Computation of the pressure inside bubbles and pores in Stokes flow

Abstract

Numerical methods are discussed for computing the pressure inside a two- or three-dimensional inviscid bubble with negligible density suspended in Stokes flow, subject to a specified rate of expansion. In the case of flow past a solitary two- or three-dimensional bubble, the bubble pressure is found by solving an integral equation of the first kind for the normal derivative of the pressure on the side of the liquid over the free surface, while requiring that the pressure field decays at a rate that is faster than the potential due to a point source. In another approach, an explicit expression for the bubble pressure is derived by applying the reciprocal theorem for the flow around the bubble and the flow due to a point source situated inside the bubble. In the case of flow past, or due to the expansion or shrinkage of, a periodic lattice of bubbles, the bubble pressure is found by solving an integral equation of the second kind for the density of an interfacial distribution of point-source dipoles, while ensuring existence and uniqueness of solution by spectrum deflation. The new methods considerably simplify the computation of the bubble pressure by circumventing the evaluation of the finite part of hypersingular integrals. Results of numerical simulations illustrate the pressure developing inside a solitary two- and three-dimensional incompressible bubble suspended in simple shear flow, and the pressure developing inside a doubly periodic array of gaseous inclusions representing shrinking pores trapped in a sintered medium.