Effects of temporal density variation and convergent geometry on nonlinear bubble evolution in classical Rayleigh-Taylor instability

Effects of temporal density variation and spherical convergence on the nonlinear bubble evolution of single-mode, classical Rayleigh-Taylor instability are studied using an analytical model based on Layzer's theory [Astrophys. J. 122, 1 (1955)]. When the temporal density variation is included, the bubble amplitude in planar geometry is shown to asymptote to integral U-t(L)(t('))rho(t('))dt(')/rho(t), where U-L=root g/(C(g)k) is the Layzer bubble velocity, rho is the fluid density, and C-g=3 and C-g=1 for the two- and three-dimensional geometries, respectively. The asymptotic bubble amplitude in a converging spherical shell is predicted to evolve as eta similar to(eta) over barm(0)(-parallel to r)\/U-L(sp)-(eta) over bar /r(0), where r(0) is the outer shell radius, (eta) over bar (t)=integral U-t(L)sp(t('))rho(t('))r(0)(2)(t('))dt(')/rho(t)r(0)(2)(t), U-L(sp)=root-r(0)(t)r(0)(t)/l, m(t)=rho(t)r(0)(3)(t), and l is the mode number.