Shear flow over a translationally symmetric cylindrical bubble pinned on a slot in a plane wall

Abstract

Steady states of a translationally-symmetric cylindrical bubble protruding from a slot in a solid wall into a liquid undergoing a simple shear flow are investigated. Deformations of and the flow past the bubble are determined by solving the nonlinear free-boundary problem comprised of the two-dimensional Navier–Stokes system by the Galerkin/finite element method. Under conditions of creeping flow, the results of finite element computations are shown to agree well with asymptotic results. When the Reynolds number Re is finite, flow separates from the free surface and a recirculating eddy forms behind the bubble. The length of the separated eddy measured in the flow direction increases with Re, whereas its width is confined to within the region that lies between the supporting solid surface and the separation point at the free surface. By tracking solution branches in parameter space with an arc-length continuation method, curves of bubble deformation versus Reynolds number are found to exhibit turning points when Re reaches a critical value Rec. Therefore, along a family of bubble shapes, solutions do not exist when Re > Rec. The locations of turning points and the structure of flow fields are found to be governed virtually by a single parameter, We = Ca Re, where We and Ca are Weber and capillary numbers. Two markedly different modes of bubble deformation are identified at finite Re. One is dominant when Re is small and is tantamount to a plain skewing or tilting of the bubble in the downstream direction; the other becomes more pronounced when Re is large and corresponds to a pure upward stretching of the bubble tip.