Numerical simulation of oscillatory Marangoni convective flow inside a cylindrical liquid zone

Abstract

The problem of the oscillatory thermocapillary convection flow inside a NaNO 3 float zone which is suspended between a pair of coaxial disks with prescribed time-dependent temperature profiles and bounded by a cylindrical free surface, has been investigated. The system of governing equations corresponding to a three-dimensional transient model was directly solved by employing a finite control volume method, fully-implicit in time, and a staggered spatial mesh in cylindrical coordinates. It has been clearly shown that for a sufficiently low temperature difference between the disks, the flow consists of a steady and perfectly axisymmetrical toroidal structure with a purely axial moyement of the fluid on the free surface and the vortex center located near that surface. Beyond the critical Marangoni number, Ma cr U ≈ 12 500, a transition from the axisymmetrical to the three-dimensional oscillatory state occurs. Under the effects of the time-dependent thermal disturbances on the free surface, the entire velocity and temperature fields rotate around the main axis following the second mode, i.e. the symmetrical mode of instability. A complete description of the flow structure and its dynamical behavior as well as a comparison with previous numerical and experimental data is given. The phenomenon of hysteresis has also been studied. It has been observed that there is a certain range of the Marangoni number where both the axisymmetrical and the oscillatory states may exist depending on whether the zone is heated up or cooled down. It has been found that the second critical Marangoni number i.e. the one corresponding to the reverse transition from the oscillatory to the axisymmetrical state, depends strongly on the temperature time-rate at which the zone is cooled.