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We present an analytical study of oscillatory laminar shear flow over acompliant viscoelastic layer on a rigid base. This problem relates tooscillating blood flow in viscoelastic vessels. The deeper motivation for thisstudy, however, is the possible use of compliant coatings for turbulent dragreduction. An analytical solution of the fluid and solid velocity is presented,and five dimensionless parameters emerge. The interaction between fluid andsolid appears to be determined by a single combined dimensionless parameter,which we call the shear interaction parameter $\chi$. The fluid satisfies ano-slip boundary condition when $\left|\chi\right| \rightarrow 0$, which occurswhen the solid is heavy, stiff and/or thin. In contrast, the fluid obeys afree-slip boundary condition when $\left|\chi\right| \rightarrow \infty$, whichcorresponds to a lightweight and/or soft solid. Three types of resonance modesare identified for an elastic solid. Two modes (odd and even) are specific tothe solid. The third mode results from the coupling with the fluid. The threemodes are less pronounced or even absent for a viscoelastic solid. Thesefindings have a twofold use. First, they help to understand the fluid and soliddynamics when shear coupling is important. Second, the presented analyticalsolution is very useful for validation of numerical fluid-structure-interactionsolvers. Future work might include the extension of the theory to multipleviscoelastic layers and the dynamic coupling of normal stresses.

Oscillatory laminar shear flow over a compliant viscoelastic layer on a rigid base