= Solution
Take constant <Darcy velocity> $U$ and work locally near a planar <porous thermal front> moving at $V_T=\Gamma U$. Let $\xi=x-V_Tt$ and perturb its position by $\eta=\eta_0e^{\sigma t+ia y}$, 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> are
$$
p'_- =A_-e^{|a|\xi}e^{\sigma t+iay},\qquad p'_+=A_+e^{-|a|\xi}e^{\sigma t+iay}.
$$
Here 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 $p^0_{-,x}=-\mu U/k$ and $p^0_{+,x}=-b\mu U/k$. Linearizing <pressure continuity> at the displaced front yields
$$
A_--A_+=-\eta_0\frac{(b-1)\mu U}{k}.
$$
Continuity of the perturbed normal <Darcy velocity> gives
$$
\delta U=-\frac{k|a|}{\mu}A_- =\frac{k|a|}{b\mu}A_+,
$$
so $A_+=-bA_-$. 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 relation>
$$
\boxed{\sigma(\lambda)=\frac{2\pi\Gamma U}{\lambda}\frac{b-1}{b+1}=\frac{2\pi V_T}{\lambda}\frac{b-1}{b+1}.}
$$
It 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 $\lambda\to0$. A resolved thermal transition and diffusion regularize that short-wave limit.
For a material interface the kinematic coefficient would be $1/\phi$ rather than $\Gamma$. Thus, if the interface of interest were instead the hot polymer–oil interface, the same calculation would give $\sigma=2\pi(U/\phi)(\mu_h-b\mu)/[\lambda(\mu_h+b\mu)]$, 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.
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