A spectral theory for small-amplitude miscible fingering

Abstract

Using the self-similar symmetry of a diffusing front, we develop a linear spectral theory for miscible fingering at inception that accurately captures the destabilization of localized disturbances (with large transverse wavelengths compared to the front width) by the unsteady front. Our theory predicts a generic selected wavelength (4πηD/U0 for gravity fingering, where η is the transverse to longitudinal dispersion ratio, and an additional factor proportional to the logarithm of the mobility ratio for viscous fingering) at the small time of O(D/U02), where D is the dispersion coefficient or diffusivity in the flow direction and U0 is the displacement velocity. This wavelength then grows in time and approaches a universal asymptotic wavelength coarsening dynamics of (η2D5/U02)1/8(t′)3/8, where t′ is the dimensional time, for all small-amplitude miscible fingering phenomena in a slot or in porous media. The 38 exponent in time is due to a unique long-wave stabilization mechanism due to transverse convection, which escapes prior quasisteady theory. Explicit and generic scalings are then derived for gravity and viscous miscible fingering phenomena and are favorably compared to experimental and numerical results on linear coarsening dynamics.