Instability in flow through elastic conduits and volcanic tremor

The stability of fluid flow through a narrow conduit with elastic walls is explored, treating the fluid as incompressible and viscous, and the walls as semi-infinite linear Hookean solids. Instabilities analogous to roll waves occur in this system; we map out the physical regime in which they are excited. For elastic wave speeds much higher than the fluid speed, a critical Reynolds number is required for instability. However, that critical value depends linearly on wavenumber, and so can be made arbitrarily small for long waves. For smaller elastic wave speeds, the critical Reynolds number is reduced still further, and Rayleigh waves can be destabilized by the fluid even at zero Reynolds number. A brief discussion is given of the nonlinear dynamics of the instabilities for large elastic wave speed, and the significance of the results to the phenomenon of volcanic tremor is presented. Although magma itself seems unlikely to generate flow-induced vibrations, the rapid flow through fractured rock of low-viscosity fluids exsolved from magma appears to be a viable mechanism for volcanic tremor.