= Buoyancy-modified Darcy fingering dispersion relation
{title2=$\sigma(\alpha)$}
Let fluid 1 lie above fluid 2, with <mass densities> $\rho_1<\rho_2$ and <dynamic viscosities> $\mu_1<\mu_2$. Define $\Delta\rho=\rho_2-\rho_1>0$, $\Delta\mu=\mu_2-\mu_1>0$, <porosity> $\phi$, <permeability of a porous medium> $k$, and downward interface speed $W$. With <Darcy flux> $\phi W$, no capillarity, and two unbounded layers, <Darcy's law> gives
$$
\sigma(\alpha)=\frac{|\alpha|[\phi W\Delta\mu-kg\Delta\rho]}{\phi(\mu_1+\mu_2)}.
$$
For a <Fourier mode> of interface displacement, the two perturbation <pressures> decay exponentially into the respective layers. The <kinematic boundary condition> makes their amplitudes proportional to $\phi\mu_1\sigma/(k|\alpha|)$ and $-\phi\mu_2\sigma/(k|\alpha|)$. Linearized <pressure continuity> adds the base-gradient difference $\phi W\Delta\mu/k-g\Delta\rho$. Eliminating the amplitudes proves the formula. The critical interface speed is $kg\Delta\rho/(\phi\Delta\mu)$; the critical <Darcy velocity> is $kg\Delta\rho/\Delta\mu$. Without <surface tension> this model has unbounded short-wave <growth rate> on its unstable side.
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