Three-dimensional simulations of hydrodynamic instability in liquid bridges: Influence of temperature-dependent viscosity

Abstract

The development of thermocapillary convection inside a cylindrical liquid bridge is investigated by using a direct numerical simulation of the three-dimensional (3D), time-dependent problem for a wide range of Prandtl numbers, Pr=1,3,4,5 and Pr=35. Above the critical value of temperature difference between the supporting disks, two counterpropagating hydrothermal waves bifurcate from the two-dimensional (2D) steady state. The existence of standing and traveling waves is discussed. The dependence of viscosity upon temperature is taken into account. The critical Reynolds number and critical frequency at which the system undergoes a transition from a 2D steady state to a 3D oscillatory flow decreases if the viscosity diminishes with temperature. The stability boundary is determined for Pr=3–5 with a viscosity contrast (νmax/νmin) up to a factor 10. Near the threshold of instability the flow organization is similar for the constant and variable viscosity cases despite the large difference in critical Reynolds numbers. The influence of variable viscosity on the flow pattern is increased when going into the supercritical region. The study of spatial-temporal behavior of oscillatory convection for the high Prandtl number, Pr=35, demonstrates a good agreement with previously published experimental results. For this high Prandtl number liquids instability begins as a standing wave with an azimuthal wave number m=1 which then switches to an oblique traveling wave ≈4%–5% above the onset of instability.