The low Reynolds number deformation of a gas bubble in shear flow: a general approach via integral equations

Abstract

The time-dependent deformation of an incompressible or compressible gas bubble in an arbitrary shear flow at low Reynolds number is formulated as a second boundary-value problem for Stokes' equations. This problem is solved via a classical Fredholm's integral equation of second kind. Contrasting with the existing previous works using the integral equation approach for this problem, the integral equation method developed here yields a unique bubble surface velocity, regardless of any flow and bubble axisymmetric property. From those previous works, the uniqueness of the surface velocity has been proved only for a spherical bubble in axisymmetric flows, and it has been conjectured that the surface velocity remains unique as long as the bubble shape and the exterior flow are axisymmetric.