A solution with free boundary for non-Newtonian fluids with Drucker–Prager plasticity criterion
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 46.

We study a free boundary problem which is motivated by a particular case of the flow of a non-Newtonian fluid, with a pressure depending yield stress given by a Drucker–Prager plasticity criterion. We focus on the steady case and reformulate the equation as a variational problem. The resulting energy has a term with linear growth while we study the problem in an unbounded domain. We derive an Euler–Lagrange equation and prove a comparison principle. We are then able to construct a subsolution and a supersolution which quantify the natural and expected properties of the solution; in particular, we show that the solution has in fact compact support, the boundary of which is the free boundary.

The model describes the flow of a non-Newtonian material on an inclined plane with walls, driven by gravity. We show that there is a critical angle for a non-zero solution to exist. Finally, using the sub/supersolutions we give estimates of the free boundary.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2018040
Classification : 76A05, 49J40, 35R35
Mots-clés : Non-Newtonian fluid, Drucker–Prager plasticity, variational inequality, free boundary
Ntovoris, E. 1 ; Regis, M. 1

1 
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     title = {A solution with free boundary for {non-Newtonian} fluids with {Drucker{\textendash}Prager} plasticity criterion},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
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Ntovoris, E.; Regis, M. A solution with free boundary for non-Newtonian fluids with Drucker–Prager plasticity criterion. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 46. doi : 10.1051/cocv/2018040. http://archive.numdam.org/articles/10.1051/cocv/2018040/

[1] T. Barker, D.G. Schaeffer, P. Bohorquez and J.M.N.T. Gray, Well-posed and ill-posed behavior of the μ(I)-rheology for granular flow. J. Fluid Mech. 779 (2015) 794–818. | DOI | MR | Zbl

[2] G. Bellettini, V. Caselles and M. Novaga, The total variation flow in ℝN. J. Differ. Equ. 184 (2002) 475–525. | DOI | MR | Zbl

[3] F. Bouchut, R. Eymard and A. Prignet, Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems. J. Evol. Equ. 14 (2014) 635–669. | DOI | MR | Zbl

[4] F. Bouchut, I.R. Ionescu and A. Mangeney, An analytic approach for the evolution of the static/flowing interface in viscoplastic granular flows. Commun. Math. Sci. 14 (2016) 14. | DOI | MR

[5] O. Cazacu and I.R. Ionescu, Compressible rigid viscoplastic fluids. J. Mech. Phys. Solids 54 (2006) 1640–1667 | DOI | MR | Zbl

[6] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics. Translated from the French by C. W. John. Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, New York (1976). | DOI | MR | Zbl

[7] L.C. Evans, Partial differential equations, 2nd edn. Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, (2010). | MR | Zbl

[8] M. Fuchs and G. Seregin, Variational methods for problems from plasticity theory and for generalized Newtonian fluids. Vol. 1749 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2000). | DOI | MR | Zbl

[9] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Vol. 105 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (1983). | MR | Zbl

[10] M.-H. Giga, Y. Giga and N. Požár, Periodic total variation flow of non-divergence type in ℝn. J. Math. Pures Appl. 102 (2014) 203–233. | DOI | MR | Zbl

[11] Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces. Adv. Differ. Equ. 21 (2016) 631–698. | MR | Zbl

[12] I.R. Ionescu, A. Mangeney, F. Bouchut and O. Roche, Viscoplastic modeling of granular column collapse with pressure-dependent rheology. J. Non-Newton. Fluid Mech. 219 (2015) 1–18. | DOI | MR

[13] P. Jop, Y. Forterre and O. Pouliquen, A constitutive law for dense granular flows. Nature 441 (2006) 727–730. | DOI

[14] D. Krejčiřík and A. Pratelli, The Cheeger constant of curved strips. Pac. J. Math. 254 (2011) 309–333. | DOI | MR | Zbl

[15] F. Krügel, Some properties of minimizers of a variational problem involving the total variation functional. Commun. Pure Appl. Anal. 14 (2015) 341–360 | DOI | MR | Zbl

[16] F. Krügel, Potential theory for the sum of the 1-Laplacian and p-Laplacian. Nonlinear Anal. 112 (2015) 165–180. | DOI | MR | Zbl

[17] G.P. Leonardi and A. Pratelli, On the Cheeger sets in strips and non-convex domains. Calc. Var. Partial Differ. Equ. 55 (2016) 15. | DOI | MR | Zbl

[18] G. Leoni, A first course in Sobolev spaces. Vol. 105 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2009). | MR | Zbl

[19] C. Lusso, F. Bouchut, A. Ern, A. Mangeney. A Free Interface Model for Static/Flowing Dynamics in Thin-Layer Flows of Granular Materials with Yield: Simple Shear Simulations and Comparison with Experiments. Appl. Sci. 7 (2017) 386. | DOI

[20] C. Lusso, A. Ern, F. Bouchut, A. Mangeney, M. Farin and O. Roche. Two-dimensional simulation by regularization of free surface viscoplastic flows with Drucker–Prager yield stress and application to granular collapse. J. Comput. Phys. 333 (2017) 387–408. | DOI | MR | Zbl

[21] J. Málek, J. Nečas, M. Rokyta and M. Ružička. Weak and measure-valued solutions to evolutionary PDEs. Vol. 13 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London (1996). | MR | Zbl

[22] O. Pouliquen, C. Cassar, P. Jop, Y. Forterre and M. Nicolas. Flow of dense granular material: towards simple constitutive laws. J. Stat. Mech.: Theo. Exp. 7 (2006) P07020.

[23] G. Seregin, Continuity for the strain velocity tensor in two-dimensional variational problems from the theory of the Bingham fluid. Ital. J. Pure Appl. Math. 2 (1998) 141–150. | MR | Zbl

[24] D.G Schaeffer, Instability in the evolution equations describing incompressible granular flow. J. Differ. Equ. 66 (1987) 19–50. | DOI | MR | Zbl

[25] F.A. Vaillo, V. Caselles and J. M. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Vol. 223 of Progress in Mathematics. Birkhäuser Verlag, Basel (2004). | MR | Zbl

[26] W.P. Ziemer, Weakly Differentiable Functions. Vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York (1989). | DOI | MR | Zbl

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