The theory for waves on a buoyant fluid conduit in a more viscous outer fluid is extended to include a visco-elastic outer fluid. The external fluid is treated as a linear Kelvin-type visco-elastic medium and a wave evolution equation is derived. This equation is identical to the purely viscous case with the exception of a new term representing the elastic effects. A conservation law is derived and used in an analytic treatment for a slowly-varying solitary wave (given initially by the exact solution to the purely viscous case) for the case of small, but non-zero, elasticity. The theory shows that the wave amplitude will decay and a shelf, required for the conservation of mass, will develop behind the wave. Numerical solutions of the evolution equation support the analytic approximation. Laboratory experiments show qualitative agreement with the analytic and numerical development. Geophysical applications suggest that these effects may be most important for melt migration in the asthenosphere.
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