  We study the large time behavior of solutions to two-dimensional Euler and
Navier-Stokes equations linearized about shear flows of the mixing layer type
in the unbounded channel $\mathbb{T} \times \mathbb{R}$. Under a simple
spectral stability assumption on a self-adjoint operator, we prove a local form
of the linear inviscid damping that is uniform with respect to small viscosity.
We also prove a local form of the enhanced viscous dissipation that takes place
at times of order $\nu^{-1/3}$, $\nu$ being the small viscosity. To prove these
results, we use a Hamiltonian approach, following the conjugate operator method
developed in the study of Schr\"odinger operators, combined with a
hypocoercivity argument to handle the viscous case.
