Bubble motion and size variation during thermal migration with phase change

An analysis of the motion of a spherical bubble in a two-phase (fluid-fluid), single component system with a vertical linear temperature gradient is presented. The model for the migration of an immiscible bubble under the effects of buoyancy and thermocapillarity, considered by Young ["The motion of bubbles in a vertical temperature gradient," J. Fluid Mech. 6, 350-356 (1959)], is modified to allow for phase change at the bubble surface. We allow the possibility of both translation of the bubble in the vertical direction and the change of bubble radius with time. Depending on the material parameters, the thermocapillary and buoyancy effects that govern the migration of an immiscible bubble can be overwhelmed by the effects of latent heat generation, resulting in a change in the mechanism driving the motion. For a water-steam system, conditions are determined for a stationary bubble in which the effects of buoyancy and thermal migration are balanced. The linear stability of the bubble is considered, and conditions are determined that correspond to small-amplitude oscillations of the position and radius of the bubble. A weakly nonlinear analysis of the solution in the vicinity of the unstable solution is performed, and the results are compared with a numerical solution of the nonlinear equations.