The buoyancy driven motion of a deformable viscous drop at intermediate Reynolds numbers has been studied using numerical techniques. The motion was assumed to be steady and rectilinear, and a pseudo-implicit method was used to solve the Navier-Stokes equations. Cases for a variety of values of the Reynolds number, Weber number, viscosity ratio and density ratio have been considered. The calculations reveal that the shape of the drop is most heavily dependent on the Weber number, attaining spheroidal capped shapes at 0(1) Reynolds numbers, and flattened ellipsoidal shapes at higher Reynolds numbers. Two mechanisms are observed for vorticity production at the interface of the drop—curvature and the no-slip condition—and the no-slip mechanism is a more efficient source of vorticity. When there is sufficient vorticity produced, a detached closed streamline wake forms at the back of the drop, in contrast to the attached wakes seen on inviscid bubbles and solid particles.
To further explore the role of vorticity production in wake formation, numerical computations were done on flow past inviscid bubbles of fixed shape. It was found that attached recirculating wakes existed at intermediate Reynolds numbers and these wakes could not be predicted by either low or high Reynolds number asymptotic theories. The numerical results indicate that the mechanism responsible for flow separation at modest Reynolds numbers may be different than that present at high Reynolds numbers.
Because of the inherent difficulties in solving the Navier-Stokes equations using successive approximation schemes, and to investigate the behavior of solutions of these equations on the dimensionless parameters, a Newton's method scheme has been developed and tested successfully on the steady buoyancy driven motion of an inviscid bubble; an arc length continuation method has also been implemented. Calculations indicate that the scheme achieves quadratic convergence.
Last, a numerical technique has been developed for the study of the creeping motion of drops and particles in the presence of a rigid plane boundary. This method is based upon the distribution of point forces on the surface of the body and the use of a Green's function to obtain the unknown velocity and stress on the body surface, without having to consider the rigid boundary.
