A dynamic multilayer shallow water model for polydisperse sedimentation

Abstract

A multilayer shallow water approach for the approximate description of polydisperse sedimentation in a viscous fluid is presented. The fluid is assumed to carry finely dispersed solid particles that belong to a finite number of species that differ in density and size. These species segregate and form areas of different composition. In addition, the settling of particles influences the motion of the ambient fluid. A distinct feature of the new approach is the particular definition of the average velocity of the mixture. It takes into account the densities of the solid particles and the fluid and allows us to recover the global mass conservation and linear momentum balance laws of the mixture. This definition motivates a modification of the Masliyah-Lockett-Bassoon (MLB) settling velocities of each species. The multilayer shallow water model allows one to determine the spatial distribution of the solid particles, the velocity field, and the evolution of the free surface of the mixture. The final model can be written as a multilayer model with variable density where the unknowns are the average velocities and concentrations in each layer, the transfer terms across each interface, and the total mass. An explicit formula of the transfer terms leads to a reduced form of the system. Finally, an explicit bound of the minimum and maximum eigenvalues of the transport matrix of the system is utilized to design a Harten-Lax-van Leer (HLL)-type path-conservative numerical method. Numerical simulations illustrate the coupled polydisperse sedimentation and flow fields in various scenarios, including sedimentation in a type of basin that is used in practice in mining industry and in a basin whose bottom topography gives rise to recirculations of the fluid and high solids concentrations.Mathematics Subject Classification. 65N06, 76T20.

A Multilayer Shallow Water Approach for Polydisperse Sedimentation with Sediment Compressibility and Mixture Viscosity

“…It is the purpose of this paper to develop a multilayer shallow water model for threedimensional polydisperse sedimentation process with the property that is possible to recover the mass and linear momentum balance laws of the mixture, which means that the mixture description is consistent with a single-phase flow model. Furthermore, and in contrast to previous efforts [10,17], sediment compressibility and viscosity of the mixture are talen into account. As in [10] we use a mass-average velocity and definition for total density of the mixture such that is possible to recover global mass and linear momentum balance of the mixture.…”

“…Furthermore, and in contrast to previous efforts [10,17], sediment compressibility and viscosity of the mixture are talen into account. As in [10] we use a mass-average velocity and definition for total density of the mixture such that is possible to recover global mass and linear momentum balance of the mixture. The novelty of the present work, besides the introduction of the compressibility and mixture viscosity terms and their numerical treatment, is the use of a rotational invariance property of the equations to design the numerical method.…”

“…In [10] we introduced a multilayer shallow water model for polydisperse sedimentation without viscous stress tensor and sediment compressibility. The goal is here to introduce these terms and to solve the resulting model numerically.…”

“…The multilayer approach consists in subdividing the computational domain into M layers in the vertical direction, which leads to a system of Saint-Venant equations. Under typical assumptions the unknowns are horizontal velocities by layer, the total height of the fluid column, and the solids concentrations for each species in each layer and in some cases as in [10] the total mass of the mixture. The vertical velocity of the mixture in each layer can be calculated by post-processing data, and we do not need to solve a partial differential equation (PDE) to obtain this quantity.…”

“…2.2 we define the solid-fluid relative velocity for each solid phase. These velocities incorporate the effective solid stress and the modified form of the Masliyah-Lockett-Bassoon (MLB; [21,22]) model introduced in [10]. The effective solid stress is only active during consolidation, that is wherever the local total solids volume fraction φ exceeds a critical concentration or gel point φ c .…”