https://www.ljll.fr/~frey/papers/Navier-Stokes/Khaled%20A.R.A.,%20The%20effect%20of%20the%20slip%20condition%20on%20Stokes%20and%20Couette%20flows%20due%20to%20an%20oscillating%20wall,%20exact%20solutions.pdf here is the analytical solution to your problem, it is a pretty tough paper to read though. here is a more indepth thesis that goes into the boundary conditions to solve such problems, and perhaps something to verify you solution to; Effect of Wall Vibrations on the Dynamics of Fluid Flow, Nafisha Nubayaatt Haq, 2025