Numerical studies of singularity formation at free surfaces and fluid interfaces in two-dimensional Stokes flow

Abstract

We consider the analytic structure of interfaces in several families of steady and unsteady two-dimensional Stokes flows, focusing on the formation of corners and cusps. Previous experimental and theoretical studies have suggested that, without surface tension, the interfaces spontaneously develop such singular points. We investigate whether and how corners and cusps actually develop in a time-dependent flow, and assess the stability of stationary cusped shapes predicted by previous authors. The motion of the interfaces is computed with high resolution using a boundary integral method for three families of flows. In the case of a bubble that is subjected to the family of straining flows devised by Antanovskii, we find that a stationary cusped shape is not likely to occur as the asymptotic limit of a transient deformation. Instead, the pointed ends of the bubble disintegrate in a process that is reminiscent of tip streaming. In the case of the flow due to an array of point-source dipoles immersed beneath a free surface, which is the periodic version of a flow proposed by Jeong & Moffatt, we find evidence that a cusped shape indeed arises as the result of a transient deformation. In the third part of the numerical study, we show that, under certain conditions, the free surface of a liquid film that is levelling under the action of gravity on a horizontal or slightly inclined surface develops an evolving corner or cusp. In certain cases, the film engulfs a small air bubble of ambient fluid to obtain a composite shape. The structure of a corner or a cusp in an unsteady flow does not have a unique shape, as it does at steady state. In all cases, a small amount of surface tension is able to prevent the formation of a singularity, but replacing the inviscid gas with a viscous liquid does not have a smoothing effect. The ability of the thin-film lubrication equation to produce mathematical singularities at the free surface of a levelling film is also discussed.

An elliptical-pore model for late-stage planar viscous sintering

“…At present, however, the theory of Greengard et al (1996) is restricted to domains of finite connectivity and generalization to the domains with infinite connectivity considered here has yet to be made. Furthermore, minor modification of the numerical method presented here can be used to compute the periodic Jeong-Moffatt flow considered by Pozrikidis (1997) using different methods. The modification required is similar to that considered in Spencer & Meiron (1994) who compute the periodic stress distribution in a semiinfinite elastic solid.…”

An Efficient Numerical Method for Studying Interfacial Motion in Two-Dimensional Creeping Flows

“…[27,34,35], for example). However, for long-time simulations where the interface changes shape significantly, the Lagrangian description may lead to either clustering or inadequate resolution of marker points.…”

“…In this section, we outline the corresponding integral equations associated with fluid interfaces in a Stokes flow. (We note here that the Sherman-Lauricella integral equations can be made to be equivalent to the integral equations used in [9,27,34,35], which are based on primitive variable formulations. This is discussed in detail in [13].…”

“…We argue, however, that the motion of marker points on the interface can become stiff if they are allowed to cluster at small spatial scales in regions of high curvature, and we demonstrate this point numerically with one particular mesh-refinement scheme. Since the cost of solving integral equations for Stokes flow can be high, many authors [27,34,35] refine the mesh in regions of high curvature in order to minimize the number of points needed for adequate resolution. The resulting stiffness cause Van de Vorst and Mattheij to use backward-difference formulae to avoid the prohibitively small step sizes needed by explicit time-stepping schemes.…”

Numerical Methods for Multiple Inviscid Interfaces in Creeping Flows

“…Standard implementations of iterative schemes (c.f. [5,29]) require O(N 2 ) operations, thus limiting these studies to only modest-sized problems in the range of 20-40 drops or bubbles (an insufficient number to determine statistical properties of evolving microstructure [5], for example). Recently, Zinchenko and Davis [42] presented an economic multipole technique for large-scale simulations of three-dimensional drops.…”