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On a Bingham fluid whose viscosity and yield limit depend on the temperature

Published online by Cambridge University Press:  22 January 2016

Yoshio Kato
Yoshio Kato*
Affiliation:
Department of Applied Physics, School of Engineering, Nagoya University, Nagoya 464-01 JAPAN
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Duvaut and Lions [2] studied the field of velocities and of temperatures in a moving incompressible Bingham fluid endowed with viscosity μ(θ) depending on the temperature θ and established the existence of a weak solution in the case of a two dimensional fluid. However, the problem of uniqueness remained unsolved. The purpose of the present paper is to give an affirmative answer to the problem, that is, to show the local existence (resp. the global existence) in the time and the uniqueness of (strong) solutions in three dimensions under the conditions that (i) the time (resp. the initial velocity and the external force) and (ii) the rate of variation of the viscosity and the yield limit with respect to the temperature are both sufficiently small. It will be easily seen that the global existence and the uniqueness also hold in the two dimensional case whenever the rate (ii) is sufficiently small.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 128 , December 1992 , pp. 1 - 14
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

References

[1] Adams, R. A., Sobolev spaces, Academic Press, 1975.Google Scholar
[2] Duvaut, G. and Lions, J. L., Transfert de chaleur dans un fluide de Bingham dont la viscosité dépend de la température, J. Funct. Anal., 11 (1972), 93–110.CrossRefGoogle Scholar
[3] Grisvard, P., Équation différentielles abstraites, Ann. Sci. Ecole Norm. Sup., (4) 2 (1969), 311–395.Google Scholar
[4] Kato, Y., Variational inequalities of Bingham type in three dimensions, to appear in Nagoya Math. J. Google Scholar
[5] Lions, J. L., Quelque méthodes de resolution des problémes aux limites non linéaires, Dunod/Gauthier-Villars, Paris, 1969.Google Scholar