Exact solutions for the evolution of a bubble in Stokes flow: A Cauchy transform approach

A Cauchy transform approach to the problem of determining the free surface evolution of a single bubble in Stokes flow is developed. A number of exact solutions to a class of problems have been derived in the literature using conformal mapping theory, and these solutions are retrieved and further generalized using the new formulation. Certain quantities which are conserved by the dynamics are also identified, the existence of which had not previously been pointed out. A principal purpose of this paper is to use the new formulation to understand when it is possible to externally specify the evolution of the bubble area in such classes of exact solution. It is found to be possible only for certain types of far-field boundary conditions.