Three-Dimensional Steady Vapor Bubbles in Rectangular Microchannels

Abstract

We develop a mathematical model of a long steady vapor bubble in a channel of rectangular cross section with given temperature distributions on the walls of the channel. Evaporation near the heated bottom of the channel is balanced by condensation in colder areas of the vapor–liquid interface near the top. An asymptotic method is developed in the limit when the shape of the bubble is dominated by capillary forces everywhere except near the walls. A lubrication-type analysis is used to find local vapor–liquid interface shapes and mass fluxes near the walls. Contact lines are present at the bottom and side walls and are modeled through a disjoining pressure. The total length of the bubble is determined from the integral mass balance after all local solutions are found and characterized. The length increases as the average heater temperature becomes higher as expected. The length of the bubble is also studied as a function of the material properties for fixed temperature distribution at the heated bottom. It is shown that longer bubbles can be obtained when kinetic effects at the vapor–liquid interface are less significant or when the side-walls conduct heat better.