Late-time asymptotic solution of nonlinear advection-diffusion equations with equal exponents

Authors:
François Fayard and T. S. Ramakrishnan

Journal:
Quart. Appl. Math. **71** (2013), 289-310

MSC (2000):
Primary 35-XX

DOI:
https://doi.org/10.1090/S0033-569X-2012-01312-8

Published electronically:
October 18, 2012

MathSciNet review:
3087424

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Abstract: We derive the late-time asymptotic solution of a nonlinear advection-diffusion equation, , where and . The equation is a more general form of the purely quadratic nonlinearity for advection and diffusion considered previously. For initial conditions with compact support, the solution has left and right moving boundaries, the distance between which is the width of the ``plume''. We show the width to grow as , with a constant correction term. The outer solution is dominated by the nonlinear advective term, the leading-order solution of which is shown to satisfy the partial differential equation and the right boundary condition exactly, but with a -dependent shifted argument. To satisfy the left boundary of vanishing plume thickness, a boundary layer is introduced, for which the inner solution may be obtained up to second order, again by using a shifted coordinate with respect to the wetting front. A leading-order composite solution for , uniformly correct to , is obtained. The first and second-order terms are correct to and respectively. The composite second-order correction involves an arbitrary constant, implying its dependence on an unknown initial condition. Numerical results that agree with the analytical solutions are given along with an expression for the unknown constant computed with an impulse initial data.

**1.**G. I. Barenblatt,*Scaling, self-similarity, and intermediate asymptotics*, Cambridge University Press (1996). MR**1426127 (98a:00005)****2.**R. de Loubens and T. S. Ramakrishnan,*Analysis and computation of gravity-induced migration in porous media*, J. Fluid Mech.**675**(2011), 60-86. MR**2801036****3.**R. de Loubens and T. S. Ramakrishnan,*Asymptotic solution of a nonlinear advection-diffusion equation*, Q. Appl. Math.**69**(2011), 389-401. MR**2816631****4.**J. Buckmaster,*Viscous sheets advancing over dry beds*, J. Fluid Mech.,**81**(1977), 737-756. MR**0455812 (56:14046)****5.**T. S. Ramakrishnan, D. Wilkinson and M. M. Dias,*Effect of capillary pressure on the approach to residual oil saturation*Trans. Porous Media,**3**(1988), 51-79.**6.**B. H. Gilding and L. A. Peletier,*The Cauchy problem for an equation in the theory of infiltration*, Arch. Rational Mech. Anal.,**61**(1976), 127-140. MR**0408428 (53:12192)****7.**B. H. Gilding,*Properties of solutions of an equation in the theory of infiltration*, Arch. Rational Mech. Anal.,**65**(1977), 203-225. MR**0447847 (56:6157)****8.**B. H. Gilding,*The occurrence of interfaces in nonlinear diffusion-advection processes*, Arch. Rational Mech. Anal.**100**(1988), 243-263. MR**918796 (89f:35104)****9.**R. E. Grundy,*Asymptotic solution of a model non-linear convective diffusion equation*, IMA J. Appl. Math.,**31**(1983), 121-137. MR**728117 (85g:76024)****10.**J. Kevorkian and J. D. Cole,*Perturbation methods in applied mathematics*, Springer-Verlag (1981). MR**608029 (82g:34082)****11.**Ph. Laurençot and F. Simondon,*Long-time behaviour for porous medium equations with convection*, Proc. Roy. Soc. Edinburgh Sect. A,**128**(1998), 315-336. MR**1621331 (99f:35108)****12.**Ph. Laurençot,*Long-Time behaviour for diffusion equations with fast convection*, Annali di Matemetica pura ed applicata,**175**(1998), 233-251. MR**1748225 (2001b:35173)****13.**D. A. Sapronov and A. E. Shishkov,*Asymptotic behaviour of supports of solutions of quasilinear many-dimensional parabolic equations of non-stationary diffusion-convection type*, Sb. Math**197**(2006), 75-790. MR**2264331 (2007f:35168)****14.**V. Vanaja and P. L. Sachdev,*Asymptotic solutions of a generalized Burgers equation*, Quarterly of Appl. Math.,**50**(1992), 627-640. MR**1193660 (93h:35184)****15.**A. D. Polyanin and V. F. Zaitsev,*Handbook of nonlinear partial differential equations*, Chapman & Hall/CRC (2000). MR**2865542****16.**A. T. Benjamin, G. O. Preton and J. J. Quinn,*A stirling encounter with harmonic numbers*, Math. Mag.,**75**(2002), 95-103. MR**1573592**

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Additional Information

**François Fayard**

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

Email:
ffayard@slb.com

**T. S. Ramakrishnan**

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

Email:
ramakrishnan@slb.com

DOI:
https://doi.org/10.1090/S0033-569X-2012-01312-8

Received by editor(s):
July 5, 2011

Published electronically:
October 18, 2012

Article copyright:
© Copyright 2012
Brown University

The copyright for this article reverts to public domain 28 years after publication.