I dedicate this article to the memory of Julian D. Cole, who has been a great inspiration to me and to the Applied Mathematics community in general for his profound impact on the development of Perturbation Methods.
Abstract
The problem of a spreading ground-water mound of liquid in a porous medium, situated on an impermeable horizontal solid layer is revisited. The mathematical formulation for this problem is given by the modified porous medium equation. A global condition in form of an energy integral is derived, describing the loss of liquid in the porous medium. This yields the necessary condition that enables the aymptotic derivation of the similarity exponents for the similarity solution of second kind. The method developed here, is further applied to the corresponding dipole problem, when instead of an energy integral another conservation law, the first moment integral, is considered.
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References
G. I. Barenblatt, Dimensional Analysis. Gordon and Breach (1987) 135 pp.
G. I. Barenblatt, V. M. Entov and V. M. Ryshik, Theory of Fluid Flows Through Natural Rocks. Dordrecht (Neth.): Kluwer (1990).
G. I. Barenblatt, Scaling. Cambridge Texts in Applied Mathematics. Cambridge: CLIP (2003) 171 pp.
P. Y. Polubarinova-Kochina, Theory of Groundwater Movement. Princeton: University Press (1962).
A. S. Kalashnikov, Some problems of qualitative theory of nonlinear second-order parabolic equations. Russ. Math. Surv. 42 (1987) 169-222
D. G. Aronson, The porous medium equation. In: A. Fasano and M. Primicerio (eds.) Some Problems of Nonlinear Diffusion. Lecture Notes in Mathematics 1224. Berlin: Springer-Verlag (1986) pp. 149
J. Hulshof and J. L. Vazquez, Maximal viscosity solutions of the modified porous medium equation and their asymptotic behavior. Eur. J. Appl. Math. 7 (1996) 453-471
I. N. Kochina, N. N. Mikhailov and M. V. Filinov, Groundwater mound damping. Int. J. Engng. Sci. 21 (1983) 413-421
B. A. Wagner and J. D. Cole, On self-similar solutions of Barenblatt’s nonlinear filtration equation. Eur. J. Appl. Math. 7 (1996) 151-167
J. Hulshof and J. L. Vazquez, Self-similar solutions of the second kind for the modified porous medium equation. Eur. J. Appl. Math. 5 (1994) 391-403
N. Goldenfeld, O. Martin, Y. Oono and L. Chen, Renormalization-group theory for the modified porous-medium equation. Phys. Rev. A 44 (1991) 6544-6550
G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations. New York: Springer (1974).
A. C. Hindmarsch, Odepack, a systematized collection of ode solvers. In: R. S. Stepleman et al. (eds.), Scientific Computing. Amsterdam: North-Holland (1983) pp. 55–64.
N. Goldenfeld, O. Martin, Y. Oono and F. Liu, Anomalous dimensions and the renormalization group in a nonlinear diffusion process. Phys. Rev. Letts. 64 (1990) 1361-1364
T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations. Stud. Appl. Math. 100 (1998) 153-193
G. I. Barenblatt and Y. B. Zeldovich, On the dipole-type solution in problems of unsteady gas filtration in the polytropic regime. Prikl. Mat. i Mekh. 21 (1957) 716-720
S. Kamin and J. L. Vazquez, Asymptotic behavior of solutions of the porous medium equation with changing sign. SIAM J. Appl. Math. 22 (1991) 34-45
J. Hulshof, Similarity solutions of the porous medium equation with sign changes. J. Math. Anal. Appl. 157 (1991) 75-111
W. L. Kath and D. S. Cohen, Waiting-time behavior in a nonlinear diffusion equation. Stud. Appl. Math. 67 (1982) 79-105
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Wagner, B. An Asymptotic Approach to Second-kind Similarity Solutions of the Modified Porous-medium Equation. J Eng Math 53, 201–220 (2005). https://doi.org/10.1007/s10665-005-6695-4
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DOI: https://doi.org/10.1007/s10665-005-6695-4