Effect of viscosity on Rayleigh-Taylor and Richtmyer-Meshkov instabilities

We consider the effect of viscosity on Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities by deriving a moment equation for fluids with arbitrary density and viscosity profiles, including surface tension. We apply our result to the classical case of two semi-infinite fluids with densities ρ₁ and ρ₂ and viscosities μ₁ and μ₂. Treating a shock as an instantaneous acceleration we find that perturbations at the interface undergo damped oscillations when viscosity and surface tension are both present. For pure viscosity the amplitude {η}(t) evolves according to {η}(t)/{η}(0)=1+({Δ}vA/2k{ν})(1-e^-2k2{ν}t) where {Δ}v is the jump velocity imparted by the shock, A=(ρ₂-ρ₁)/(ρ₂+ρ₁), {ν}=(μ₁+μ₂)/(ρ₁+ρ₂), k=2{π}/{λ} is the wave number of the perturbation, and t is time. We also consider the turbulent energy in accelerating fluids and calculate the reduction in Eturbulent as a function of {ν}, and propose experiments to measure the effect of viscosity on RT and RM instabilities.