Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T06:05:49.298Z Has data issue: false hasContentIssue false
  • English
  • Français

The blocking of an inhomogeneous Bingham fluid.Applications to landslides

Published online by Cambridge University Press:  15 January 2003

Patrick Hild ,
Ioan R. Ionescu ,
Thomas Lachand-Robert  and
Ioan Roşca
Patrick Hild
Affiliation:
Laboratoire de Mathématiques, UMR CNRS 5127, Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France. [email protected]., [email protected]. Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France. [email protected].
Ioan R. Ionescu
Affiliation:
Laboratoire de Mathématiques, UMR CNRS 5127, Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France. [email protected]., [email protected].
Thomas Lachand-Robert
Affiliation:
Laboratoire de Mathématiques, UMR CNRS 5127, Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France. [email protected]., [email protected].
Ioan Roşca
Affiliation:
Department of Mathematics, University of Bucharest, Str. Academiei, 14, 70109 Bucharest, Romania. [email protected].
Get access

Abstract

This work is concerned with the flow of a viscousplastic fluid. We choose a model of Bingham typetaking into account inhomogeneous yield limit of thefluid, which is well-adapted in the description oflandslides. After setting the generalthreedimensional problem, the blocking property isintroduced. We then focus on necessary andsufficient conditions such that blocking of the fluidoccurs.The anti-plane flow intwodimensional andonedimensional cases is considered.A variational formulation in terms of stresses isdeduced. More fine properties dealing with localstagnant regions as well as local regions where thefluid behaves like a rigid body are obtained indimension one.

Keywords

Viscoplastic fluidinhomogeneous Bingham modellandslidesblockingpropertynondifferentiable variationalinequalitieslocal qualitative properties.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 6 , November 2002 , pp. 1013 - 1026
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

E.C. Bingham, Fluidity and plasticity. Mc Graw-Hill, New-York (1922).
Cazacu, O. and Cristescu, N., Constitutive model and analysis of creep flow of natural slopes. Ital. Geotech. J. 34 (2000) 44-54.
Cristescu, N., Plastical flow through conical converging dies, using viscoplastic constitutive equations. Int. J. Mech. Sci. 17 (1975) 425-433. CrossRef
Cristescu, N., On the optimal die angle in fast wire drawing. J. Mech. Work. Technol. 3 (1980) 275-287. CrossRef
N. Cristescu, A model of stability of slopes in Slope Stability 2000. Proceedings of Sessions of Geo-Denver 2000, D.V. Griffiths, G.A. Fenton and T.R. Martin (Eds.). Geotechnical special publication 101 (2000) 86-98.
N. Cristescu, O. Cazacu and C. Cristescu, A model for slow motion of natural slopes. Can. Geotech. J. (to appear).
DiPerna, R.J. and Lions, P.-L., Ordinary differential equations, Sobolev spaces and transport theory. Invent. Math. 98 (1989) 511-547. CrossRef
G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).
R. Glowinski, Lectures on numerical methods for nonlinear variational problems. Notes by M.G. Vijayasundaram and M. Adimurthi. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 65. Tata Institute of Fundamental Research, Bombay; Springer-Verlag, Berlin-New York (1980).
R. Glowinski, J.-L. Lions and R. Trémolières, Analyse numérique des inéquations variationnelles. Tome 1 : Théorie générale et premières applications. Tome 2 : Applications aux phénomènes stationnaires et d'évolution. Méthodes Mathématiques de l'Informatique, 5. Dunod, Paris (1976).
Ionescu, I. and Sofonea, M., The blocking property in the study of the Bingham fluid. Int. J. Engng. Sci. 24 (1986) 289-297. CrossRef
I. Ionescu and M. Sofonea, Functional and numerical methods in viscoplasticity. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1993).
Ionescu, I. and Vernescu, B., A numerical method for a viscoplastic problem. An application to the wire drawing. Internat. J. Engrg. Sci. 26 (1988) 627-633. CrossRef
J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure. Appl. Math. XX (1967) 493-519.
P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol 1: Incompressible models. Oxford University Press (1996).
P.P. Mosolov and V.P. Miasnikov, Variational methods in the theory of the fluidity of a viscous-plastic medium. PPM, J. Mech. and Appl. Math. 29 (1965) 545-577. CrossRef
P.P. Mosolov and V.P. Miasnikov, On stagnant flow regions of a viscous-plastic medium in pipes. PPM, J. Mech. and Appl. Math. 30 (1966) 841-854. CrossRef
P.P. Mosolov and V.P. Miasnikov, On qualitative singularities of the flow of a viscoplastic medium in pipes. PPM, J. Mech and Appl. Math. 31 (1967) 609-613. CrossRef
Nouri, A. and Poupaud, F., An existence theorem for the multifluid Navier-Stokes problem. J. Differential Equations 122 (1995) 71-88. CrossRef
Oldroyd, J.G., A rational formulation of the equations of plastic flow for a Bingham solid. Proc. Camb. Philos. Soc. 43 (1947) 100-105. CrossRef
Suquet, P., Un espace fonctionnel pour les équations de la plasticité. Ann. Fac. Sci. Toulouse Math. (6) 1 (1979) 77-87. CrossRef
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam (1979).
R. Temam, Problèmes mathématiques en plasticité. Gauthiers-Villars, Paris (1983).
Temam, R. and Strang, G., Functions of bounded deformation. Arch. Rational Mech. Anal. 75 (1980) 7-21. CrossRef