The neutral stability of surface-tension driven cavity flows subject to buoyant forces—I. Transverse and longitudinal disturbances

Abstract

A Couette—Poiseuille surface-tension driven return flow is sustained in a horizontal cavity by a temperature gradient which exist along the free surface. The zone, shaped long and wide with respect to its depth, is filled with an inert single-component fluid. The zone is allowed to be positioned either right-side-up or upside-down with respect to gravity; the zone is subject to various buoyant forces by altering either the zone depth or orientation. The hydrodynamic stability of the zone is considered by superimposing a small perturbation upon the steady two-dimensional basic flow. The longitudinal and transverse periodic instabilities are considered for fluids with Prandtl number values in the range 0.01–10, subject to various Bond number environments in both zone orientations. Inclusion of both zone orientations allows one to consider a wide range of basic state features to explore with respect to their resultant stability. For a given material (Pr), the physical properties and gravitational environment dictate the critical temperature gradients, zone depth, and type of the most dangerous disturbance likely to occur for a given steady return flow. This, in turn, determines the container design limitations for a given system.