In this talk we discuss the long-time dynamics of linearized Navier Stokes equations near spectrally stable shear flows on a torus. Physically, there are two significant phenomena: inviscid damping of velocity fields over time and depletion of vorticity from critical points of the shear flow. In recent years the mechanisms for both phenomena have been gradually understood in the context of ideal fluids, e.g. when the viscosity of the fluid is assumed to be zero. A natural question is whether the same phenomena would occur in the viscous case when the viscosity is small. We will discuss some recent progress on this problem and present an upcoming result which establishes inviscid damping and vorticity depletion that are uniform with respect to viscosity. This is joint work with Shan Chen and Hao Jia