Vert ^{2}\leq\frac{\delta_{0}^{2}}{k_{0}} \oint_{\varGamma} \Vert \nabla_{s}G\Vert ^{2}dA. $$

(30)

Next, from the Poincaré’s inequality,

$$ \Vert \nabla G\Vert ^{2}\geq\lambda_{1}\Vert G\Vert ^{2}-c_{1} \oint_{\varGamma} G^{2}dA, $$

(31)

for a constant c ₁>0, and so we employ this in (30) to derive

$$ \Vert G\Vert ^{2}\leq\frac{c_{1}}{\lambda_{1}} \oint_{\varGamma} G^{2}dA+ \frac{\delta_{0}^{2}}{\lambda_{1}k_{0}} \oint_{\varGamma} \vert \nabla_{s}G\vert ^{2}dA. $$

(32)

Finally, we replace G _α in (7) by the equivalent time derivative G _,t . In this way we may derive from (32)

$$ \Vert G_{,t}\Vert ^{2}\leq\frac{c_{1}}{\lambda_{1}}\oint_{\varGamma} G_{,t}^{2}dA+ \frac{\delta_{0}^{2}}{\lambda_{1}k_{0}} \oint_{\varGamma} \vert \nabla_{s}G_{,t} \vert ^{2}dA. $$

(33)