Mathematical Modeling of Constrained Vapor Bubbles

We develop a mathematical model of a long vapor bubble in a micro-channel with given temperature distributions on the walls. We assume that the shape of the bubble is dominated by capillary forces everywhere except near the walls of the channel and use a lubrication-type analysis to find the local vapor-liquid interface shapes and mass fluxes near the walls. Both two- and three-dimensional steady-state solutions are found such that evaporation near the heated bottom is balanced by condensation in colder areas of the vapor-liquid interface. The total length in this steady regime is found from the integral mass balance and investigated as a function of heating conditions. Steady-state conditions can no longer be satisfied when the intensity of heating is above a certain level. In this regime the bubble is expanding. We investigate such expansion in the framework of a two-dimensional model in the limit of small capillary number.