Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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: We carry out an asymptotic analysis as $ t\rightarrow\infty$ for the nonlinear advection-diffusion equation, $ \partial_t u = 2\alpha u \partial_x u + \partial_x(u \partial_x u) $, where $ \alpha$ is a constant. This equation describes the movement of a buoyancy-driven plume in an inclined porous medium, with $ \alpha$ 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 $ \mathcal{O}(\sqrt{t})$. 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 $ \mathcal{O}(1/\sqrt{t})$. 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 $ \mathcal{O}(1)$, as well as a composite solution correct to $ \mathcal{O}(1/t)$. 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

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