Viscid contributions to the hydrodynamic flows past a rising spherical gas bubble with a time-dependent radius

We study the effect of the viscosity on the hydrodynamic flow fields past the interface of a spherical deforming gas bubble impulsively started at a constant velocity in a viscous liquid of large extent at rest. Exact solutions for the unsteady inner and outer flow fields within the boundary layers are obtained making appropriate scalings on the position, velocity and time variables in the non-linear Navier–Stokes equations. These theoretical results apply to any slowly deforming fluid sphere, whatever the time-dependence of its radius, provided that the internal circulation is complete, the flow separation is negligible, the Reynolds numbers are large and the bubble retains its spherical shape. A numerical application to the case of deforming air bubble in water is discussed.