  We present an analytical study of oscillatory laminar shear flow over a
compliant viscoelastic layer on a rigid base. This problem relates to
oscillating blood flow in viscoelastic vessels. The deeper motivation for this
study, however, is the possible use of compliant coatings for turbulent drag
reduction. An analytical solution of the fluid and solid velocity is presented,
and five dimensionless parameters emerge. The interaction between fluid and
solid appears to be determined by a single combined dimensionless parameter,
which we call the shear interaction parameter $\chi$. The fluid satisfies a
no-slip boundary condition when $\left|\chi\right| \rightarrow 0$, which occurs
when the solid is heavy, stiff and/or thin. In contrast, the fluid obeys a
free-slip boundary condition when $\left|\chi\right| \rightarrow \infty$, which
corresponds to a lightweight and/or soft solid. Three types of resonance modes
are identified for an elastic solid. Two modes (odd and even) are specific to
the solid. The third mode results from the coupling with the fluid. The three
modes are less pronounced or even absent for a viscoelastic solid. These
findings have a twofold use. First, they help to understand the fluid and solid
dynamics when shear coupling is important. Second, the presented analytical
solution is very useful for validation of numerical fluid-structure-interaction
solvers. Future work might include the extension of the theory to multiple
viscoelastic layers and the dynamic coupling of normal stresses.
