Stabilization of Kelvin-Voigt viscoelastic Fuid Fow model

TL;DR

This paper proves the stabilization and convergence of solutions for the Kelvin-Voigt viscoelastic fluid flow model, including uniform results as the regularization parameter approaches zero, linking to Navier-Stokes equations.

Contribution

It establishes stabilization results for the Kelvin-Voigt model and demonstrates uniform convergence to Navier-Stokes system as the regularization parameter tends to zero.

Findings

Proved convergence of unsteady solutions to steady state.

Derived power and exponential convergence under various conditions.

Results are uniform as the regularization parameter approaches zero.

Abstract

In this article, stabilization result for the viscoelastic fluid flow problem governed by Kelvin-Voigt model, that is, convergence of the unsteady solution to a steady state solution is proved under the assumption that linearized self-adjoint steady state eigenvalue problem has a minimal positive eigenvalue. Both power and exponential convergence results are derived under various conditions on the forcing function. It is shown that results are valid uniformly in the time relaxation or some times called regularization parameter $\kappa$ as $\kappa\to 0$, which in turn, establishes results for the Navier-Stokes system.