# The blocking of an inhomogeneous Bingham fluid. Applications to landslides

Patrick Hild; Ioan R. Ionescu; Thomas Lachand-Robert; Ioan Roşca^{[1]}

- [1] Department of Mathematics, University of Bucharest, Str. Academiei, 14, 70109 Bucharest, Romania.

- Volume: 36, Issue: 6, page 1013-1026
- ISSN: 0764-583X

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topHild, Patrick, et al. "The blocking of an inhomogeneous Bingham fluid. Applications to landslides." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.6 (2002): 1013-1026. <http://eudml.org/doc/244777>.

@article{Hild2002,

abstract = {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.},

affiliation = {Department of Mathematics, University of Bucharest, Str. Academiei, 14, 70109 Bucharest, Romania.},

author = {Hild, Patrick, Ionescu, Ioan R., Lachand-Robert, Thomas, Roşca, Ioan},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {viscoplastic fluid; inhomogeneous Bingham model; landslides; blocking property; nondifferentiable variational inequalities; local qualitative properties; anti-plane flow; variational formulation; stagnant regions},

language = {eng},

number = {6},

pages = {1013-1026},

publisher = {EDP-Sciences},

title = {The blocking of an inhomogeneous Bingham fluid. Applications to landslides},

url = {http://eudml.org/doc/244777},

volume = {36},

year = {2002},

}

TY - JOUR

AU - Hild, Patrick

AU - Ionescu, Ioan R.

AU - Lachand-Robert, Thomas

AU - Roşca, Ioan

TI - The blocking of an inhomogeneous Bingham fluid. Applications to landslides

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2002

PB - EDP-Sciences

VL - 36

IS - 6

SP - 1013

EP - 1026

AB - 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.

LA - eng

KW - viscoplastic fluid; inhomogeneous Bingham model; landslides; blocking property; nondifferentiable variational inequalities; local qualitative properties; anti-plane flow; variational formulation; stagnant regions

UR - http://eudml.org/doc/244777

ER -

## References

top- [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. Zbl0309.73033
- [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). Zbl1140.74411
- [7] R.J. DiPerna and P.-L. Lions, Ordinary differential equations, Sobolev spaces and transport theory. Invent. Math. 98 (1989) 511–547. Zbl0696.34049
- [8] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). Zbl0298.73001MR464857
- [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). Zbl0456.65035MR597520
- [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). Zbl0358.65091
- [11] I. Ionescu and M. Sofonea, The blocking property in the study of the Bingham fluid. Int. J. Engng. Sci. 24 (1986) 289–297. Zbl0575.76011
- [12] I. Ionescu and M. Sofonea, Functional and numerical methods in viscoplasticity. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1993). Zbl0787.73005MR1244578
- [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. Zbl0637.73047
- [14] J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure. Appl. Math. XX (1967) 493–519. Zbl0152.34601
- [15] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol 1: Incompressible models. Oxford University Press (1996). Zbl0866.76002MR1422251
- [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. Zbl0168.45505
- [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. Zbl0168.45601
- [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. Zbl0236.76006
- [19] A. Nouri and F. Poupaud, An existence theorem for the multifluid Navier-Stokes problem. J. Differential Equations 122 (1995) 71–88. Zbl0842.35079
- [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. Zbl0029.32702
- [21] P. Suquet, Un espace fonctionnel pour les équations de la plasticité. Ann. Fac. Sci. Toulouse Math. (6) 1 (1979) 77–87. Zbl0405.46027
- [22] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam (1979). Zbl0426.35003MR603444
- [23] R. Temam, Problèmes mathématiques en plasticité. Gauthiers-Villars, Paris (1983). Zbl0547.73026MR711964
- [24] R. Temam and G. Strang, Functions of bounded deformation. Arch. Rational Mech. Anal. 75 (1980) 7–21. Zbl0472.73031

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