Steady viscous flow within a parabolic boundary

The steady flow of a viscous incompressible fluid within a parabolic boundary is considered. The boundary wall is assumed to slide with a prescribed speed which is equal to the surface speed of an inviscid uniform flow past a parabola without incidence. First, the full Navier-Stokes equations are solved numerically for several values of Reynolds number up to 1000. At a large Reynolds number, both the fast motion of the fluid and the vorticity are localized in a thin layer near the wall and the motion of the fluid exterior to the layer is irrotational and very slow. Second, a solution in the form of an asymptotic expansion for a large Reynolds number is obtained by boundary-layer approximation. The boundary-layer solution agrees with the exact Navier-Stokes solution at a large Reynolds number and gives the correct limit of the exact solution at the infinite Reynolds number.