Dispersive Shock Waves in Viscously Deformable Media

Abstract

The viscously dominated, low Reynolds' number dynamics of multi-phase, compacting media can lead to nonlinear, dissipationless/dispersive behavior 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 modeled 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 behaviors 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.

32 pages, 10 figures

References in corpus (6)

• On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics
• Observation of a gradient catastrophe generating solitons
• Undular bore theory for the Gardner equation
• Degenerate dispersive equations arising in the study of magma dynamics
• Painleve Analysis and Similarity Reductions for the Magma Equation
• Self-Similar Solutions of the Non-Strictly Hyperbolic Whitham Equations for the KdV Hierarchy

Cited by in corpus (22)

• Dispersive shock waves and modulation theory
• Negative mass hydrodynamics in a Spin-Orbit--Coupled Bose-Einstein Condensate
• Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
• Soliton gas in integrable dispersive hydrodynamics
• Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves
• Solitonic dispersive hydrodynamics: theory and observation
• Observation of dispersive shock waves, solitons, and their interactions in viscous fluid conduits
• Shock waves in dispersive hydrodynamics with non-convex dispersion
• Radiating dispersive shock waves in nonlocal optical media
• Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves
• Dynamic soliton-mean flow interaction with nonconvex flux
• Discontinuous shock solutions of the Whitham modulation equations as dispersionless limits of traveling waves
• Modulations of viscous fluid conduit periodic waves
• Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure
• Dispersive Riemann problem for the Benjamin-Bona-Mahony equation
• Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals
• Theory of quasi-simple dispersive shock waves and number of solitons evolved from a nonlinear pulse
• Dispersive shock waves in systems with nonlocal dispersion of Benjamin-Ono type
• Interactions of large amplitude solitary waves in viscous fluid conduits
• Solitary wave fission of a large disturbance in a viscous fluid conduit
• Whitham Shocks and Resonant Dispersive Shock Waves Governed by the Higher Order Korteweg-de Vries Equation
• Solitary wave-mean flow interaction in strongly nonlineardispersive shallow water waves