Instabilities of Thermocapillary Flows between Counter-rotating Disks

Thermocapillary flows are important in many applications, such as the floating-zone and Czochralski crystal growth techniques. In production of crystals by the floating zone method, the feed and crystal rods are often rotating in order to suppress the azimuthal asymmetry. We perform the linear stability analysis of the thermocapillary flows between counter-rotating disks. The basic flow and temperature solutions are obtained by using the pseudo-spectral Chebyshev method. The perturbation equations are solved with Chebyshev polynomial expansions in the radial and vertical directions. When no rotation is applied, the instability depends on the Prandtl number. For small Prandtl number liquids (Pr <= 0.1), the first instability of the axisymmetric flow is a stationary secondary flow. When a rotation is applied, the bifurcation is from the axisymmetric state to an oscillatory state. The most unstable mode is a traveling wave. The critical frequencies changes with the rotation Reynolds number significantly, and the direction of wave propagation can be opposite for high rotation Reynolds numbers. The flow is destabilized by weak rotation but stabilized by strong rotation. As the rotation Reynolds number increases, the appearance of the secondary vortex in the basic flow can decrease the growth rate of perturbation significantly. Energy analysis shows that the perturbation energy consists of the viscous dissipation, the work done by Marangoni forces and the interaction between the perturbation flow and the basic flow, respectively. For Prandtl numbers lower than 0.01, the perturbation energy mainly comes from the third part, which suggests that the perturbation is hydrodynamic. When Prandtl number is larger than 0.1, the second part becomes more important, and the perturbation consists of hydrothermal waves, which shows that the thermocapillary effects are important for large Prandtl number. The work done by Marangoni forces decrease with the rotation Reynolds number.