On a temperature-dependent Hele-Shaw flow in one dimension

Antonio Fasano; Laura Pezza

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2001)

  • Volume: 12, Issue: 1, page 57-67
  • ISSN: 1120-6330

Abstract

top
A model is presented for a Hele-Shaw flow with variable temperature in one space dimension. The problem to be solved is a free boundary problem for a parabolic equation with a non-linear and non-local free boundary condition. Existence and uniqueness are proved.

How to cite

top

Fasano, Antonio, and Pezza, Laura. "On a temperature-dependent Hele-Shaw flow in one dimension." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 12.1 (2001): 57-67. <http://eudml.org/doc/252423>.

@article{Fasano2001,
abstract = {A model is presented for a Hele-Shaw flow with variable temperature in one space dimension. The problem to be solved is a free boundary problem for a parabolic equation with a non-linear and non-local free boundary condition. Existence and uniqueness are proved.},
author = {Fasano, Antonio, Pezza, Laura},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Hele-Shaw flows; Free boundary problems; Nonlocal conditions; free boundary problems; nonlocal conditions},
language = {eng},
month = {3},
number = {1},
pages = {57-67},
publisher = {Accademia Nazionale dei Lincei},
title = {On a temperature-dependent Hele-Shaw flow in one dimension},
url = {http://eudml.org/doc/252423},
volume = {12},
year = {2001},
}

TY - JOUR
AU - Fasano, Antonio
AU - Pezza, Laura
TI - On a temperature-dependent Hele-Shaw flow in one dimension
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2001/3//
PB - Accademia Nazionale dei Lincei
VL - 12
IS - 1
SP - 57
EP - 67
AB - A model is presented for a Hele-Shaw flow with variable temperature in one space dimension. The problem to be solved is a free boundary problem for a parabolic equation with a non-linear and non-local free boundary condition. Existence and uniqueness are proved.
LA - eng
KW - Hele-Shaw flows; Free boundary problems; Nonlocal conditions; free boundary problems; nonlocal conditions
UR - http://eudml.org/doc/252423
ER -

References

top
  1. Alexander, A.N. - Entov, V.M., On the steady-state advancement of fingers and bubbles in a Hele-Shaw cell filled by a non-Newtonian fluid. EJAM, 8, 1997, 73-88. Zbl0883.76003MR1431414
  2. Combescot, R. - Hakim, V. - Dombre, T. - Pomeau-Plumir, Y. - Shape, A., Selection for Saffman-Taylor fingers. Phys. Rev. Lett., 56, 1986, 2036-2039. 
  3. Elliott, C.M. - Ockendon, J.R., Weak and variational methods for free and moving boundary problems. Res. Notes Math., 59, Pitman, London1992. Zbl0476.35080
  4. Entov, V.M. - Etingof, P.J., Viscous flows with time dependent free boundaries in a non-planar Hele-Shaw cell. EJAM, 8, 1997, 23-36. Zbl0878.76023MR1431411
  5. Friedman, A., Partial Differential Equations of Parabolic Type. Prentice Hall Inc., 1964. Zbl0144.34903MR181836
  6. Galin, L.A., Unsteady filtration with a free surface. Dokl. Akad. Nauk. SSSR, 47, 1945, 246-249. Zbl0061.46202MR14004
  7. Green, W.H. - Ampt, G.A., Studies on soil physics. The flow of air and water through soils. J. Agric. Sci., 4, 1911, 1-24. 
  8. Hele-Shaw, H.S., The flow of water. Nature, vol. 58, n. 1489, 1898, 33-36. JFM29.0646.01
  9. Helfrich, K.R., Thermo-viscous fingering of flow in a thin gap: a model of magma flow in dikes and fissures. J. Fluid Mech., 305, 1995, 219-238. Zbl0945.76028
  10. Howison, S.D., Complex variable methods in Hele-Shaw moving boundary problems. Euro J. Appl. Math., 3, 1992, 209-224. Zbl0759.76022MR1182213DOI10.1017/S0956792500000802
  11. Ladyženskaja, O.A. - Solonnikov, V.A. - Ural'ceva, N.N., Linear and Quasilinear Equations of Parabolic Type. AMSTranslations of Mathematical Monographs, 23, 1968. Zbl0174.15403MR241822
  12. Ya, P., On the motion of the oil contour. Dokl. Akad. Nauk. S.S.S.R., vol. 47, 1945, 254-257. Zbl0061.46112
  13. Richardson, S., Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. JFM, 56, 1972, 606-618. Zbl0256.76024
  14. Saffman, P.G., Selection mechanisms and stability of fingers and bubbles in Hele-Shaw cells. IMA J. Appl. Math., 46, 1991, 137-145. Zbl0718.76113MR1106258DOI10.1093/imamat/46.1-2.137
  15. Saffman, P.G. - Taylor, G.I., The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. Roy. Soc., A245, 1958, 312-329. Zbl0086.41603MR97227
  16. Wylie, J.J. - Lister, J.R., The effects of temperature-dependent viscosity on flow in a cooled channel with application to basaltic fissure eruptions. J. Fluid Mech., 305, 1995, 239-261. Zbl0879.76028
  17. Wylie, J.J. - Lister, J.R., Stability of straining flow with surface cooling and temperature-dependent viscosity. J. Fluid Mech., 365, 1998, 369-381. Zbl0907.76036

NotesEmbed ?

top

You must be logged in to post comments.