zbMATH — the first resource for mathematics

Rarefactive solitary waves in two-phase fluid flow of compacting media. (English) Zbl 0761.76097
Rarefactive solitary wave solutions of a third order nonlinear partial differential equation derived by D. R. Scott and D. J. Stevenson [Geophys. Res. Lett. 11, No. 11, 1161-1164 (1984)] to describe the one-dimensional migration of melt under the action of gravity through the Earth’s mantle are investigated. The partial differential equation contains two parameters, \(n\) and \(m\), which are the exponents in power laws relating, respectively, the permeability of the medium and the bulk and shear viscosities of the solid matrix to the voidage. It is proved that, for any value of \(m\), rarefactive solitary wave solutions satisfying certain physically reasonable boundary conditions always exist if \(n>1\) but do not exist if \(0\leq n\leq 1\). It is also proved that the speed of the solitary wave is an increasing function of the amplitude of the wave. Six new exact rarefactive solitary wave solutions, four of which are expressed in terms of elementary functions and two in terms of elliptic integrals, are derived.

76T99 Multiphase and multicomponent flows
35Q51 Soliton equations
86A60 Geological problems
Full Text: DOI
[1] Turcotte, D.L., Magma migration, Ann. rev. Earth planet. sci., 10, 397-408, (1982)
[2] Scott, D.R.; Stevenson, D.J., Magma solitons, Geophys. res. lett., 11, 1161-1164, (1984)
[3] McKenzie, D., The generation and compaction of partially molten rock, J. petrol., 25, 713-765, (1984)
[4] Richter, F.M.; McKenzie, D., Dynamical models for melt segregation from a deformable matrix, J. geology, 92, 729-740, (1984)
[5] Fowler, A.C., A mathematical model of magma transport in the asthenosphere, Geophys. astrophys. fluid dynamics, 33, 63-96, (1985) · Zbl 0568.76100
[6] Scott, D.R.; Stevenson, D.J., Magma ascent by porous flow, J. geophys. res., 91, 9283-9296, (1986)
[7] Barcilon, V.; Richter, F.M., Nonlinear waves in compacting media, J. fluid mech., 164, 429-448, (1986) · Zbl 0587.76165
[8] McKenzie, D.P., The compaction of igneous and sedimentary rocks, J. geol. soc. lond., 144, 299-307, (1987)
[9] Takahashi, D.; Satsuma, J., Explicit solutions of magma equation, J. phys. soc. Japan, 57, 417-421, (1988)
[10] Scott, D.R., The competition between percolation and circulation in a deformable porous medium, J. geophys. res., 93, 6451-6462, (1988)
[11] Barcilon, V.; Lovera, O.M., Solitary waves in magma dynamics, J. fluid mech., 204, 121-133, (1989) · Zbl 0674.76093
[12] Fowler, A.C., Generation and creep of magma in the Earth, SIAM J. appl. math., 49, 231-245, (1989) · Zbl 0665.76110
[13] Takahashi, D.; Sachs, J.R.; Satsuma, J., Properties of the magma and modified magma equations, J. phys. soc. Japan, 59, 1941-1953, (1990)
[14] Nakayama, M.; Meson, D.P., Compressive solitary waves in compacting media, Int. J. non-linear mech., 26, 631-640, (1991) · Zbl 0754.76085
[15] Arzi, A.A., Critical phenomena in the rheology of partially molten rocks, Tectonophysics, 44, 173-184, (1978)
[16] Gillespie, R.P.; Gillespie, R.P.; Gillespie, R.P., Integration, (), 25 · Zbl 0060.13101
[17] Gradshteyn, I.S.; Ryzhik, I.M.; Gradshteyn, I.S.; Ryzhik, I.M., Tables of integrals, series and products, (), 905 · Zbl 0918.65002
[18] Abramowitz, M.; Stegun, I.A.; Abromowitz, M.; Stegun, I.A., (), 590
[19] Turcotte, D.L.; Schubert, G., Geodynamics applications of continuum physics to geological problems, (), 413-414
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.