Figure 1: Schematic diagram of Stokes second problem.
1. (Analytic solution)
(1) Starting from the three-dimensional Navier-Stokes equations, derive the simplified governing equation for the Stokes second problem. Provide all of the assumptions that are used in your derivation.
(1) (2) Show that the solution to the Stokes second problem is
u(y,t) = U0e−ηs cos(nt − ηs), where (2)
2. (Numerical analysis)
Consider two infinitely long plates placed at y = 0 and y = L. The bottom plate (y = 0) is oscillating with u(0,t) = cos(nt), while the top plate (y = L) is stationary. We aim to obtain velocity profiles u(y,t) between two plates by solving Eq. 1 under the assumptions ν = 1, n = 2, U0 = 1 and L = 10.
(2) Repeat (1) using the Crank-Nicolson (C-N) scheme in time.
(4) Repeat (2) with L = 2. Discuss the effect of the gap distance between the two plates on the velocity profiles.
