= Thermal-front viscous-fingering dispersion relation
{title2=$\sigma=\Gamma U|a|(b-1)/(b+1)$}
For a sharp thermal front moving at fraction $\Gamma$ of the <Darcy velocity> $U$, a cold fluid of viscosity $\mu$ displacing its hot counterpart of viscosity $b\mu$ has the displayed growth rate in the local, unbounded planar approximation without <thermal conduction>. The harmonic pressure perturbations decay on either side. <Pressure continuity> and continuity of perturbed <Darcy flux> give the velocity response $\delta U=U|a|(b-1)\eta/(b+1)$; the thermal front's kinematic law is $\eta_t=\Gamma\delta U$. The prefactor is the thermal-front speed, not the material <pore velocity>. Increasing viscosity with heating therefore makes the trailing thermal interface unstable even when the leading displacement of the original fluid is stable.
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