  The unsteady motion of a two-layer fluid induced by oscillatory motion of a
flat plate along its length is mathematically analyzed. Two cases are
considered: (i) the two-layer fluid is bounded only by the oscillating plate
(Stokes' second problem), (ii) the two-layer fluid is confined between two
parallel plates, one of which oscillates while the other is held stationary
(oscillatory Couette flow). In each of the Stokes' and Couette cases, both
cosine and sine oscillations of the plate are considered. It is assumed that
the fluids are immiscible, and that the flat interface between the fluids
remains flat for all times. Solutions to the initial-boundary value problems
are obtained using the Laplace transform method. Steady periodic and transient
velocity fields are explicitly presented. Transient and steady-state shear
stresses at the boundaries of the flows are calculated. The results derived in
this paper retrieve previously known results for corresponding single-layer
flows. Further, illustrative example of each of the Stokes' problem and the
Couette flow is presented and discussed. Again, the results obtained could also
be applicable to a problem of heat conduction in a composite solid with
sinusoidal temperature variation on the surface.
