Instabilities in a cylindrical cavity heated from below with a free surface. I. Effect of Biot and Marangoni numbers

Multiple instabilities of thermocapillary flow in a cylindrical pool with a rotating disk on the free surface

The thermocapillary flow instabilities of silicon melt in a cylindrical pool with a rotating disk on the free surface (a simplified model of the Czochralski crystal growth) are numerically investigated by using the linear stability analysis. The complete neutral or critical stability curves are determined. Results show that the neutral stability curves form a closed region in the parameter plane, in which the steady axisymmetric flow is linearly stable. Two types of rotating wave (RW1 and RW2) instabilities and two types of hydrothermal wave (HTW1 and HTW2) instabilities are found. The energy analysis shows that all the instabilities are hydrodynamic (inertial) in nature. Specifically, RW1 and RW2 are caused by the azimuthal shear induced by the crystal rotation, while HTW1 and HTW2 are caused by the radial shear induced by the thermocapillary force.

Three-dimensional continuation study of convection in a tilted rectangular enclosure

A continuation method developed from a three-dimensional spectral finite element code is used to study natural convection in a tilted rectangular cavity. The cavity has its length equal to two times the side of its square cross section and it contains a fluid with a Prandtl number Pr = 1. A detailed bifurcation diagram is first obtained in the case without inclination in order to get the sequence of the different branches of solutions and determine the stable solutions. The focus is then put on the stable solutions in the inclined cavity, when the tilt occurs around its longest axis. The subtle changes induced by the tilt on the convective system are clarified. Three different stable solutions are obtained: the longitudinal roll L- solution (with the same sense of rotation as the inclination angle), which develops smoothly from zero Rayleigh number on the leading branch; the longitudinal roll L+ solution (with a sense of rotation opposite to the inclination angle), which is on a disconnected branch and is stabilized beyond a secondary bifurcation point; the oblique roll O ± solutions (corresponding to transverse roll solutions perturbed by the longitudinal flow induced by the tilt), which quickly appear beyond saddle-node points on new disconnected branches. The domain of existence of these stable solutions is eventually obtained and described in the Rayleigh number-inclination parameter space. Finally, the Nusselt number is determined as a function of the inclination at a constant Rayleigh number for the different stable solutions. The Nusselt number is maximum at an inclination of 49.55° for the leading longitudinal roll L- solution.

Instabilities in a cylindrical cavity heated from below with a free surface. II. Effect of a horizontal magnetic field

The effect of a constant and uniform horizontal magnetic field on the flow in a cylindrical cavity heated from below, with a free surface at the top, is numerically investigated. The azimuthal modes, which usually trigger convection in a cylinder, are changed by the horizontal magnetic field to oriented modes, either parallel or perpendicular to the magnetic field direction. The corresponding primary thresholds increase with the Hartmann number Ha. This increase, however, depends on the structure of the modes and is the weakest for the parallel modes and the strongest for the perpendicular modes. The changes that affect the evolution of the primary thresholds with the aspect ratio for nonzero Ha are also emphasized. The nonlinear evolution of the convection with a horizontal magnetic field is presented through bifurcation diagrams for different values of the Prandtl number Pr. For Pr=1 and small values of Ha, the structuring effect of the horizontal magnetic field, which involves modifications of the flow structures and bifurcation points, is put into light. Results are finally shown for smaller Pr values corresponding to liquid metals.