The pressure inside a suspended gas bubble

An integral representation is derived for the pressure inside an inviscid, compressible or incompressible bubble suspended in a Newtonian fluid, in terms of the instantaneous bubble shape and structure of the ambient velocity field. The starting point in deriving this representation is the reciprocal theorem for Navier-Stokes flow, applied for the flow around the bubble and the flow due to a point source situated inside the bubble. The integral representation is used to evaluate the pressure inside a spherical bubble in potential flow to leading order with respect to the Weber number, and the pressure inside a circular bubble in Stokes flow to leading order with respect to the capillary number. The results are consistent with and generalize numerical and analytical solutions derived by previous authors.