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Take constant Darcy velocity $U$ and work locally near a planar porous thermal front moving at $V_T=\Gamma U$. Let $\xi =x-V_{T}t$ and perturb its position by $\eta ={\eta }_0{e}^{\sigma t+iay}$, with wavelength $\lambda =2\pi /|a|$. Neglect thermal conduction and regard the adjoining layers as semi-infinite on this disturbance scale. In each region, Darcy's law and incompressibility imply that the pressure perturbation is harmonic, so the bounded normal modes areHere the minus side is cold with dynamic viscosity $\mu$, and the plus side is hot with dynamic viscosity $b\mu$. The base pressure gradients are $p_{-,x}^0=-\mu U/k$ and $p_{+,x}^0=-b\mu U/k$. Linearizing pressure continuity at the displaced front yieldsContinuity of the perturbed normal Darcy velocity givesso $A_{+}=-b{A}_{-}$. Therefore $\delta U=U|a|(b-1){\eta }_0/(b+1)$. The thermal-front kinematic law is $\sigma {\eta }_0=\Gamma \delta U$, proving the thermal-front viscous-fingering dispersion relationIt is positive for $b>1$, zero for $b=1$, and negative for $b<1$. Without thermal conduction the model has no finite fastest wavelength: the positive growth rate increases without bound as $\lambda \to 0$. A resolved thermal transition and diffusion regularize that short-wave limit.

For a material interface the kinematic coefficient would be $1/\varphi$ rather than $\Gamma$. Thus, if the interface of interest were instead the hot polymer–oil interface, the same calculation would give $\sigma =2\pi (U/\varphi )({\mu }_h-b\mu )/[\lambda ({\mu }_h+b\mu )]$, the usual planar viscous-fingering dispersion relation. This distinguishes the two interfaces and their speeds. The boxed thermal result uses the local planar approximation; distant wells or the other front modify the pressure eigenfunctions for wavelengths comparable to those separations.