A semi-infinite bubble advancing into a planar tapered channel

We consider two problems in which a semi-infinite bubble moves into a uniformly convergent two-dimensional channel under creeping-flow conditions. In the first (steady) problem, the bubble is stationary with respect to the channel walls, which move away from the channel vertex parallel to themselves, each entraining a liquid film of uniform thickness; the appropriate flux is provided by a line source at the channel vertex. In the second (unsteady) problem, the channel walls are fixed and the bubble is drawn at a constant speed towards the channel vertex by a line sink. In this case the bubble interface assumes a self-similar shape, and wedge-shaped films are deposited on the channel walls. The boundary-element method, supported by asymptotic approximations, is used to compute flows over a range of bubble speeds (measured by a capillary number Ca) and wedge angles α. In the steady problem, the deposited film thickness increases monotonically with α at low Ca, but diminishes with increasing α at sufficiently high Ca. In the unsteady problem, the film thickness was found always to increase with both α and Ca. In both cases, the dimensionless pressure drop across the bubble tip can be nonmonotonic in α. Implications of these results in modeling coating and peeling flows are discussed.