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

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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

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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

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  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
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  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
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  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

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