Resonant thermocapillary and buoyant flows with finite frequency gravity modulation

Abstract

Interaction between a base thermocapillary flow and a time-dependent buoyant force is studied for a slot geometry. A temperature gradient applied along a fluid-filled slot with thermocapillarity at a free surface produces a base parallel flow. The system is subjected to streamwise gravitational acceleration that varies harmonically in time. Grassia and Homsy [Phys. Fluids. 10, 1273 (1998)] have shown that in the limit of zero frequency modulation, coupling of the thermocapillary flow with long wave convective modes leads to singularities at critical points corresponding to the Rayleigh-Benard eigenvalues. In the case of small but finite frequency modulation studied here, inertial effects moderate the singularities which are replaced by a response that scales exponentially with the inverse of the dimensionless modulation frequency. An O(1) delay is observed in the onset of the resonant response even for small modulation frequencies. The response is also found to scale exponentially with the inverse Prandtl number for large Prandtl numbers and to be independent of Prandtl number for small Prandtl numbers. Relaxation oscillations are observed in certain parameter ranges as a result of the coupling between the fluid and thermal fields. A Galerkin approximation is used to reduce the problem to an equivalent dynamical system, the analysis of which gives analytical support to and insight into the numerical results. (C) 1999 American Institute of Physics. [S1070-6631(99)03809-X].