On a two-dimensional exterior Stokes flow with mixed boundary conditions

Summary

When the motion of a viscous fluid around a gas bubble is discussed, it is frequently assumed, especially for flows at low Reynolds numbers, that the bubble takes on a spherical shape in three dimensions or a circular cross-section in a two-dimensional flow. If this assumption is made, arid the gas within the bubble is assumed to have negligible density and viscosity, then the problem of finding the exterior flow is mathematically overdetermined and it is not obvious that a solution to the problem exists. Moreover, if such a solution does exist, then the over-determination of the system should, in general, give rise to relationships between the flow parameters, that is, certain conditions must be satisfied to ensure the existence of a solution. It is the purpose of this paper to derive these conditions in the case of a two-dimensional Stokes flow. The problem is generalised to the extent that part of the circular boundary is taken to be rigid, on which the no-slip condition is to be satisfied and part is to be a free streamline, on which stress conditions are to be satisfied. The conditions for the existence of a solution to this problem are derived and the solution is found in closed form. The method of solution is that of reducing the problem to one of a mixed boundary-value problem in analytic function theory. The classical solutions for the Stokes flow around a circular bubble and around a rigid circle are then easily derived as limiting cases.