Energy dissipation near the outflow boundary in the vanishing viscosity limit

We consider the incompressible Navier-Stokes and Euler equations in a bounded domain with non-characteristic boundary condition, and study the energy dissipation near the outflow boundary in the zero-viscosity limit. We show that in a general setting, the energy dissipation rate is proportional to $\bar{U}{\bar{V}}^2$, where $\bar{U}$ is the strength of the suction and $\bar{V}$ is the tangential component of the difference between the Euler and the Navier-Stokes solutions on the outflow boundary. Moreover, we show that the enstrophy within a layer of order $\nu \mathrm{/}\bar{U}$ is comparable with the total enstrophy. The rate of enstrophy production near the boundary is inversely proportional to $\nu$.