Transient shear wave propagation in a solid-liquid coupled system*

We present a novel closed-form analytical solution for the response of a forced solid-fluid system. It consists of a wide finite elastic solid located underneath a Newtonian fluid. The bottom surface of the solid is forced horizontally for a finite time period. The system is governed by coupled partial differential equations, describing the shear displacement in the elastic solid and the shear velocity in the fluid. The boundary conditions represent the continuity of shear velocity and shear stress at the solid-fluid boundary, the lower-boundary shear forcing by continuous imposed displacement or shear rate, and the decay of the shear velocity of the fluid at large distance from the solid-liquid interface. Initial conditions of zero displacement and velocity are used.

The transient response of the system is obtained analytically using integral transform techniques. The solution clearly shows a set of superposed elastic waves in the solid: an incident wave engendered by the driving action at the lower boundary, partially reflected waves at the solid-fluid interface, and perfectly reflected waves at the lower boundary. The transmitted waves which enter the viscous fluid via the interface coupling have an exponentially damped oscillatory profile, with a penetration depth that depends on the fluid viscosity. This response is highly reminiscent of the classic periodic ‘Stokes layer’, and the transient extensions of similar problems.

These waves have interesting applications in active mixing, turbulent drag reduction, and sensing for biological and chemical flows. The analytical solution gives physical insight which cannot be obtained through a purely numerical solution of the governing system, and is more comprehensive than studies of shear wave propagation using simplified eigenmode analysis, or analogical models not derived from first-principles physical laws.