  We prove a stability threshold theorem for 2D Navier-Stokes on three
unbounded domains: the whole plane $\mathbb{R} \times \mathbb{R}$, the half
plane $\mathbb{R} \times [0,\infty)$ with Navier boundary conditions, and the
infinite channel $\mathbb{R} \times [-1, 1]$ with Navier boundary conditions.
Starting with the Couette shear flow, we consider initial perturbations
$\omega_{in}$ which are of size $\nu^{1/2}(1+\ln(1/\nu)^{1/2})^{-1}$ in an
anisotropic Sobolev space with an additional low frequency control condition
for the planar cases. We then demonstrate that such perturbations exhibit
inviscid damping of the velocity, as well as enhanced dissipation at
$x$-frequencies $|k| \gg \nu$ with decay time-scale $O(\nu^{-1/3}|k|^{-2/3})$.
On the plane and half-plane, we show Taylor dispersion for $x$-frequencies $|k|
\ll \nu$ with decay time-scale $O(\nu |k|^{-2})$, while on the channel we show
low frequency dispersion for $|k| \ll \nu$ with decay time-scale $O(\nu^{-1})$.
Generalizing the work of arXiv:2311.00141 done on $\mathbb{T} \times [-1,1]$,
the key contribution of this paper is to perform new nonlinear computations at
low frequencies with wave number $|k| \lesssim \nu$ and at intermediate
frequencies with wave number $\nu \lesssim |k| \leq 1$, and to provide the
first enhanced dissipation result for a fully-nonlinear shear flow on an
unbounded $x$-domain.
