Steady creeping motion of a liquid bubble in an immiscible viscous fluid bounded by a vertical porous cylinder of finite thickness

The creeping vertical motion of a fluid sphere (drop or gas) or liquid bubbles of different shapes in another immiscible fluid confined by porous boundaries is encountered in several situations in industry and technology. Such flows axe generally multi-phase in nature. In this work, we have considered a flow field comprising a non-Newtonian bubble region surrounded by a liquid film of Newtonian fluid. This inner region is bounded by a permeable cylindrical medium pervaded by the same Newtonian fluid. We have studied the interaction features of this multiphase flow in terms of certain practically important geometrical and physical parameters. We have carried out an exact analysis of the governing equations in the three flow fields - Non-Newtonian, Newtonian. film and porous regions. The effects of pressure gradient, permeability and rheological parameters on the bubble velocity and the flow in different regions have been discussed.