Solutocapillary convection in spherical shells

Abstract

A linear stability study of solutocapillary driven Marangoni instabilities in small spherical shells is presented. The shells contain a binary fluid with an evaporating solvent. The viscosity is a strong function of the solvent concentration, the inner surface of the shell is assumed impermeable and stress free, while nonlinear boundary conditions are modeled and prescribed at the receding outer boundary. A time-dependent diffusive state is possible and may lose stability through the Marangoni mechanism due to surface tension dependence on solvent concentration (buoyant forces are negligible in this microscale problem). A frozen-time or quasisteady state linear stability analysis is performed to compute the critical Reynolds number and degree of surface harmonics, as well as the maximum growth rate of perturbations at specified parameters. The development of maximum growth rates in time was also computed by solving the initial value problem with random initial conditions. Results from both approaches are in good agreement except at short times where there is dependence on initial conditions. The physical problem models the manufacturing of spherical shells used as targets in inertial confinement fusion experiments where perfect sphericity is demanded for efficient fusion ignition. It is proposed that the Marangoni instability might be the source of observed surface roughness. Comparisons with the available experiments are made with reasonable qualitative and quantitative agreement.