The unsteady expansion and contraction of a long two-dimensional vapour bubble between superheated or subcooled parallel plates

Abstract

In an attempt to model the growth and collapse of a vapour bubble in nucleate boiling this paper investigates the unsteady expansion and contraction of a long two-dimensional vapour bubble confined between superheated or subcooled parallel plates whose motion is driven by mass-transfer effects due to evaporation from the liquid to the vapour and condensation from the vapour to the liquid. It is shown that in the asymptotic limit of strong surface tension (small capillary number) the solution consists of two capillary-statics regions (in which the bubble interface is semicircular at leading order) and two thin films attached to the plates, connected by appropriate transition regions. This generalization of the steady and isothermal problem addressed by Bretherton (1961) has a number of interesting physical and mathematical features. Unlike in Bretherton's problem, the bubble does not translate but can change in size. Furthermore, the thin films are neither spatially nor temporally uniform and may dry out locally, possibly breaking up into disconnected patches of liquid. Furthermore, there is a complicated nonlinear coupling with a delay character between the profiles of the thin films and the overall expansion or contraction of the bubble which means that the velocity with which the bubble expands or contracts is typically not monotonic. This coupling is investigated for three different combinations of thermal boundary conditions and two simple initial thin-film profiles. It is found that when both plates are superheated equally the bubble always expands, and depending on the details of the initial thin-film profiles, this expansion may either continue indefinitely or stop in a finite time. When both plates are subcooled equally the bubble always contracts, and the length of the thin-film region always approaches zero asymptotically. When one plate is superheated and the other subcooled with equal magnitude the bubble may either expand or contract initially, but eventually the bubble always contracts just as in the pure-condensation case.