Singular limit of 2D second grade fluid past an obstacle

TL;DR

This paper analyzes the behavior of 2D second grade non-Newtonian fluids past an obstacle, proving convergence to Euler fluids under specific parameter conditions and estimating the convergence rate.

Contribution

It establishes the singular limit of second grade fluids to Euler fluids as parameters tend to zero, with explicit convergence rate estimates.

Findings

Convergence of second grade fluids to Euler fluids under the condition  = o(/3)

Explicit estimates of the convergence rate as  and  approach zero

Validation of the second grade fluid model's behavior in the zero-parameter limit

Abstract

In this paper, we consider the 2D second grade fluid past an obstacle satisfying the standard non-slip boundary condition at the surface of the obstacle. Second grade fluid model is a well-known non-Newtonian model, with two parameters: $\alpha$ representing length-scale, while $\nu > 0$ corresponding to viscosity. We prove that, under the constraint condition $\nu = {o}(\alpha^\frac{4}{3})$, the second grade fluid with a suitable initial velocity converges to the Euler fluid as $\alpha$ tends to zero. Moreover, we estimate the convergence rate of the solution of second grade fluid equations to the one of Euler fluid equations as $\nu$ and $\alpha$ approach zero.