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 rateThe 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, $X=Ut/\varphi$ makes the rate time dependent; constant exponential growth is a frozen-position approximation.