Asymptotic solution of a nonlinear advection-diffusion equation

Authors:
R. de Loubens and T. S. Ramakrishnan

Journal:
Quart. Appl. Math. **69** (2011), 389-401

MSC (2000):
Primary 35-XX

DOI:
https://doi.org/10.1090/S0033-569X-2011-01214-X

Published electronically:
March 10, 2011

MathSciNet review:
2816631

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Abstract | References | Similar Articles | Additional Information

Abstract: We carry out an asymptotic analysis as for the nonlinear advection-diffusion equation, , where is a constant. This equation describes the movement of a buoyancy-driven plume in an inclined porous medium, with having a specific physical significance related to the bed inclination. For compactly supported initial data, the solution is characterized by two moving boundaries propagating with finite speed and spanning a distance of . We construct an exact outer solution to the PDE that satisfies the right boundary condition. The vanishing condition at the left boundary is enforced by introducing a moving boundary layer, for which we obtain a closed-form expression. The leading-order composite solution is uniformly correct to . A higher-order correction to the inner and the composite solutions is also derived analytically. As a result, we obtain late-time asymptotic expansions for the two moving boundaries, correct to , as well as a composite solution correct to . The findings of this paper are illustrated and verified by numerical computations.

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Additional Information

**R. de Loubens**

Affiliation:
Schlumberger-Doll Research, 1 Hampshire Street, Cambridge, Massachusetts 02139

**T. S. Ramakrishnan**

Affiliation:
Schlumberger-Doll Research, 1 Hampshire Street, Cambridge, Massachusetts 02139

DOI:
https://doi.org/10.1090/S0033-569X-2011-01214-X

Received by editor(s):
September 23, 2009

Published electronically:
March 10, 2011