Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension

Abstract

We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface. The motion is driven by a constant surface tension acting at the free boundary so that, with the effects of gravity ignored, one expects the boundary to approach a circular form as time evolves. It is shown that, if at some initial instant the region occupied by the fluid is given by a rational conformal map of the unit disc, then it must retain this property as long as the region remains simply-connected. Moreover, its evolution may be described analytically; in simple cases this description is explicit, but in more complicated problems the numerical integration of a system of first order differential equations may be required.

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

“…We therefore consider how it might be removed. We can for example, as was done in [41], insist that φ(0, t) = 0 (or, although we do not yet do so, any other specified function of t). This purely mathematical assumption is clearly physically appropriate in cases where the flow is symmetric about, for example, the x-and y-axes, for then in the absence of an externally-imposed uniform translation the non-singular velocity at z = 0 vanishes.…”

“…Let us now return to our outline of the theory. Referring to [41] and [10] for the details (though we give a brief outline in §4), when we reformulate the problem in the ζ-plane, the boundary conditions may be analytically continued off the unit circle to give functional identities holding globally in the ζ-plane (equations (2.18) and (2.19) of [41]). In terms of the functions X (ζ, t) and Φ(ζ, t) introduced above, these equations are most conveniently expressed as follows:…”

“…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/ .…”

Drawing of micro-structured fibres: circular and non-circular tubes

“…But, remarkably, it happens that the two-dimensional problem for the quasi-steady evolution of free-surface Stokes flows driven purely by surface tension is known to admit a rich variety of analytical solutions and we have chosen to focus on such solutions in this paper. Besides the geometrically simple case of a concentric annular tube there are known analytical solutions involving non-trivial geometries due to Hopper (1990), Richardson (1992), Crowdy & Tanveer (1998a,b), Cummings, Howison & King (1997), Richardson (2000), Crowdy (2003), among others. Without exception, these exact solutions derive from a complex-variable formulation of two-dimensional Stokes flow, one involving so-called Goursat functions common in plane elasticity (Langlois 1964;Muskhelishvili 1977).…”

Exact results with the J-integral applied to free-boundary flows

“…These are integrals over a two-dimensional domain whereas the conserved integrals that we discuss here are along a one-dimensional path within a two-dimensional domain. For further discussion of this and related conservation integrals, see Richardson (1972Richardson ( , 1992, Tanveer & Vasconcelos (1995), Cummings, King & Howison (1997), Crowdy & Tanveer (1998) and Crowdy (1999).…”