Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We derive the late-time asymptotic solution of a nonlinear advection-diffusion equation, $ u_t = [\alpha /(q-1)](u^q)_x + (1/q) (u^q)_{xx}$, where $ \alpha \ne 0$ and $ q > 2$. 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 $ t^{1/q}$, 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 $ t$-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 $ u$, uniformly correct to $ O(1/t^{1/q})$, is obtained. The first and second-order terms are correct to $ O((1/t^{2/q})\ln t)$ and $ O(1/t^{2/q})$ 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.


References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35-XX

Retrieve articles in all journals with MSC (2000): 35-XX


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.

American Mathematical Society