Two-dimensional bubbles in slow viscous flows

Abstract

The representation of a biharmonic function in terms of analytic functions is used to transform a problem of two-dimensional Stokes flow into a boundary-value problem in analytic function theory. The relevant conditions to be satisfied at a free surface, where there is a given surface tension, are derived.A method for dealing with the difficulties of such a free surface is demonstrated by obtaining solutions for a two-dimensional, in viscid bubble in (a) a shear flow, and (b) a pure straining motion. In both cases the bubble is found to have an elliptical cross-section.The solutions obtained can be shown to be unique only if certain restrictive assumptions are made, and if these are relaxed the same methods may give further solutions. Experiments on three-dimensional inviscid bubbles (Rumscheidt & Mason 1961; Taylor 1934) demonstrate that angular points appear in the bubble surface, and an analysis is presented to show that such a discontinuity in a two-dimensional free surface is necessarily a genuine cusp and the nature of the flow about such a point is examined.

Two-dimensional Stokes and Hele-Shaw flows with free surfaces

“…In terms of these functions, the conditions (3.2) are easily seen to give the single complex boundary condition [37] …”

“…The moments are 37) though the integrand in (4.37) can be generalised to any analytic function ofζ; generalisations of this kind may again be useful for rational maps (cf. [8]).…”

“…Stokes flows with free surfaces, on the other hand, have until recently received much less attention, but there has been much more activity following the demonstration by Hopper [20,21] and Richardson [41] that explicit unsteady solutions could be constructed (steady solutions were given earlier in [4,17,37,39,49]). A bibliography listing more than 600 papers in both areas can be found at the web address http://www.maths.ox.ac/~howison/ .…”

Semi-analytical solutions for two-dimensional elastic capsules in Stokes flow

“…In this paper, we derive semi-analytical solutions for the dynamics of a twodimensional capsule in Stokes flow by generalizing a well-established complex variable technique for time-evolving bubbles in Stokes flow (Richardson 1968;Tanveer & Vasconcelos 1995;Bazant & Crowdy 2005). One motivation for this work is to provide analytical solutions to help validate the accuracy of new numerical methods for elastic membranes in flow, such as the two-dimensional studies of Veerapaneni et al (2009).…”

Hydrodynamic singularities

“…Since the flow remains two-dimensional, this corresponds to a line of singularities of the surface. Assuming that it is a true cusp, the original authors analyzed the flow using a solution due to Richardson (1968). This local solution leads to a logarithmic divergence of the dissipation, and thus cannot be consistent with continuum theory or a finite driving power.…”