A Model for Displacements Between Parallel Plates That Shows Change of Type from Hyperbolic to Elliptic
Abstract
We consider miscible displacement between parallel plates, where the viscosity is a function of the concentration. By selecting a piecewise representation, the problem can be considered as ``threephase'' flow. Assuming a lubricationtype approximation, the mathematical description is in terms of two quasilinear hyperbolic equations. When the mobility of the middle phase is smaller than its neighbors, the system is genuinely hyperbolic and can be solved analytically. However, when it is larger, an elliptic region develops. This changeoftype behavior is for the first time proved here based on sound physical principles. Numerical solutions with a small diffusion are presented. Good agreement is obtained outside the elliptic region, but not inside, where the numerical results show unstable behavior. We conjecture that for the solution of the real problem in the mixedtype case, the full higherdimensionality problem must be considered inside the elliptic region, in which the lubrication (parallelflow) approximation is no longer appropriate. This is discussed in a companion presentation.
 Publication:

APS Division of Fluid Dynamics Meeting Abstracts
 Pub Date:
 November 2003
 Bibcode:
 2003APS..DFD.KQ002S