Michael J. Shelley

Hele - Shaw flow and pattern formation in a time-dependent gap

We consider flow in a Hele - Shaw cell for which the upper plate is being lifted uniformly at a specified rate. This lifting puts the fluid under a lateral straining flow, sucking in the interface and causing it to buckle. The resulting short-lived patterns can resemble a network of connections with triple junctions. The basic instability - a variant of the Saffman - Taylor instability - is found in a version of the two-dimensional Darcys law, where the divergence condition is modified to account for the lifting of the plate. For analytic data, we establish the existence, uniqueness and regularity of solutions when the surface tension is zero. We also construct some exact analytic solutions, both with and without surface tension. These solutions illustrate some of the possible behaviours of the system, such as cusp formation and bubble fission. Further, we present the results of numerical simulations of the bubble motion, examining in particular the distinctive pattern formation resulting from the Saffman - Taylor instability, and the effect of surface tension on a bubble evolution that in the absence of surface tension would fission into two bubbles. AMS classification scheme numbers: 76E30, 76D45

Instabilities and singularities in Hele-Shaw flow

A mechanism by which smooth initial conditions evolve towards a topological reconfiguration of fluid interfaces is studied in the context of Darcy’s law. In the case of thin fluid layers, nonlinear PDEs for the local thickness are derived from an asymptotic limit of the vortex sheet representation. A particular example considered is the Rayleigh–Taylor instability of stratified fluid layers, where the instability of the system is controlled by a Bond number B. It is proved that, for a range of B and initial data “subharmonic” to it, interface pinching must occur in at least infinite time. Numerical simulations suggest that “pinching” singularities occur generically when the system is unstable, and in particular immediately above a bifurcation point to instability. Near this bifurcation point an approximate analytical method describing the approach to a finite-time singularity is developed. The method exploits the separation of time scales that exists close to the first instability in a system of finite ex...

SINGULARITY FORMATION IN THIN JETS WITH SURFACE TENSION

We derive and study asymptotic models for the dynamics of a thin jet of fluid that is separated from an outer immiscible fluid by fluid interfaces with surface tension. Both fluids are assumed to be in- compressible, inviscid, irrotational, and density matched. One such thin jet model is a coupled system of PDEs with nonlocal terms { Hilbert transforms { that result from expansion of a Biot-Savart integral. In order to make the asymptotic model well-posed, the Hilbert transforms act upon time derivatives of the jet thickness, making the system implicit. Within this thin jet model, we demonstrate numerically the formation of nite-time pinching singularities, where the width of the jet collapses to zero at a point. These singularities are driven by the surface tension, and are very similar to those observed previously by Hou, Lowengrub, and Shelley in large-scale simulations of the Kelvin-Helmholtz instability with surface tension, and in other related studies. Dropping the nonlocal terms of the model, we also study a much simpler local model. For this local model we can preclude analytically the formation of certain types of singularities, though not those of pinching type. Surprisingly, we nd that this local model forms pinching singularities of a very similar type to those of the nonlocal thin jet model.

High-order and efficient methods for the vorticity formulation of the Euler equations

In this work, the authors develop new methods for the accurate and efficient solution of the two-dimensional, incompressible Euler equations in the vorticity form. Here, the velocity is recovered directly from the Biot–Savart relation with vorticity, and the vorticity is evolved through its transport equation. Using a generalized Poisson summation formula, the full asymptotic error expansion is constructed for the second-order point vortex approximation to the Biot–Savart integral over a rectangular grid. The expansion is in powers of

Domain of convergence of perturbative solutions for Hele-Shaw flow near interface collapse

h^2

Spatial and temporal stability issues for interfacial flows with surface tension

, and its coefficients depend linearly upon only local derivatives of the vorticity. In particular, the second-order term depends only upon the vorticity gradient. Except at second-order, the coefficients also involve rapidly convergent, two-dimensional lattice sums. At second-order, the sum is conditionally convergent, but can be calculated easily and rapidly. Therefore, we can remove the second-order term explicitly from the point vortex approximation to obtain a fou...

The convergence of an exact desingularization for vortex methods

Recent work [Phys. Fluids 10, 2701 (1998)] has shown that for Hele-Shaw flows sufficiently near a finite-time pinching singularity, there is a breakdown of the leading-order solutions perturbative in a small parameter e controlling the large-scale dynamics. To elucidate the nature of this breakdown we study the structure of these solutions at higher order. We find a finite radius of convergence that yields a new length scale exponentially small in e. That length scale defines a ball in space and time, centered around the incipient singularity, inside of which perturbation theory fails. Implications of these results for a possible matching of outer solutions to inner scaling solutions are discussed.

Removing the stiffness from interfacial flows with surface tension

Many physically interesting problems involve the propagation of free surfaces in fluids with surface tension effects. Surface tensions is an ever-present physical effect that is often neglected due to the difficulties associated with its inclusion in the equations of motion. Accurate simulation of these interfaces presents a problem of considerable difficulty on several levels. First, even for stably stratified flows like water waves, it turns out that straightforward spatial discretizations (of the boundary integral formulation) generate numerical instability. Second, surface tension introduces a large number of derivatives through the Laplace-Young boundary condition. This induces severe time step restrictions for explicit time integration methods. In this paper, we present a class of stable spatial discretizations and we present a reformulation of the equations of motion that make apparent how to remove the high order time step restrictions introduced by the surface tension. This paper is a review of the results given in [1,2].

Models of non-Newtonian Hele-Shaw flow

Expanding upon an observation of Hou [Math. Comp., submitted], exact desingularizations are presented of the Euler equations in two and three dimensions for which the singularity within the Biot–Savart integrand is reduced by one order. The reformulated equations are then solved numerically using either the point vortex method or the vortex blob method. The increased smoothness of the Biot–Savart integrand allows us to prove convergence of these methods in the maximum norm. Our numerical experiments show that discretization of the reformulated equations display increased stability relative to discretizations of the original equations. The improvement in stability is manifested as a more slowly growing error in time.