Explicit solutions to a free interface model for the static/flowing transition in thin granular flows

Lusso, Christelle; Bouchut, François; Ern, Alexandre; Mangeney, Anne

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S369-S395

This work is devoted to an analytical description of the dynamics of the static/flowing interface in thin dry granular flows. Our starting point is the asymptotic model derived by Bouchut et al. [Comm. Math. Sci. 14 (2016) 2101–2126] from a free surface incompressible model with viscoplastic rheology including a Drucker–Prager yield stress. This asymptotic model is based on the thin-layer approximation (the flow is thin in the direction normal to the topography compared to its down-slope extension), but the equations are not depth-averaged. In addition to the velocity, the model includes a free surface at the top of the flow and a free time-dependent static/flowing interface at the bottom. In the present work, we simplify this asymptotic model by decoupling the space coordinates, and keeping only the dependence on time and on the normal space coordinate Z. We introduce a time- and Z-dependent source term, assumed here to be given, which represents the opposite of the net force acting on the flowing material, including gravity, pressure gradient, and internal friction. We prove several properties of the resulting simplified model that has a time- and Z-dependent velocity and a time-dependent static/flowing interface as unknowns. The crucial advantage of this simplified model is that it can provide explicit solutions in the inviscid case, for different shapes of the source term. These explicit inviscid solutions exhibit a rich behaviour and qualitatively reproduce some physical features observed in granular flows.

Classification : 35R35, 74N20
Keywords: Granular flows, static/flowing transition, Drucker–Prager yield stress, thin-layer flow, interface dynamics

C. Ancey and B. M. Bates, Stokes’ third problem for Herschel-Bulkley fluids. J. Non-Newtonian Fluid Mech. 243 (2017) 27–37. | MR | DOI

J. E. Andrade, Q. Chen, P. H. Le, C. F. Avila and T. M. Evans, On the rheology of dilative granular media: bridging solid- and fluid-like behavior. J. Mech. Phys. Solids 60 (2012) 1122–1136. | MR | DOI

A. Aradian, E. Raphael and P.-G. De Gennes, Surface flow of granular materials: a short introduction to some recent models. C. R. Phys. 3 (2002) 187–196. | DOI

I. S. Aranson and L. S. Tsimring, Continuum theory of partially fluidized granular flows. Phys. Rev. E 65 (2002) 061303.

I. S. Aranson, L. S. Tsimring, F. Malloggi and E. Clement, Nonlocal rheological properties of granular flows near a jamming limit. Phys. Rev. E 78 (2008) 031303. | DOI

N. J. Balmforth and R. V. Craster, A consistent thin-layer theory for Bingham plastics. J. Non-Newtonian Fluid Mech. 84 (1999) 65–81. | Zbl | DOI

M. Barbolini, A. Biancardi, F. Cappabianca, L. Natale and M. Pagliardi, Laboratory study of erosion processes in snow avalanches. Cold Reg. Sci. Technol. 43 (2005) 1–9.

T. Barker and J. M. N. T. Gray, Partial regularisation of the incompressible $\mu \left(I\right)$-rheology for granular flow. J. Fluid Mech. 828 (2017) 5–32.

T. Barker, D. G. Schaeffer, P. Bohorquez and J. M. N. T. Gray, Well-posed and ill-posed behaviour of the $\mu \left(I\right)$-rheology for granular flow. J. Fluid Mech. 779 (2015) 794–818.

T. Barker, D. G. Schaeffer, M. Shearer and J. M. N. T. Gray, Well-posed continuum equations for granular flow with compressibility and $\mu \left(I\right)$-rheology. Proc. R. Soc. A 473 (2017) 20160846.

J.-P. Bouchaud, M. E. Cates, J. R. Prakash and S. F. Edwards, A model for the dynamics of sandpile surface. J. Phys. Paris I 4 (1994) 1383–1410.

F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography. Comm. Math. Sci. 2 (2004) 359–389. | MR | Zbl

F. Bouchut, A. Mangeney-Castelnau, B. Perthame and J.-P. Vilotte, A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows. C. R. Math. Acad. Sci. Paris 336 (2003) 531–536. | MR | Zbl

F. Bouchut, E. Fernández-Nieto, A. Mangeney and P.-Y. Lagrée, On new erosion models of Savage-Hutter type for avalanches. Acta Mech. 199 (2008) 181–208. | Zbl

F. Bouchut, I. R. Ionescu and A. Mangeney, An analytic approach for the evolution of the static/flowing interface in viscoplastic granular flows. Comm. Math. Sci. 14 (2016) 2101–2126. | MR

T. Boutreux, E. Raphael and P.-G. De Gennes, Surface flows of granular materials: a modified picture for thick avalanches. Phys. Rev. E 58 (1998) 4692–4700. | MR

H. Capart, C.-Y. Hung and C. P. Stark, Depth-integrated equations for entraining granular flows in narrow channels. J. Fluid Mech. 765 (2015) R4. | MR

J. Chauchat and M. Médale, A three-dimensional numerical model for dense granular flows based on the $\mu \left(I\right)$ rheology. J. Comput. Phys. 256 (2014) 696–712. | MR

D. C. Drucker and W. Prager, Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 10 (1952) 157–165. | MR | Zbl

R. Delannay, A. Valance, A. Mangeney, O. Roche and P. Richard, Granular and particle-laden flows: from laboratory experiments to field observations. J. Phys. D: Appl. Phys. 50 (2017) 053001.

S. Douady, B. Andreotti and A. Daerr, On granular surface flow equations. Eur. Phys. J. B 11 (1999) 131–142.

A. N. Edwards and J. M. N. T. Gray, Erosion-deposition waves in shallow granular free-surface flows. J. Fluid Mech. 762 (2015) 35–67. | MR

M. Farin, A. Mangeney and O. Roche, Fundamental changes of granular flow dynamics, deposition, and erosion processes at high slope angles: insights from laboratory experiments. J. Geophys. Res. Earth Surf. 119 (2014) 504–532.

E. D. Fernández-Nieto, J. Garres-Diaz, A. Mangeney and G. Narbona-Reina, A multilayer shallow model for dry granular flows with the $\mu \left(I\right)$ rheology: application to granular collapse on erodible beds. J. Fluid. Mech. 798 (2016) 643–681. | MR

J. M. N. T. Gray, Granular flow in partially filled slowly rotating drums. J. Fluid Mech. 441 (2001) 1–29.

J. M. N. T. Gray and A. N. Edwards, A depth-averaged $\mu \left(I\right)$-rheology for shallow granular free-surface flows. J. Fluid Mech. 755 (2014) 503–534.

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

R. M. Iverson, Elementary theory of bed-sediment entrainment by debris flows and avalanches. J. Geophys. Res. 117 (2012) F03006.

R. M. Iverson, C. Ouyang, Entrainment of bed material by Earth-surface mass flows: review and reformulation of depth-integrated theory. Rev. Geophys. 53 (2015) 27–58.

P. Jop, Y. Forterre and O. Pouliquen, Crucial role of sidewalls in granular surface flows: consequences for the rheology. J. Fluid Mech. 541 (2005) 167–192. | Zbl

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

D. V. Khakhar, A. V. Orpe, P. Andresen and J. M. Ottino, Surface flow of granular materials: model and experiments in heap formation. J. Fluid Mech. 441 (2001) 225–264.

P.-Y. Lagrée, L. Staron and S. Popinet, The granular column collapse as a continuum: validity of a two-dimensional Navier-Stokes model with a $\mu \left(I\right)$-rheology. J. Fluid Mech. 686 (2011) 378–408. | MR | Zbl

J.-L. Lions, Remarks on some nonlinear evolution problems arising in Bingham flows. Proc. Int. Symp. Partial Differ. Equ. Geom. Normed Linear Spaces (Jerusalem, 1972) Isr. J. Math. 13 (1972) 155–172. | MR | Zbl

C. Lusso, F. Bouchut, A. Ern and 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.

A. Mangeney, L. S. Tsimring, D. Volfson, I. S. Aranson and F. Bouchut, Avalanche mobility induced by the presence of an erodible bed and associated entrainment. Geophys. Res. Lett. 34 (2007) L22401.

A. Mangeney, F. Bouchut, N. Thomas, J.-P. Vilotte and M.-O. Bristeau, Numerical modeling of self-channeling granular flows and of their levee/channel deposits. J. Geophys. Res. Earth Surf. 112 (2007) F02017.

A. Mangeney, O. Roche, O. Hungr, N. Mangold, G. Faccanoni and A. Lucas, Erosion and mobility in granular collapse over sloping beds. J. Geophys. Res. Earth Surf. 115 (2010) F03040.

A. Mangeney-Castelnau, F. Bouchut, J.-P. Vilotte, E. Lajeunesse, A. Aubertin and M. Pirulli, On the use of Saint Venant equations to simulate the spreading of a granular mass. J. Geophys. Res. Solid Earth 110 (2005) B09103.

N. Martin, I. R. Ionescu, A. Mangeney, F. Bouchut and M. Farin, Continuum viscoplastic simulation of a granular column collapse on large slopes: $\mu \left(\mu I\right)$ rheology and lateral wall effects. Phys. Fluids 29 (2017) 013301.

GdR MIDI, On dense granular flows. Eur. Phys. J. E Soft. Matter 14 (2004) 341–365.

S. Parez, E. Aharonov and R. Toussaint, Unsteady granular flows down an inclined plane. Phys. Rev. E 93 (2016) 042902.

E. B. Pitman, C. C. Nichita, A. K. Patra, A. C. Bauer, M. Bursik and A. Webb, A model of granular flows over an erodible surface. Discrete Continuous Dyn. Syst. – B 3 (2003) 589–599. | MR | Zbl

S. B. Savage and K. Hutter, The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199 (1989) 177–215. | MR | Zbl

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

D. G. Schaeffer, T. Barker, D. Tsuji, P. Gremaud, M. Shearer and J. M. N. T. Gray, Constitutive relations for compressible granular flow in the inertial regime. J. Fluid Mech. 874 (2019) 926–951. | MR

N. Taberlet, P. Richard, A. Valance, R. Delannay, W. Losert, J. M. Pasini and J. T. Jenkins, Super stable granular heap in thin channel. Phys. Rev. Lett. 91 (2003) 264301.

T. Trinh, P. Boltenhagen, R. Delannay and A. Valance, Erosion and deposition processes in surface granular flows. Phys. Rev. E 96 (2017) 042904.