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Dispersive shock waves in viscously deformable media

Published online by Cambridge University Press:  08 February 2013

Nicholas K. Lowman  and
M. A. Hoefer
Nicholas K. Lowman*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
M. A. Hoefer
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
*
†Email address for correspondence: nklowman@ncsu.edu
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Abstract

The viscously dominated, low-Reynolds-number dynamics of multi-phase, compacting media can lead to nonlinear, dissipationless/dispersive behaviour when viewed appropriately. In these systems, nonlinear self-steepening competes with wave dispersion, giving rise to dispersive shock waves (DSWs). Example systems considered here include magma migration through the mantle as well as the buoyant ascent of a low-density fluid through a viscously deformable conduit. These flows are modelled by a third-order, degenerate, dispersive, nonlinear wave equation for the porosity (magma volume fraction) or cross-sectional area, respectively. Whitham averaging theory for step initial conditions is used to compute analytical, closed-form predictions for the DSW speeds and the leading edge amplitude in terms of the constitutive parameters and initial jump height. Novel physical behaviours are identified including backflow and DSW implosion for initial jumps sufficient to cause gradient catastrophe in the Whitham modulation equations. Theoretical predictions are shown to be in excellent agreement with long-time numerical simulations for the case of small- to moderate-amplitude DSWs. Verifiable criteria identifying the breakdown of this modulation theory in the large jump regime, applicable to a wide class of DSW problems, are presented.

JFM classification

Geophysical and Geological Flows: Magma and lava flow Compressible Flows: Shock waves Waves/Free-surface Flows: Solitary waves

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Type
Papers
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Journal of Fluid Mechanics , Volume 718 , 10 March 2013 , pp. 524 - 557
Copyright
©2013 Cambridge University Press

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