Flow instabilities in thermocapillary-buoyant liquid pools

Abstract

The linear stability of the incompressible axisymmetric flow in a buoyant-thermocapillary liquid pool is considered which is heated from above by a heat flux with a parabolic radial profile. Buoyancy forces and radial thermocapillary stresses due to the inhomogeneous surface temperature distribution drive a toroidal vortex. In the absence of buoyancy and for low Prandtl numbers the basic flow becomes unstable typically by a stationary centrifugal instability. At moderate Prandtl numbers the rotational symmetry is broken by hydrothermal waves. In the limit of vanishing Prandtl number two other critical modes are found if the pool is very shallow. One mode is a centrifugally destabilized rotating wave with high azimuthal wavenumber. The other mode is steady and it is driven by the deceleration of the radial inward return flow as it approaches the axis. The deceleration results from an entrainment of fluid into the thin layer of rapid radial outward surface flow. The centrifugal instability of the toroidal vortex flow is assisted by buoyancy in the low-Prandtl-number limit, because the cooling from the sidewall augments the thermocapillary driving. For moderately high Prandtl numbers a stable thermal stratification suppresses the hydrothermal-wave instabilities.