Justification of lubrication approximation: an application to fluid/solid interactions

We consider the stationary Stokes problem in a three-dimensional fluid domain $\mathcal F$ with non-homogeneous Dirichlet boundary conditions. We assume that this fluid domain is the complement of a bounded obstacle $\mathcal B$ in a bounded or an exterior smooth container $\Omega$. We compute sharp asymptotics of the solution to the Stokes problem when the distance between the obstacle and the container boundary is small

Universality in Fluid Domain Coarsening: The case of vapor-liquid transition

Domain growth during the kinetics of phase separation is studied following vapor-liquid transition in a single component Lennard-Jones fluid. Results are analyzed after appropriately mapping the continuum snapshots obtained from extensive molecular dynamics simulations to a simple cubic lattice. For near critical quench interconnected domain morphology is observed. A brief period of slow diffusive growth is followed by a linear viscous hydrodynamic growth that lasts for an extended period of time. This result is in contradiction with earlier inclusive reports of late time growth exponent 1/2 that questions the uniqueness of the non-equilibrium universality for liquid-liquid and vapor-liquid transitions.Comment: 6 pages, 5 figure

Global diffeomorphism of the Lagrangian flow-map defining Equatorially trapped water waves

The aim of this paper is to prove that a three dimensional Lagrangian flow which defines equatorially trapped water waves is dynamically possible. This is achieved by applying a mixture of analytical and topological methods to prove that the nonlinear exact solution to the geophysical governing equations, derived by Constantin in J. Geophys. Res., 117 (2012), is a global diffeomorphism from the Lagrangian labelling variables to the fluid domain beneath the free surface

Ergodicity of Stochastically Forced Large Scale Geophysical Flows

We investigate the ergodicity of 2D large scale quasigeostrophic flows under random wind forcing. We show that the quasigeostrophic flows are ergodic under suitable conditions on the random forcing and on the fluid domain, and under no restrictions on viscosity, Ekman constant or Coriolis parameter. When these conditions are satisfied, then for any observable of the quasigeostrophic flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long

Vanishing viscosity limit for an expanding domain in space

We study the limiting behavior of viscous incompressible flows when the fluid domain is allowed to expand as the viscosity vanishes. We describe precise conditions under which the limiting flow satisfies the full space Euler equations. The argument is based on truncation and on energy estimates, following the structure of the proof of Kato's criterion for the vanishing viscosity limit. This work complements previous work by the authors, see [Kelliher, Comm. Math. Phys. 278 (2008), 753-773] and [arXiv:0801.4935v1].Comment: 23 pages, submitted for publicatio

Splash singularities for the one-phase Muskat problem in stable regimes

This paper shows finite time singularity formation for the Muskat problem in a stable regime. The framework we found is with a dry region, where the density and the viscosity are set equal to $0$ (the gradient of the pressure is equal to $(0,0)$) in the complement of the fluid domain. The singularity is a splash-type: a smooth fluid boundary collapses due to two different particles evolve to collide at a single point. This is the first example of a splash singularity for a parabolic problem.Comment: Minor comments added, 26 pages, 1 figur

Non-linear fate of internal wave attractors

We present a laboratory study on the instability of internal wave attractors in a trapezoidal fluid domain filled with uniformly stratified fluid. Energy is injected into the system via standing-wave-type motion of a vertical wall. Attractors are found to be destroyed by parametric subharmonic instability (PSI) via a triadic resonance which is shown to provide a very efficient energy pathway from long to short length scales. This study provides an explanation why attractors may be difficult or impossible to observe in natural systems subject to large amplitude forcing