  The second-grade fluid equations are a model for viscoelastic fluids, with
two parameters: $\alpha > 0$, corresponding to the elastic response, and $\nu >
0$, corresponding to viscosity. Formally setting these parameters to $0$
reduces the equations to the incompressible Euler equations of ideal fluid
flow. In this article we study the limits $\alpha, \nu \to 0$ of solutions of
the second-grade fluid system, in a smooth, bounded, two-dimensional domain
with no-slip boundary conditions. This class of problems interpolates between
the Euler-$\alpha$ model ($\nu = 0$), for which the authors recently proved
convergence to the solution of the incompressible Euler equations, and the
Navier-Stokes case ($\alpha = 0$), for which the vanishing viscosity limit is
an important open problem. We prove three results. First, we establish
convergence of the solutions of the second-grade model to those of the Euler
equations provided $\nu = \mathcal{O}(\alpha^2)$, as $\alpha \to 0$, extending
the main result in [19]. Second, we prove equivalence between convergence (of
the second-grade fluid equations to the Euler equations) and vanishing of the
energy dissipation in a suitably thin region near the boundary, in the
asymptotic regime $\nu = \mathcal{O}(\alpha^{6/5})$, $\nu/\alpha^2 \to \infty$
as $\alpha \to 0$. This amounts to a convergence criterion similar to the
well-known Kato criterion for the vanishing viscosity limit of the
Navier-Stokes equations to the Euler equations. Finally, we obtain an extension
of Kato's classical criterion to the second-grade fluid model, valid if $\alpha
= \mathcal{O}(\nu^{3/2})$, as $\nu \to 0$. The proof of all these results
relies on energy estimates and boundary correctors, following the original idea
by Kato.
