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The classics: Understanding the basic behavior of the "McKenzie" Equations

Presented by: 
Marc Spiegelman
Tuesday 16th February 2016 - 10:00 to 11:00
INI Seminar Room 1
Given a set of consistent conservation equations for two-phase flow in deformable porous media, this presentation strives to develop a better physical intuition into their behavior through a series of classical model problems. These problems illustrate the basic behavior of the equations as well as highlighting some of the open questions and continuing mathematical/computational issues involved with their solution.

We begin by emphasizing that the solid velocity field can be usefully decomposed into incompressible and compressible components. Incompressible flow is governed by Stokes equation for porosity driven convection, while compressible flow is governed by a non-linear wave equation for the evolution of porosity. The driving force for non-linear waves is the non-linearity between porosity and permeability. However, the waveforms and propagation speed depend on the solid bulk rheology. Depending on choice of solid rheology, these waves can manifest as kinematic shocks, non-local dispersive solitary waves and wave-trains (poro-viscous) or diffusive porosity waves (poro-elastic). However, these waves all arise to accommodate variations in melt flux on scales much larger than the compaction length. We will also consider other localization phenomena that occur on scales smaller than the compaction length including the development of melt-bands due to shear of a solid with poro sity weakening shear viscosity, and channel formation due to simplified reactive flow.

These equations demonstrate a rich array of behavior, all arising from the rheological response of a deformable solid to variations in fluid flux. These solutions suggest that magma transport in the mantle is strongly localized and time-dependent. While considerable uncertainties in the formulations remain, the general framework is computationally tractable and provides a quantitative framework for investigating coupled fluid/solid mechanics in the Earth.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons