= Planar viscous-fingering dispersion relation
{title2=$\sigma(\alpha)$}
For two fluids with <dynamic viscosities> $\mu_1<\mu_2$, uniform <permeability of a porous medium> $k$, <porosity> $\phi$, and imposed <Darcy velocity> $U$, the <Saffman–Taylor instability> of a planar interface has growth rate $\sigma=\alpha U(\mu_2-\mu_1)/[\phi(\mu_2+\mu_1)]$ without capillarity. With constant interfacial tension $\gamma$ and the stabilizing pressure-jump convention, a perturbation of transverse <wavenumber> $\alpha>0$ instead has
$$
\sigma=\frac{\alpha}{\phi(\mu_1+\mu_2)}[(\mu_2-\mu_1)U-k\gamma\alpha^2].
$$
The fastest-growing mode satisfies $\alpha_m^2=(\mu_2-\mu_1)U/(3k\gamma)$.
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