Equilibration and convected limit in 2D-1D corotational Oldroyd's fluid-structure interaction
Prince Romeo Mensah
TL;DR
The article studies a 2D corotational Oldroyd fluid with solute diffusion strongly coupled to a 1D viscoelastic shell on a moving domain. It first proves exponential decay to equilibrium for the stress, shell, and velocity components, with explicit rates depending on diffusion, viscosity, and geometric constants. Then it performs a vanishing-diffusion limit (equivalently, infinite relaxation time) to a diffusionless convected-derivative Oldroyd system, establishing convergence via a relative-energy framework on a transformed domain and proving a weak-strong uniqueness principle in the limit. The approach combines moving-domain reformulations (Hanzawa transform), energy-dissipation estimates, and Grönwall-type arguments to connect the diffusive-damped model to its convected-derivative counterpart, providing rigorous justification for reduced models in polymeric fluid-structure interaction.
Abstract
We consider a solute-solvent-structure mutually coupled system of equations given by an Oldroyd-type model for a two-dimensional dilute corotational polymer fluid with solute diffusion and damping that is interacting with a one-dimensional viscoelastic shell. Firstly, we give the rate at which its solution decays exponentially in time to the equilibrium solution, independent of the choice of the initial datum. Secondly, as the polymer relaxation time goes to infinity (or, equivalently, the center-of mass diffusion goes to zero), we show that any family of strong solutions of the system described above, that is parametrized by the relaxation time, converges to an essentially bounded weak solution of a corotational polymer fluid-structure interaction system whose solute evolves according to the convected time derivative of its extra stress tensor. A consequence of this is a weak-strong uniqueness result.