On the existence of shock propagation in a flow through deformable porous media

E. Comparini; M. Ughi

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 2, page 321-347
  • ISSN: 0392-4041

Abstract

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We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered to be functions of the flux intensity. The medium is initially dry and we neglect capillarity, so that a sharp wetting front proceeds into the medium. We consider the open problem of the continuation of the solution in the case of onset of singularities, which can be interpreted as a local collapse of the medium, in the general case of convex boundary pressure. We study the behaviour of the solution after the development of a singularity and we study the existence of the solution after the time at which the shock line reaches the surface

How to cite

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Comparini, E., and Ughi, M.. "On the existence of shock propagation in a flow through deformable porous media." Bollettino dell'Unione Matematica Italiana 5-B.2 (2002): 321-347. <http://eudml.org/doc/195500>.

@article{Comparini2002,
abstract = {We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered to be functions of the flux intensity. The medium is initially dry and we neglect capillarity, so that a sharp wetting front proceeds into the medium. We consider the open problem of the continuation of the solution in the case of onset of singularities, which can be interpreted as a local collapse of the medium, in the general case of convex boundary pressure. We study the behaviour of the solution after the development of a singularity and we study the existence of the solution after the time at which the shock line reaches the surface},
author = {Comparini, E., Ughi, M.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {321-347},
publisher = {Unione Matematica Italiana},
title = {On the existence of shock propagation in a flow through deformable porous media},
url = {http://eudml.org/doc/195500},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Comparini, E.
AU - Ughi, M.
TI - On the existence of shock propagation in a flow through deformable porous media
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/6//
PB - Unione Matematica Italiana
VL - 5-B
IS - 2
SP - 321
EP - 347
AB - We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered to be functions of the flux intensity. The medium is initially dry and we neglect capillarity, so that a sharp wetting front proceeds into the medium. We consider the open problem of the continuation of the solution in the case of onset of singularities, which can be interpreted as a local collapse of the medium, in the general case of convex boundary pressure. We study the behaviour of the solution after the development of a singularity and we study the existence of the solution after the time at which the shock line reaches the surface
LA - eng
UR - http://eudml.org/doc/195500
ER -

References

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  1. COMPARINI, E.- UGHI, M., Shock propagation in a one-dimensional flow through deformable porous media, MECCANICA, 35 (2000), 119-132. Zbl0970.76051MR1797363
  2. EVANS, L. C., Partial Differential Equations, Berkeley Mathematics Lecture Notes, 1994. Zbl0902.35002
  3. FASANO, A.- TANI, P., Penetration of a Wetting Front in a Porous Medium with Flux Dependent Hydraulic Parameters, K. Cooke et al. eds, SIAM, 1995. Zbl0887.35124MR2410605
  4. GREEN, W.- AMPT, G., Studies on soil physics. The flow of air and water through soils, J. Agric. Sci., 4 (1911), 1-24. 
  5. TA-TSIEN , LI- WEN-CI, YU, Boundary Value Problem for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985. Zbl0627.35001MR823237
  6. TA-TSIEN, LI, Global classical solutions for Quasilinear Hyperbolic Systems, Masson/ J. Wiley, 1994. Zbl0841.35064MR1291392
  7. TA-TSIEN, LI- YAN-CHUN, ZHAO, Global shock solutions to a class of piston problems for a system of one-dimensional isentropic flow, Chin. Ann. of Math., 12 B (1991), 495-499. Zbl0748.76069MR1154636
  8. MANCINI, A., A discontinuous in time stabilized Galerkin approach for an Hyperbolic system with a free boundary, to appear. 
  9. RHEE, H. K.- ARIS, R.- AMUDSON, N. R., First Order Partial Differential Equations, Prentice Hall, Englewood Cliffs, New Jersey 07632, 1986. Zbl0699.35002
  10. TANI, P., Fronti di saturazione in mezzi porosi con caratteristiche idrauliche variabili, Tesi, Univ. Firenze (1994). 
  11. TERRACINA, A., A free boundary problem for scalar conservation laws, SIAM J. of Math. Anal., 30 5 (1999), 985-1009. Zbl0936.35202MR1709784

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