The granular monoclinal wave: a dynamical systems survey. (English) Zbl 1469.76137

Summary: The theoretical existence of the granular monoclinal wave, based on the Saint-Venant equations for flowing granular matter, was reported recently by D. Razis et al. [ibid. 843, 810–846 (2018; Zbl 1444.76124)]. The present paper focuses on the mathematical interpretation of its behaviour, treating the equation of motion that describes any granular waveform as a dynamical system, taking also into consideration the Froude number offset \(\varGamma\) introduced by Y. Forterre and O. Pouliquen [ibid. 486, 21–50 (2003; Zbl 1156.76458)]. The critical value of the Froude number below which stable uniform flows are observed is determined directly from the stability analysis of the aforementioned dynamical system. It is shown that the granular monoclinal wave, represented as a heteroclinic orbit in phase space, can be categorized into two classes: (i) the mild class, for which the exact form of the waveform can be approximated by the non-viscous (first-order) adaptation of the granular Saint-Venant equations, and (ii) the steep class, for a description of which a second-order (viscous) term in the Saint-Venant equations is absolutely needed to capture the dynamics of the wave. The mathematical criterion that distinguishes the two classes is the changing sign of the trace of the Jacobian matrix evaluated at the fixed point corresponding to the waveform’s lower plateau.


76T25 Granular flows


channel flow
Full Text: DOI


[1] Balmforth, N.J. & Mandre, S.2004Dynamics of roll waves. J. Fluid Mech.514, 1-33. · Zbl 1067.76009
[2] Börzsönyi, T., Hasley, T.C. & Ecke, R.E.2005Two scenarios for avalanche dynamics in inclined granular layers. Phys. Rev. Lett.94, 208001.
[3] Edwards, A.N. & Gray, J.M.N.T.2015Erosion-deposition waves in shallow granular free-surface flows. J. Fluid Mech.762, 35-67.
[4] Edwards, A.N., Viroulet, S., Kokelaar, B.P. & Gray, J.M.N.T.2017Formation of levees, troughs and elevated channels by avalanches on erodible slopes. J. Fluid Mech.823, 278-315. · Zbl 1415.76709
[5] Edwards, A.N., Russell, A.S., Johnson, C.G. & Gray, J.M.N.T.2019Frictional hysteresis and particle deposition in granular free-surface flows. J. Fluid Mech.875, 1058-1095. · Zbl 1430.76476
[6] Ferrick, M.G.2005Simple wave and monoclinal wave models: river flow surge applications and implications. Water Resour. Res.41, W11402.
[7] Forterre, Y. & Pouliquen, O.2003Long-surface-wave instability in dense granular flows. J. Fluid Mech.486, 21-50. · Zbl 1156.76458
[8] Forterre, Y.2006Kapiza waves as a test for three-dimensional granular flow rheology. J. Fluid Mech.563, 123-132. · Zbl 1100.76067
[9] Fowler, A.2011Mathematical Geoscience. Springer.
[10] 2004On dense granular flows. Eur. Phys. J. E14, 341-365.
[11] Gray, J.M.N.T. & Edwards, A.N.2014A depth-averaged \(\mu (I)\)-rheology for shallow granular free-surface flows. J. Fluid Mech.755, 503-534. · Zbl 1330.76137
[12] Kranenburg, C.1992On the evolution of roll waves. J. Fluid Mech.245, 249-261. · Zbl 0765.76011
[13] Lagrée, P.-Y., Saingier, G., Deboeuf, S., Staron, L. & Popinet, S.2017 Granular front for flow down a rough incline: about the value of the shape factor in depths averaged models. In Proceedings ‘Powders & Grains 2017’, EPJ Web of Conferences, vol. 140, p. 03046.
[14] Le Méhauté, B.1976An Introduction to Hydrodynamics and Water Waves. Springer. · Zbl 0327.76001
[15] Moots, E.E. & Mavis, F.T.1938A Study in Flood Waves, vol. 14. University of Iowa Studies in Engineering.
[16] Pouliquen, O.1999Scaling laws in granular flows down rough inclined planes. Phys. Fluids11, 542-548. · Zbl 1147.76477
[17] Pouliquen, O. & Forterre, Y.2002Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane. J. Fluid Mech.453, 113-151. · Zbl 0987.76522
[18] Razis, D., Edwards, A.N., Gray, J.M.N.T. & Van Der Weele, K.2014Arrested coarsening of granular roll waves. Phys. Fluids26, 123305. · Zbl 1323.76118
[19] Razis, D., Kanellopoulos, G. & Van Der Weele, K.2018The granular monoclinal wave. J. Fluid Mech.843, 810-846. · Zbl 1444.76124
[20] Razis, D., Kanellopoulos, G. & Van Der Weele, K.2019A dynamical systems view of granular flow: from monoclinal flood waves to roll waves. J. Fluid Mech.869, 143-181. · Zbl 1415.76717
[21] Russell, A.S., Johnson, C.G., Edwards, A.N., Viroulet, S., Rocha, F.M. & Gray, J.M.N.T.2019Retrogressive failure of a static granular layer on an inclined plane. J. Fluid Mech.869, 313-340. · Zbl 1415.76718
[22] Saingier, G., Deboeuf, S. & Lagrée, P.-Y.2016On the front shape of an inertial granular flow down a rough incline. Phys. Fluids28, 053302.
[23] Savage, S.B. & Hutter, K.1989The motion of a finite mass of granular material down a rough incline. J. Fluid Mech.199, 177-215. · Zbl 0659.76044
[24] Schiesser, W.E.1991The Numerical Method of Lines: Integration of Partial Differential Equations. Academic Press. · Zbl 0763.65076
[25] Shome, M.L. & Steffler, P.M.2006Flood plain filling by a monoclinal flood wave. J. Hydraul. Engng ASCE132, 529-532.
[26] Viroulet, S., Baker, J.L., Edwards, A.N., Johnson, C.G., Gjaltema, C., Clavel, P. & Gray, J.M.N.T2017Multiple solutions for granular flow over a smooth two-dimensional bump. J. Fluid Mech.815, 77-116. · Zbl 1383.76526
[27] Whitham, G.B.1974Linear and Nonlinear Waves. John Wiley & Sons.
[28] Yu, J. & Kevorkian, J.1992Nonlinear evolution of small disturbances into roll waves in an inclined open channel. J. Fluid Mech.243, 575-594. · Zbl 0755.76031
[29] Yu, J., Kevorkian, J. & Haberman, R.2000Weak nonlinear long waves in channel flow with internal dissipation. Stud. Appl. Maths105, 143-163. · Zbl 1136.76352
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.