  We study the $L^{\infty}$ stability of the 2D Navier-Stokes equations with a
viscosity-dependent Navier boundary condition around shear profiles which are
linearly unstable for the Euler equation. The dependence from the viscosity is
given in the Navier boundary condition as $\partial_y u = \nu^{-\gamma}u$ for
some $\gamma\in\mathbb{R}$, where $u$ is the tangential velocity. With the
no-slip boundary condition, which corresponds to the limit $\gamma \to
+\infty$, a celebrated result from E. Grenier provides an instability of order
$\nu^{1/4}$. M. Paddick proved the same result in the case $\gamma=1/2$,
furthermore improving the instability to order one. In this paper, we extend
these two results to all $\gamma \in \mathbb{R}$, obtaining an instability of
order $\nu^{\theta}$, where $$\theta:=\begin{cases} \frac{1}{4} &\text{if }
\gamma \geq \frac{3}{4};\\ \gamma - \frac{1}{2} &\text{if } \frac{1}{2}<\gamma
< \frac{3}{4};\\ 0 &\text{if } \gamma \leq \frac{1}{2}. \end{cases}$$ When
$\gamma \geq 1/2$, the result denies the validity of the Prandtl boundary layer
expansion around the chosen shear profile.
