Uniform Stabilization to a Non-trivial Equilibrium of a Fluid-viscoelastic Structure Interaction Model

Graduate #54
Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics
Session: 4
Room: Park Tower 8219

Tony Haines - Old Dominion University

In this project, we aim to study the stabilization to a non-trivial equilibrium of a fluid-viscoelastic structure interaction (FVI) model. FVI studied here belongs to a subcategory of fluid-structure interaction (FSI) where the interaction between fluid and a structure submerged in the fluid is at the heart of the matter. Some examples of FSI are a submarine submerged in the ocean or the blood cells in the human blood. The underlying system of PDEs that determines the dynamical behavior of the FVI comprised of the full nonlinear Navier-Stokes equation and a wave equation. The interaction between the fluid and the viscoelastic structure occurs at an interface between the two medium. The coupled system is subject to a time independent external force, which generates non-trivial and possibly unstable equilibrium. The goal is to stabilize the coupled system via feedback mechanism. The solid in the interactive system is made from viscoelastic material. As is well known, viscoelastic material could dissipate heat and is widely used to stabilize mechanical systems. The fluid is subject to an additional proportional feedback. An import feature of the proposed feedback mechanism is that there are no feedback on the interface. We run numerical simulation to verify the theoretical result that FVI subject to the aforementioned feedback mechanism is uniformly stable. The numerical scheme is constructed based on the finite element method and the discontinuous Galerkin method. We apply the monolithic approach to unify the velocity of the fluid and the velocity of the solid through the transmission condition on the interface. The numerical simulation shows that the energy of the coupled system does not decay if there are no damping in the system; the interior damping on the fluid has to be strong enough in order to achieve an exponential decay rate of the energy functional around the non-trivial equilibrium. In the future, we will run more numerical simulation with different parameters of fluid viscosity and we will also find a quantitative relationship between the decay rate of the energy and the strength of the interior damping on the fluid.

Funder Acknowledgement(s): Funding was provided by an NSF Grant No. 1601127 to Yongjin Lu.

Faculty Advisor: Yongjin Lu, ylu@vsu.edu

Role: The numerical simulation, variation of certain parameters and writing of some parts of the codes was conducted by me.