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Permeability models for magma flow through the Earth’s mantle: a Lie group analysis. (English) Zbl 1266.86006
Summary: The migration of melt through the mantle of the Earth is governed by a third-order nonlinear partial differential equation for the voidage or volume fraction of melt. The partial differential equation depends on the permeability of the medium which is assumed to be a function of the voidage. It is shown that the partial differential equation admits, as well as translations in time and space, other Lie point symmetries provided the permeability is either a power law or an exponential law of the voidage or is a constant. A rarefactive solitary wave solution of the partial differential equation is derived in the form of a quadrature for the exponential law for the permeability.

MSC:
86A99 Geophysics
74J35 Solitary waves in solid mechanics
35Q86 PDEs in connection with geophysics
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[1] D. R. Scott and D. J. Stevenson, “Magma solitons,” Geophysical Research Letters, vol. 4, pp. 1161-1164, 1984.
[2] D. Mckenzie, “The generation and compaction of partially molten rock,” Journal of Petrology, vol. 25, no. 3, pp. 713-765, 1984. · doi:10.1093/petrology/25.3.713
[3] V. Barcilon and F. M. Richter, “Nonlinear waves in compacting media,” Journal of Fluid Mechanics, vol. 164, pp. 429-448, 1986. · Zbl 0587.76165 · doi:10.1017/S0022112086002628
[4] D. Takahashi and J. Satsuma, “Explicit solutions of magma equation,” Journal of the Physical Society of Japan, vol. 57, no. 2, pp. 417-421, 1988. · doi:10.1143/JPSJ.57.417
[5] M. Nakayama and D. P. Mason, “Rarefactive solitary waves in two-phase fluid flow of compacting media,” Wave Motion, vol. 15, no. 4, pp. 357-392, 1992. · Zbl 0761.76097 · doi:10.1016/0165-2125(92)90054-6
[6] S. E. Harris, “Conservation laws for a nonlinear wave equation,” Nonlinearity, vol. 9, no. 1, pp. 187-208, 1996. · Zbl 0892.35098 · doi:10.1088/0951-7715/9/1/006
[7] G. H. Maluleke and D. P. Mason, “Optimal system and group invariant solutions for a nonlinear wave equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 9, no. 1, pp. 93-104, 2004. · Zbl 1036.35010 · doi:10.1016/S1007-5704(03)00018-2
[8] S. E. Harris and P. A. Clarkson, “PainlevĂ© analysis and similarity reductions for the magma equation,” Symmetry, Integrability and Geometry, vol. 2, paper 068, 17 pages, 2006. · Zbl 1132.35328 · doi:10.3842/SIGMA.2006.068 · emis:journals/SIGMA/2006/Paper068/ · eudml:53831
[9] G. H. Maluleke and D. P. Mason, “Derivation of conservation laws for a nonlinear wave equation modelling melt migration using Lie point symmetry generators,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 4, pp. 423-433, 2007. · Zbl 1110.35076 · doi:10.1016/j.cnsns.2005.05.010
[10] M. Nakayama and D. P. Mason, “Compressive solitary waves in compacting media,” International Journal of Non-Linear Mechanics, vol. 26, no. 5, pp. 631-640, 1991. · Zbl 0754.76085 · doi:10.1016/0020-7462(91)90015-L
[11] M. Nakayama and D. P. Mason, “On the existence of compressive solitary waves in compacting media,” Journal of Physics A, vol. 27, no. 13, pp. 4589-4599, 1994. · Zbl 0841.76087 · doi:10.1088/0305-4470/27/13/032
[12] M. Nakayama and D. P. Mason, “On the effect of background voidage on compressive solitary waves in compacting media,” Journal of Physics A, vol. 28, no. 24, pp. 7243-7261, 1995. · Zbl 0877.35110 · doi:10.1088/0305-4470/28/24/021
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