Michael Shearer

Undercompressive shocks in thin film flows

Abstract Equations of the type ht+(h2−h3)x=−ϵ3(h3hxxx)x arise in the context of thin liquid films driven by the competing effects of a thermally induced surface tension gradient and gravity. In this paper, we focus on the interaction between the fourth order regularization and the nonconvex flux. Jump initial data, from a moderately thick film to a thin precurser layer, is shown to give rise to a double wave structure that includes an undercompressive wave. This wave, which approaches an undercompressive shock as ϵ→0, is an accumulation point for a countable family of compressive waves having the same speed. The family appears through a series of bifurcations parameterized by the shock speed. At each bifurcation, a pair of traveling waves is produced, one being stable for the PDE, the other unstable. The conclusions are based primarily on numerical results for the PDE, and on numerical investigations of the ODE describing traveling waves. Fourth order linear regularization is observed to produce a similar bifurcation structure of traveling waves.

EXISTENCE OF UNDERCOMPRESSIVE TRAVELING WAVES IN THIN FILM EQUATIONS

We consider undercompressive traveling wave solutions ofthe partial differential equation ∂th + ∂xf (h )= −∂x(h 3 ∂ 3h )+ D∂x(h 3 ∂xh), when the flux function f has the nonconvex form f (h )= h2 − h3. In numerical simulations, these waves appear to play a central role in the dynamics ofthe PDE; they also explain unusual phenomena in experiments ofdriven contact lines modeled by the PDE. We prove existence ofan undercom- pressive traveling wave solution for sufficiently small nonnegative D and nonexistence when D is sufficiently large.

Riemann problems for nonstrictly hyperbolic 2×2 systems of conservation laws

On resout le probleme de Riemann pour des systemes 2×2 de lois de conservation hyperboliques ayant des fonctions flux quadratiques

Stability of compressive and undercompressive thin film travelling waves

Recent studies of liquid lms driven by competing forces due to surface tension gradients and gravity reveal that undercompressive travelling waves play an important role in the dynamics when the competing forces are comparable. In this paper, we provide a theoretical framework for assessing the spectral stability of compressive and undercompressive travelling waves in thin lm models. Associated with the linear stability problem is an Evans function which vanishes precisely at eigenvalues of the linearized operator. The structure of an index related to the Evans function explains computational results for stability of compressive waves. A new formula for the index in the undercompressive case yields results consistent with stability. In considering stability of undercompressive waves to transverse perturbations, there is an apparent inconsistency between long-wave asymptotics of the largest eigenvalue and its actual behaviour. We show that this paradox is due to the unusual structure of the eigenfunctions and we construct a revised long-wave asymptotics. We conclude with numerical computations of the largest eigenvalue, comparisons with the asymptotic results, and several open problems associated with our ndings.

Shear-driven size segregation of granular materials: modeling and experiment.

Granular materials segregate by size under shear, and the ability to quantitatively predict the time required to achieve complete segregation is a key test of our understanding of the segregation process. In this paper, we apply the Gray-Thornton model of segregation (developed for linear shear profiles) to a granular flow with an exponential shear profile, and evaluate its ability to describe the observed segregation dynamics. Our experiment is conducted in an annular Couette cell with a moving lower boundary. The granular material is initially prepared in an unstable configuration with a layer of small particles above a layer of large particles. Under shear, the sample mixes and then resegregates so that the large particles are located in the top half of the system in the final state. During this segregation process, we measure the velocity profile and use the resulting exponential fit as input parameters to the model. To make a direct comparison between the continuum model and the observed segregation dynamics, we map the local concentration (from the model) to changes in packing fraction; this provides a way to make a semiquantitative comparison with the measured global dilation. We observe that the resulting model successfully captures the presence of a fast mixing process and relatively slower resegregation process, but the model predicts a finite resegregation time, while in the experiment resegregation occurs only exponentially in time.

Secondary Bifurcation Near a Double Eigenvalue

General conditions are formulated under which secondary bifurcation is rigorously established for a family of bifurcation problems depending continuously on a real auxiliary parameter. With more specific conditions, it is shown that, although the presence of secondary bifurcation renders the problem a priori degenerate, a full local bifurcation analysis is still possible.The results of this paper demonstrate the prime importance of symmetry (or more generally, invariance) to the mechanism by which secondary bifurcation points are created as the auxiliary parameter is varied.

Nonlinear evolution equations that change type

This volume should be of interest to applied mathematicians, to researchers in partial differential equations, and to those involved in fluid dynamics and numerical analysis examining models for viscoelastic flows, porous medium and granular flows, and flows exhibiting phase transitions. As papers in this volume indicate, physical processes whose simplest models may involve change of type occur also in other dynamic contexts, such as in the simulation of oil reservoirs, involving multiphase flow in a porous medium, and in granular flow.

Time-dependent solutions for particle-size segregation in shallow granular avalanches

Rapid shallow granular free-surface flows develop in a wide range of industrial and geophysical flows, ranging from rotating kilns and blenders to rock-falls, snow slab-avalanches and debris-flows. Within these flows, grains of different sizes often separate out into inversely graded layers, with the large particles on top of the fines, by a process called kinetic sieving. In this paper, a recent theory is used to construct exact time-dependent two-dimensional solutions for the development of the particle-size distribution in inclined chute flows. The first problem assumes the flow is initially homogeneously mixed and is fed at the inflow with homogeneous material of the same concentration. Concentration shocks develop during the flow and the particles eventually separate out into inversely graded layers sufficiently far downstream. Sections with a monotonically decreasing shock height, between these layers, steepen and break in finite time. The second problem assumes that the material is normally graded, with the small particles on top of the coarse ones. In this case, shock waves, concentration expansions, non-centred expanding shock regions and breaking shocks develop. As the parameters are varied, nonlinearity leads to fundamental topological changes in the solution, and, in simple-shear, a logarithmic singularity prevents a steady-state solution from being attained.

Loss of real characteristics for models of three-phase flow in a porous medium

In this paper we examine the generalized Buckley-Leverett equations governing threephase immiscible, incompressible flow in a porous medium, in the absence of gravitational and diffusive/dispersive effects. We consider the effect of the relative permeability models on the characteristic speeds in the flow. Using a simple idea from projective geometry, we show that under reasonable assumptions on the relative permeabilities there must be at least one point in the saturation triangle at which the characteristic speeds are equal. In general, there is a small region in the saturation triangle where the characteristic speeds are complex. This is demonstrated with the numerical results at the end of the paper.

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion is known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is nonconvex. This review compares the structure of solutions of Riemann problems for a conservation law with nonconvex, cubic flux regularized by two different mechanisms: (1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and (2) a combination of diffusion and dispersion in the mKdV--Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs), and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both class...