  In this article we consider the 2D Navier-Stokes equations with variable
viscosity depending on the vertical position. As our main result we establish
linear enhanced dissipation near the non-affine stationary states replacing
Couette flow. Moreover it turns out that the shear flow overcompensates for
weakening viscosity: decreasing viscosity leads to stronger enhanced
dissipation and increasing viscosity leads to weaker dissipation than in the
constant viscosity case.
