Thermocapillary Flow Analysis of Levitated Thin Liquid Disk

Experiments with thermocapillary flow and temperature measurements in a horizontally levitated, laser-heated thin glycerin disk have been successfully carried out and reported. Such a heated disk has the advantage of a relatively low gravitational potential over the thickness, thus mitigating the buoyancy effects, and helping isolate the thermocapillary driven flows. For the purpose of predicting the thermal properties from these measurements, it is necessary to develop a theoretical model of the thermal processes. Such a model is being developed, and based on the observed shape, the thickness is taken to be a minimum at the center with a gentle parabolic profile at both the top and the bottom surfaces. This minimum thickness is much smaller than the radius of disk drop and the ratio of thickness to radius becomes much less than unity. It is heated by laser beam in normal direction to the edge. A general three-dimensional momentum equation is transformed into a two-dimensional vorticity equation. For the very viscous small liquid drop, the Stokes equation can be used to describe the fluid motion with viscous force and pressure force. Considering boundary conditions on the surface where the surface stresses are balanced with the thermocapillary force corresponding to temperature gradient, we take an average over thickness of disk drop and obtain a two-dimensional momentum equation which includes surface tension terms. To avoid difficulties with handling pressure terms, it is formulated in terms of the vorticity and some additional terms due to a dimpled surface. This vorticity equation has temperature gradient (corresponding to a Marangoni number) in the plane of the disk which brings about its coupling with the energy equation. In the same way, the three-dimensional energy equation is averaged over the disk thickness. With convection boundary condition at the surfaces, we integrate a general three dimensional energy equation to get an averaged two dimensional energy equation which has convection terms, conduction terms, and additional source terms corresponding to a Biot number. A numerical approach is used to solve these steady state governing equations in cylindrical coordinate. These governing equations are discretized into finite difference equations by using central difference approximation. In a cylindrical coordinate system, there could be many singularities at the center. The calculations yield the temperature distribution and the thermally-driven flow field. These results will be used to formulate a model that, in conjuction with experiments, will enable development of a method for the non-contact thermophysical property measurement of liquids.