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SUMMARY:Continuum Models of Two-Phase Flow in Porous Media - Shearer\, M (
 North Carolina State University)
DTSTART:20130611T110000Z
DTEND:20130611T120000Z
UID:TALK45725@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:I discuss two models of two-phase fluid flow in which undercom
 pressive shock waves have been discovered recently. In the first part of t
 he talk\, the focus is on two-phase flow in porous media. Plane waves  are
  modeled by the one-dimensional Buckley-Leverett equation\, a  scalar cons
 ervation law.  The Gray-Hassanizadeh model for rate-dependent capillary pr
 essure  adds  dissipation and a BBM-type dispersion\, giving rise to under
 compressive waves.  Two-phase flow in porous media is notoriously subject 
 to fingering instabilities\, related to the classic Saffman-Taylor instabi
 lity. However\,   a  two dimensional linear stability analysis of sharp pl
 anar interfaces reveals a criterion  predicting that weak Lax shocks may b
 e stable or unstable to long-wave two-dimensional perturbations. This surp
 rising result is related to the hyperbolic-elliptic nature of the system o
 f  linearized equations.   Numerical simulations of the full nonlinear sys
 tem of equations\, including dissipation and dispersion\,  verify the stab
 ility predictions at the hyperbolic level.   In the second part of the tal
 k\, I describe a phase field model of a resident fluid being displaced by 
 injected air in a thin tube (a microscopic pore).  PDE simulations reveal 
 the appearance of a rarefaction wave together with a faster undercompressi
 ve wave that terminates at the spherical cap tip of the injected air.  Pre
 liminary analysis and ODE simulations help to explain this structure.\n
LOCATION:Seminar Room 1\, Newton Institute
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