On variable viscosity and enhanced dissipation

In this article we consider the two-dimensional 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. For instance, these shear flows may grow exponentially. Moreover it turns out that, in contrast to the constant viscosity case, decreasing viscosity leads to stronger enhanced dissipation and increasing viscosity leads to weaker dissipation.