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

Kanellopoulos, Giorgos

J. Fluid Mech. 921, Paper No. A6, 20 p. (2021).

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.

MSC:

76T25

Granular flows

Keywords:

channel flow

Citations:

Zbl 1444.76124; Zbl 1156.76458

References:

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