Andrew J. Bernoff

Stability of Self-similar Solutions for Van der Waals Driven Thin Film Rupture

Recent studies of pinch-off of filaments and rupture in thin films have found infinite sets of first-type similarity solutions. Of these, the dynamically stable similarity solutions produce observable rupture behavior as localized, finite-time singularities in the models of the flow. In this letter we describe a systematic technique for calculating such solutions and determining their linear stability. For the problem of axisymmetric van der Waals driven rupture (recently studied by Zhang and Lister), we identify the unique stable similarity solution for point rupture of a thin film and an alternative mode of singularity formation corresponding to annular “ring rupture.”

Dynamics of Three-Dimensional Thin Film Rupture

Abstract We consider the problem of thin film rupture driven by van der Waals forces. A fourth-order nonlinear PDE governs the low Reynolds number lubrication model for a viscous liquid on a solid substrate. Finite-time singularities in this equation model rupture leading to formation of dry spots in the film. Our study addresses the problem of rupture in the full three-dimensional geometry. We focus on stability and selection of the dynamics determined by the initial conditions on small finite domains with planar and axisymmetric geometries. We also address the final stages of the dynamics — self-similar dynamics for point, line, and ring rupture. We will demonstrate that line and ring rupture are unstable and will generically destabilize to produce axisymmetric rupture at isolated points.

A Primer of Swarm Equilibria

We study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Beginning with a discrete dynamical model, we derive a variational description of the corresponding continuum population density. Equilibrium solutions are extrema of an energy functional and satisfy a Fredholm integral equation. We find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal Lagrangian displacements of mass. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. These agree closely with numerical simulations of the underlying discrete model. The exact solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework. The equilibria typically are compactly supported and may contain δ-concentrations or jump discontinuities at the edge of the support. We apply our methods to a model of locust swarms, which in nature are observed to consist of a concentrated population on the ground separated from an airborne group. Our model can reproduce this configuration; quasi-two-dimensionality of the model plays a critical role.

Transient anomalous diffusion in Poiseuille flow

We revisit the classical problem of dispersion of a point discharge of tracer in laminar pipe Poiseuille flow. For a discharge at the centre of the pipe we show that in the limit of small non-dimensional diffusion, D , tracer dispersion can be divided into three regimes. For small times ( t [Lt ] D −1/3 ), diffusion dominates advection yielding a spherically symmetric Gaussian dispersion cloud. At large times ( t [Gt ] D −1 ), the flow is in the classical Taylor regime, for which the tracer is homogenized transversely across the pipe and diffuses with a Gaussian distribution longitudinally. However, in an intermediate regime (D −1/3 [Gt ] t [Gt ] D −1 ), the longitudinal diffusion is anomalous with a width proportional to t [Lt ] Dt 2 and a distinctly asymmetric longitudinal distribution. We present a new solution valid in this regime and verify our results numerically. Analogous results are presented for an off-centre release; here the distribution width scales as D 1/2 t 3/2 in the anomalous regime. These results suggest that anomalous diffusion is a hallmark of the shear dispersion of point discharges at times earlier than the Taylor regime.

Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition

Abstract Phase transitions can be modeled by the motion of an interface between two locally stable phases. A modified Kuramoto-Sivashinsky equation, h t + ∇ 2 h + ∇ 4 h = (1 − λ )|∇ h | 2 ± λ (∇ 2 h ) 2 + δλ ( h xx h yy − h xy 2 ), describes near planar interfaces which are marginally long-wave unstable. We study the question of finite-time singularity formation in this equation in one and two space dimensions on a periodic domain. Such singularity formation does not occur in the Kuramoto-Sivashinsky equation ( λ = 0). For all 1 ≥ λ > 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite-time blow up in which a second derivative of h becomes unbounded. Using a bifurcation theory analysis, we show a parallel between the stability of steady periodic solutions and the question of finite-time blow up in one dimension. Finally, we consider the local structure of the blow up in the one-dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self-similar grid. The simulations resolve the singularity to over 25 decades in |h xx | l ∞ and indicate that the singularities are all locally described by a unique self-similar profile in h xx . We discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics.

Rapid relaxation of an axisymmetric vortex

In this paper it is argued that a two‐dimensional axisymmetric large Reynolds number (Re) monopole when perturbed will return to an axisymmetric state on a time scale (Re1/3) that is much faster than the viscous evolution time scale (Re). It is shown that an arbitrary perturbation can be broken into three pieces; first, an axisymmetric piece corresponding to a slight radial redistribution of vorticity; second, a translational piece which corresponds to a small displacement of the center of the original vortex; and finally, a nonaxisymmetric perturbation which decays on the Re1/3 time scale due to a shear/diffusion averaging mechanism studied by Rhines and Young [J. Fluid Mech. 133, 133 (1983)] for a passive scalar and Lundgren [Phys. Fluids 25, 2193 (1982)] for vorticity. This mechanism is verified numerically for the canonical example of a Lamb monopole. This result suggests a physical explanation for the persistence of monopole structures in large Reynolds flows, such as decaying turbulence.

Quasi-steady monopole and tripole attractors for relaxing vortices

Using fully nonlinear simulations of the two-dimensional Navier–Stokes equations at large Reynolds number (Re), we bracket a threshold amplitude above which a perturbed Gaussian monopole will relax to a quasi-steady, rotating tripole, and below which will relax to an axisymmetric monopole. The resulting quasi-steady structures are robust to small perturbations. We propose a means of measuring the decay rate of disturbances to asymptotic vortical structures wherein streamlines and lines of constant vorticity correspond in some rotating or translating frame. These experiments support the hypothesis that small or moderate deviations from asymptotic structures decay through inviscid and viscous mixing.

A model for rolling swarms of locusts

Abstract.We construct an individual-based kinematic model of rolling migratory locust swarms. The model incorporates social interactions, gravity, wind, and the effect of the impenetrable boundary formed by the ground. We study the model using numerical simulations and tools from statistical mechanics, namely the notion of H-stability. For a free-space swarm (no wind and gravity), as the number of locusts increases, the group approaches a crystalline lattice of fixed density if it is H-stable, and in contrast becomes ever denser if it is catastrophic. Numerical simulations suggest that whether or not a swarm rolls depends on the statistical mechanical properties of the corresponding free-space swarm. For a swarm that is H-stable in free space, gravity causes the group to land and form a crystalline lattice. Wind, in turn, smears the swarm out along the ground until all individuals are stationary. In contrast, for a swarm that is catastrophic in free space, gravity causes the group to land and form a bubble-like shape. In the presence of wind, the swarm migrates with a rolling motion similar to natural locust swarms. The rolling structure is similar to that observed by biologists, and includes a takeoff zone, a landing zone, and a stationary zone where grounded locusts can rest and feed.

Self-similar asymptotics for linear and nonlinear diffusion equations

The long-time asymptotic solutions of initial value problems for the heat equation and the nonlinear porous medium equation are self-similar spreading solutions. The symmetries of the governing equations yield three-parameter families of these solutions given in terms of their mass, center of mass, and variance. Unlike the mass and center of mass, the variance, or “time-shift,” of a solution is not a conserved quantity for the nonlinear problem. We derive an optimal linear estimate of the long-time variance. Newmans Lyapunov functional is used to produce a maximum entropy time-shift estimate. Results are applied to nonlinear merging and time-dependent, inhomogeneously forced diffusion problems.

Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation

Abstract Following the ideas of Howard and Kopell [9] a perturbation theory is developed for slowly varying fully nonlinear wavetrains (i.e. solutions which appear locally as travelling waves, but with frequencies and wavelengths which may vary widely on long length and time scales). This perturbation theory is applied to the Ginzburg-Landau equation. The motion and stability of slowly varying wavetrains is shown to be governed by a simple wave equation which can develop shocks corresponding to rapid changes in wavenumber. Numerical results supporting this theory are presented. A shock structure is proposed and numerically verified. These results together with a winding invariant valid in the limit of slow variation suggest that over a large range of parameters many initial conditions relax to uniform wavetrains. The evolution of a marginally diffusively stable wavetrain is also examined; it is argued that the evolution is governed by a perturbed Korteweg-de Vries equation.