Thermocapillary Motion of Bubbles Inside Drops
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THERMOCAPILLARY MOTION OF BUBBLES INSIDE DROPS
N. SHANKAR, ROBERT COLE, AND R. SHANKAR SUBRAMANIAN Department of Chemical Engineering, Clarkson College of Technology, Potsdam, New York 13676
ABSTRACT The quasi-static thermocapillary migration of a bubble located inside a drop in free fall is considered for arbitrary axisymmetric temperature fields prescribed on the drop surface. Some results are presented, and an approximation based on the superposition of simpler solutions is discussed. INTRODUCTION The near free fall environment aboard orbiting spacecrafts will permit the processing of materials without a container. Such containerless processing has been cited as an important advantage in producing certain high technology glasses out of materials that would normally crystallize on the container walls due to heterogeneous nucleation (1,2). However, it is well known that, in glass manufacture, bubbles are formed due to chemical reactions as well as due to the gas entrapped in the grains of the raw materials. On earth, these gas bubbles are removed by a combination of dissolution and buoyant rise, thus producing the desired bubble-free glass. In orbit, due to the reduction of buoyant effects, alternate mechanisms must be developed to homogenize and fine (remove bubbles from) the glass melt. One such mechanism is thermocapillarity. At Clarkson, with financial support from NASA, we are planning experiments to be flown aboard the space shuttle in which thermocapillary bubble migration inside a liquid drop in a space laboratory will be studied (3,4). It is well known that bubbles migrate in a fluid in the presence of a temperature gradient (5,6). In space experiments, a non-uniform temperature field may be induced in a drop containing the bubbles, by, say, local heating of the drop surface with a radiant source of energy. As a consequence, interfacial tension gradients will be induced on the drop and bubble surfaces. The resulting tangential stresses at these surfaces will drive motion in the fluid, and also cause migration of the bubbles. We have constructed a theoretical description of such thermocapillary bubble motion due to an arbitrary axially symmetric temperature field on the surface of the drop when the bubble is located along the symmetry axis. Analytical solutions of the energy and the momentum equations have been developed in the quasi-static limit using bispherical coordinates and the velocity of the bubble along with the temperature and velocity fields inside the drop obtained. Details of these calculations are reported elsewhere (7,8). At the instant when the bubble crosses the drop center, bispherical coordinates cannot be used due to the singularity in the bipolar coordinate system. In this limit, spherical polar coordinates may be employed. Again, analytical solutions have been developed in the quasi-static limit, and the details reported elsewhere (8,9). As the governing equations are linear a modal analysis has been performed where the effects of pure Legendre modes of
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