Suppression of the Rayleigh-Taylor instability in a confined geometry

We study the Rayleigh-Taylor instability of two miscible fluids in a Hele-Shaw geometry; confined in a thin gap, of size $b$, between two large flat plates. Using this geometry, we inject a fluid into another with a different density as to produce an unstable situation in which a heavy fluid initially resides above a layer of lighter fluid. Below a critical gap spacing, $b_c$, we find that no Rayleigh-Taylor fingers form despite the fluid density gradient that typically instigates the instability. We use simulations, validated by comparison with experiments, to determine $b_c$ as a function of the difference of fluid densities $\Delta\rho$, the viscosities $\eta$, and diffusivities $D$. We argue that this critical confinement scale is set by a competition between destabilizing buoyancy forces and stabilizing effects of viscosity and diffusion. An argument based on dimensional analysis gives scaling exponents consistent with the observed results, $b_c \sim (D\eta / g \Delta\rho)^{1/3}$. In addition to the critical gap, we measure the characteristic wavelength and onset time in this confined geometry and compare it to the theoretical predictions for the Rayleigh-Taylor instability in open space.