Guided Waves Along Fluid-Filled Cracks in Elastic Solids and Instability at High Flow Rates

We characterize wave propagation along an infinitely long crack or conduit in an elastic solid containing a compressible, viscous fluid. Fluid flow is described by quasi-one-dimensional mass and momentum balance equations with a barotropic equation of state, and the wall shear stress is written as a general function of width-averaged velocity, density, and conduit width. Our analysis focuses on small perturbations about steady flow, through a constant width conduit, at an unperturbed velocity determined by balancing the pressure gradient with drag from the walls. Short wavelength disturbances propagate relative to the fluid as sound waves with negligible changes in conduit width. The elastic walls become more compliant at longer wavelengths since strains induced by opening or closing the conduit are smaller, and the fluid compressibility becomes negligible. As wavelength increases, the sound waves transition to crack waves propagating relative to the fluid at a slower phase velocity that is inversely proportional to the square-root of wavelength. Associated with the waves are density, velocity, pressure, and width perturbations that alter drag. At sufficiently fast flow rates, crack waves propagating in the flow direction are destabilized when drag reduction from opening the conduit exceeds the increase in drag from increased fluid velocity. This instability may explain the occurrence of self-excited oscillations in fluid-filled cracks.