Exact Solutions for Two Bubbles in Stokes Flow

Abstract

The evolution of two interacting bubbles in creeping fluid flow (Stokes flow) is considered in the presence of surface tension and far-field strain, when the flow and shapes have reflectional symmetries about both x and y axis. We find a large family of exact solutions with fixed bubble area but increasing straining field, or fixed straining field and decreasing bubble area. Two methods introduced include conformal mapping and Cauchy Transform approach. An infinite number of conserved quantities are identified without surface tension. With nonzero surface tension, there is a canonical choice of variables that make the mathematical structure have a triangular property that ensure exact solution for any truncation of the series associated with the initial conformal map. The Cauchy transform, which is the generating function for the moments of the domain, is found to have the curious mathematical property that if initially meromorphic, remains meromorphic for later times. The consequence of that will be discussed.