  In this paper, we establish vanishing viscosity limit of the 2D Navier-Stokes
equations in a horizontally periodic strip. On the vertical direction, the
horizontal component of the velocity is subjected to two different types of
boundary conditions: at the lower boundary, we give the degenerate zero
boundary condition, while at the upper boundary, a small smooth perturbation of
non-zero constant is prescribed. Due to this different boundary condition
setting, the boundary layer effects are different near the lower and upper
boundaries, which result in different thickness of boundary layer and different
leading order boundary layer equations. We will construct an approximate
solution to this 2D Navier-Stokes equations by using higher order asymptotic
approximation and show the validity of the boundary layer expansion. The
leading order of the Euler solution is the Couette flow $(Ay,0)$ for some
suitable constant $A$, which is determined by using the principle of the
Prandtl-Batchelor theory.
