Special Session 65: Recent Progress in Free Boundary Problems in Fluid Flow and Fluid-Structure Interactions

Global Well-posedness of Viscous Water-Waves

Hantaek Bae Ulsan National Institute of Science and Technology Korea Co-Author(s):    Woojae Lee, Jaeyong Shin

Abstract: In this talk, we discuss a nonlinear model of the viscous water-waves system proposed by Dias-Dychenko-Zakharov (DDZ). We first study the linear model of DDZ. We then derive a new approximated model of DDZ and show how to find a global-in time solution.

On a Nonlinear Acoustics - Structure Interaction Model

Barbara Kaltenbacher University of Klagenfurt Austria Co-Author(s):    Amjad Tuffaha

Abstract: In this talk we consider a coupled system of nonlinear acoustic structure interactions. The model consists of the nonlinear Westervelt equation on a bounded domain with non homogeneous boundary conditions, coupled with a 4th order linear equation defined on a lower dimensional interface occupying part of the boundary of the domain, with transmission boundary conditions matching acoustic velocities and acoustic pressures. While the well-posedness of the Westervelt model has been well studied in the literature, we here present new work on the analysis of this coupled problem. We establish local-in-time and global in time well-posedness for small data. Another contribution of this work, is a novel variational weak formulation of the linearized system and a consideration of various boundary conditions. If time permits, we will also provide an outlook on a control-shape optimization problem for this coupled system.

On the dynamics of the interface between two incompressible fluids in a porous media

Omar Lazar New-York Abu Dhabi university United Arab Emirates Co-Author(s):

Abstract: Consider two incompressible fluids with different densities and same viscosity in a porous media. We shall consider surface tension effect (in addition to gravity) and show why the dynamic of the interface between these two immiscible fluids can be arbitrarily large Lipschitz.

Stability of the Stokes immersed boundary problem with bending and stretching energy

Hui Li New York University Abu Dhabi United Arab Emirates Co-Author(s):

Abstract: In this talk, we present stability results for an elastic string with bending and stretching energy immersed in a 2-D Stokes flow. We introduce the curve`s tangent angle function and the stretching function to describe the different deformations of the elastic string. These two functions are defined on the arc-length coordinate and the material coordinate respectively. Reformulating the problem into a parabolic system via the fundamental solution of the Stokes equation, we establish local well-posedness in Sobolev space under non-self-intersecting and well-stretched initial configurations. For initial configurations close to an evenly parametrized circle, we prove that the solution can be extended globally and the global solution will converge exponentially to the equilibrium state.

Variational theory of a incompressible heat conducting bi-fluid system involving an elastic interface.

Sourav Mitra IIT Indore India Co-Author(s):    Dr. Sourav Mitra, Dr. Sebastian Schwarzacher

Abstract: In this talk I will present our recent result on the global existence of weak solutions of a fluid structure interaction problem involving two Newtonian incompressible heat conducting fluids separated by an elastic plate/ shell. The elastic plate involved allows heat transfer between the two fluids. We show the global existence of weak solution to our model until the Koiter energy degenerates or the structure undergoes a self intersection or touches the rigid boundary $\partial\Omega$. The two fluids involved are assumed to have different viscosities and they depend on the respective temperatures. The temperature dependence of the viscosities is modeled by the celebrated Vogel-Fulcher-Tammann equation. We use a variational strategy in order to construct weak solutions which involves regularization of the Koiter energy, adding artificial dissipation and discretization using two different time scales: velocity scale and the acceleration scale.