= Surface-tension-gradient stabilization of viscous fingering
Suppose the apparent <capillary pressure> is $p_1-p_2=\gamma(1+\beta x)(\kappa_0+\nabla\cdot\mathbf n)$, with normal towards the displaced fluid and positive apparent tension at the flat front $x=X$. Linearizing the <Darcy law> interface problem gives the local growth rate
$$
\sigma(\alpha;X)=\frac{\alpha}{\phi(\mu_1+\mu_2)}[(\mu_2-\mu_1)U-k\gamma\kappa_0\beta-k\gamma(1+\beta X)\alpha^2].
$$
The pressure increase experienced by a forward protrusion is stabilizing when $\kappa_0\beta>0$. It suppresses every positive <wavenumber> when $U\le k\gamma\kappa_0\beta/(\mu_2-\mu_1)$. For a fixed spatial gradient, $X=Ut/\phi$ makes the rate time dependent; constant exponential growth is a frozen-position approximation.
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