The blocking of an inhomogeneous Bingham fluid. Applications to landslides
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 6, pp. 1013-1026.

This work is concerned with the flow of a viscous plastic fluid. We choose a model of Bingham type taking into account inhomogeneous yield limit of the fluid, which is well-adapted in the description of landslides. After setting the general threedimensional problem, the blocking property is introduced. We then focus on necessary and sufficient conditions such that blocking of the fluid occurs. The anti-plane flow in twodimensional and onedimensional cases is considered. A variational formulation in terms of stresses is deduced. More fine properties dealing with local stagnant regions as well as local regions where the fluid behaves like a rigid body are obtained in dimension one.

Classification : 49J40,  76A05
Mots clés : viscoplastic fluid, inhomogeneous Bingham model, landslides, blocking property, nondifferentiable variational inequalities, local qualitative properties
     author = {Hild, Patrick and Ionescu, Ioan R. and Lachand-Robert, Thomas and Ro\c{s}ca, Ioan},
     title = {The blocking of an inhomogeneous Bingham fluid. Applications to landslides},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {1013--1026},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {6},
     year = {2002},
     doi = {10.1051/m2an:2003003},
     zbl = {1057.76004},
     mrnumber = {1958656},
     language = {en},
     url = {}
Hild, Patrick; Ionescu, Ioan R.; Lachand-Robert, Thomas; Roşca, Ioan. The blocking of an inhomogeneous Bingham fluid. Applications to landslides. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 6, pp. 1013-1026. doi : 10.1051/m2an:2003003.

[1] E.C. Bingham, Fluidity and plasticity. Mc Graw-Hill, New-York (1922).

[2] O. Cazacu and N. Cristescu, Constitutive model and analysis of creep flow of natural slopes. Ital. Geotech. J. 34 (2000) 44-54.

[3] N. Cristescu, Plastical flow through conical converging dies, using viscoplastic constitutive equations. Int. J. Mech. Sci. 17 (1975) 425-433. | Zbl 0309.73033

[4] N. Cristescu, On the optimal die angle in fast wire drawing. J. Mech. Work. Technol. 3 (1980) 275-287.

[5] 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.

[6] N. Cristescu, O. Cazacu and C. Cristescu, A model for slow motion of natural slopes. Can. Geotech. J. (to appear).

[7] R.J. Diperna and P.-L. Lions, Ordinary differential equations, Sobolev spaces and transport theory. Invent. Math. 98 (1989) 511-547. | Zbl 0696.34049

[8] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). | MR 464857 | Zbl 0298.73001

[9] 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). | MR 597520 | Zbl 0456.65035

[10] 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). | Zbl 0358.65091

[11] I. Ionescu and M. Sofonea, The blocking property in the study of the Bingham fluid. Int. J. Engng. Sci. 24 (1986) 289-297. | Zbl 0575.76011

[12] I. Ionescu and M. Sofonea, Functional and numerical methods in viscoplasticity. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1993). | MR 1244578 | Zbl 0787.73005

[13] I. Ionescu and B. Vernescu, A numerical method for a viscoplastic problem. An application to the wire drawing. Internat. J. Engrg. Sci. 26 (1988) 627-633. | Zbl 0637.73047

[14] J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure. Appl. Math. XX (1967) 493-519. | Zbl 0152.34601

[15] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol 1: Incompressible models. Oxford University Press (1996). | MR 1422251 | Zbl 0866.76002

[16] 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. | Zbl 0168.45505

[17] 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. | Zbl 0168.45601

[18] 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. | Zbl 0236.76006

[19] A. Nouri and F. Poupaud, An existence theorem for the multifluid Navier-Stokes problem. J. Differential Equations 122 (1995) 71-88. | Zbl 0842.35079

[20] J.G. Oldroyd, A rational formulation of the equations of plastic flow for a Bingham solid. Proc. Camb. Philos. Soc. 43 (1947) 100-105. | Zbl 0029.32702

[21] P. Suquet, Un espace fonctionnel pour les équations de la plasticité. Ann. Fac. Sci. Toulouse Math. (6) 1 (1979) 77-87. | Numdam | Zbl 0405.46027

[22] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam (1979). | MR 603444 | Zbl 0426.35003

[23] R. Temam, Problèmes mathématiques en plasticité. Gauthiers-Villars, Paris (1983). | MR 711964 | Zbl 0547.73026

[24] R. Temam and G. Strang, Functions of bounded deformation. Arch. Rational Mech. Anal. 75 (1980) 7-21. | Zbl 0472.73031