Open Access

ARTICLE

Solutocapillary Convection in Spherical Shells with a Receding and Deforming Interface

Pravin Subramanian¹, Abdelfattah Zebib¹

1 Mechanical & Aerospace Engineering, Rutgers University, Piscataway, NJ, 08854

Fluid Dynamics & Materials Processing 2008, 4(3), 139-162. https://doi.org/10.3970/fdmp.2008.004.139

Abstract

A theoretical and computational 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 non-linear 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 micro-scale problem). The Capillary number (Ca) provides a measure of the deviation from sphericity and to leading order in the limit Ca → 0 the outer surface evolves with time in a convective state as it does in the diffusive state. We model the motion in this limit and compute supercritical, nonlinear, time-dependent, axisymmetric and three-dimensional, infinite Schmidt number solutocapillary convection. The normal stress balance imposes compatibility restrictions and allows two admissible states: axisymmetric hemispherical convection and three-dimensional solutions exhibiting cubic symmetry. We employ global mass conservation to compute upper bounds on the companion O(Ca) free surface deformations.

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