On the Rayleigh-Taylor instability of a diffuse interface

Abstract

We present a formulation for the linear stability of the miscible Rayleigh-Taylor instability, retaining viscosity, Fickian mass diffusion, and non-Boussinesq effects. Rather than adopting the diffusive corrections of Duff, Harlow, and Hirt (1962) or relying on the viscous eigenvalue problem of Chandrasekhar (1961), the present formulation is derived directly from the incompressible variable-density governing equations. Both approaches are recovered as limiting cases, extending Morgan, Likhachev, and Jacobs (2016). The governing equations admit a time-evolving self-similar diffusive base state in which an error-function density profile is accompanied by a nonzero vertical base velocity required by the non-solenoidal constraint. Linearization yields a reduced coupled formulation for the vertical velocity and density perturbations, suitable for both theoretical analysis and computation. Within the quasi-steady-state approximation, we show that the formulation reduces, in the infinite-Schmidt-number limit, to the classical viscous immiscible theory and recovers the Chandrasekhar eigenvalue problem. At finite Schmidt number, diffusion alters the growth-rate spectrum and its high-wavenumber structure. Beyond the spectral cutoff we establish the scaling $ω_r \sim -λ_m k^2$, where $λ_m$ is determined by the competition between the locally stratified kinematic viscosity and the reference diffusive scale. By isolating the effect of diffusion, we show that its stabilizing influence weakens with increasing density stratification. Buoyancy production becomes bimodal and shifts toward the lighter-fluid side, while pressure production grows to rival buoyancy and acts as an energy source on the heavier side of stratification. Together, these results clarify when classical diffusive corrections remain applicable and when diffusion and non-Boussinesq coupling must be retained.