Monthly Archives: March 2017

Steve Shkoller (UC Davis)

The Analysis and PDE seminar will take place on Monday, April 3rd, in Evans 740 from 16:10 to 17:00.

Title: Models for Rayleigh-Taylor mixing and interface turnover

Abstract: The instability of a heavy fluid layer supported by a light one is generally known as Rayleigh-Taylor (RT) instability. It can occur under gravity and, equivalently, under an acceleration of the fluid system in the direction toward the denser fluid. Whenever the pressure is higher in the lighter fluid, the differential acceleration causes the two fluids to mix.

The Euler equations serve as the basic mathematical model for RT instability and mixing between two fluids. This highly unstable system of conservation laws is both difficult to analyze (as it is ill-posed in the absence of surface tension and viscosity) and simulate; DNS of RT can be prohibitively expensive. In this talk, I will describe a novel framework to derive a hierarchy of asymptotic models that can be used to predict the location and shape of the RT interface as well as the mixing of the two fluids.

The models are derived in two very different asymptotic regimes. The first regime assumes that the fluid interface is a graph with size restrictions on the slope of the interface. The model PDE inherits the RT stability condition from the Euler equations, and in the stable regime, it is both locally and globally well-posed with precise asymptotic behavior that predicts nonlinear saturation for bubble growth. In the second asymptotic regime the interface can turnover, and there are no size restrictions on the amplitude or slope of the interface.

I will describe these models and show numerical simulations and comparisons with well-known RT experiments and simulations. I will then show results of fluid mixing, and discuss current work, advancing both modeling strategies. This is joint work with Rafa Granero.

Hung Tran (University of Wisconsin-Madison)

The Analysis and PDE seminar will take place on Monday, March 20, in room 740, Evans hall, from 16:10 to 17:00.

Title: Homogenization: Beyond well-posedness theory.

Abstract: I will describe some recent progress on going beyond the well-posedness theory in homogenization of Hamilton-Jacobi equations. In particular, I will focus on the decomposition method to find the formula of the effective Hamiltonian in some situations. Joint work with Qian and Yu.

Herbert Koch (University of Bonn)

The Analysis and PDE Seminar will take place on Monday, March 13, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Stationary solutions to the 2d Euler equation

Abstract: The two dimensional Euler equation has a large number of stationary solutions. Distribution functions of the vorticity are preserved under the flow.
I will explain a parametrization of Arnold stable stationary solutions by distribution functions of their vorticity. This is joint work with Antoine Chiffrut.