Hyperbolic systems of conservation laws in gravity-driven, particle-laden thin-film flows

Mavromoustaki, Aliki; Bertozzi, Andrea L.

doi:10.1007/s10665-014-9688-3

Cited by 23 publications

(19 citation statements)

References 22 publications

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“…We see that so long as X R is not too large, the 3-curve will indeed intersect the surface. From the (h, φ) projection (see right panel in Figure 5, it is clear that we also require (h R , φ R ) to be within a certain region, in agreement with the monodisperse theory [9]. If (h R , φ R ) lies outside this region, then the 3-connection is instead a rarefaction:…”

Section: Second and Third Shockssupporting

confidence: 73%

“…Analysis of hyperbolic systems arising for bidisperse settling have been extensively studied in the context of other models, for instance in the case of settling in a quiescent fluid [2], though not in the incline geometry where we are interested in the Email addresses: jtwong@math.ucla.edu (Jeffrey T. Wong), bertozzi@math.ucla.edu (Andrea L. Bertozzi) long-term dynamics of the fronts rather than the settling of particles to the substrate. The Riemann problem has been studied for the monodisperse model [9,20], showing the existence of double shock solutions in the settled regime and a transition to singular shocks that occurs in the ridged regime, where particles accumulate at the front. The corresponding rarefaction-singular shock solutions that arise for constant volume initial conditions have also been studied [19].…”

Section: Introductionmentioning

confidence: 99%

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A conservation law model for bidensity suspensions on an incline

Wong

Bertozzi

2016

Physica D: Nonlinear Phenomena

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We study bidensity suspensions of a viscous fluid on an incline. The particles migrate within the fluid due to a combination of gravity-induced settling and shear induced migration. We propose an extension a recent model [N. Murisic, B. Pausader, D. Peschka, and A. L. Bertozzi, Dynamics of particle settling and resuspension in viscous liquid films, Journal of Fluid Mechanics 717 (2013), 203-231] for monodisperse suspensions to two species of particles, resulting in a hyperbolic system of three conservation laws for the height and particle concentrations. We analyze the Riemann problem and show that the system exhibits three-shock solutions representing distinct fronts of particles and liquid traveling at different speeds as well as singular shock solutions for sufficiently large concentrations, for which the mechanism is essentially the same as the single-species case. We also consider initial conditions describing a fixed volume of fluid, where solutions are rarefaction-shock pairs, and present a comparison to recent experimental results. The long-time behavior of solutions is identified for settled mono-and bidisperse suspensions and some leading-order asymptotics are derived in the single-species case for moderate concentrations.

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Section: Second and Third Shockssupporting

confidence: 73%

Section: Introductionmentioning

confidence: 99%

A conservation law model for bidensity suspensions on an incline

Wong

Bertozzi

2016

Physica D: Nonlinear Phenomena

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“…We have a good understanding of the flow of monodisperse slurries [3,4] which is well-supported by mathematical analysis both in low particle concentrations [18] and highly-packed suspensions [19]. In order to gain some physical insight on the bidisperse slurry dynamics, we examine here the mathematical model for the particle flux, J, employed in monodisperse suspensions.…”

Section: Resultsmentioning

confidence: 95%

Experimental investigation of bidensity slurries on an incline

Lee¹,

Mavromoustaki²,

Urdaneta³

et al. 2014

Granular Matter

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We investigate the dynamics of bidensity slurries on an incline. The particle-fluid mixture consists of two species of negatively buoyant particles that have roughly the same size but significantly variant densities. This mismatch in particle densities induces or prevents settling depending on the relative amount of heavy to light particles, leading to complex regimes also found in the monodisperse case. In addition, when settling effects dominate within the thin film, we observe the phase separation down the incline between the particles and the liquid, as well as between two particle types.

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“…Lubrication models built on the equilibrium theory recently showed promise in describing the dynamic problem on a planar slope [24,25] and can be extended to the current spiral geometry.…”

Section: Discussionmentioning

confidence: 99%

Behavior of a particle-laden flow in a spiral channel

2014

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Spiral gravity separators are devices used in mineral processing to separate particles based on their specific gravity or size. The spiral geometry allows for the simultaneous application of gravitational and centripetal forces on the particles, which leads to segregation of particles. However, this segregation mechanism is not fundamentally understood, and the spiral separator literatures does not tell a cohesive story either experimentally or theoretically. While experimental results vary depending on the specific spiral separator used, present theoretical works neglect the significant coupling between the particle dynamics and the flow field. Using work on gravity-driven monodisperse slurries on an incline that empirically accounts for this coupling, we consider a monodisperse particle slurry of small depth flowing down a rectangular channel that is helically wound around a vertical axis. We use a thin-film approximation to derive an equilibrium profile for the particle concentration and fluid depth and find that, in the steady state limit, the particles concentrate towards the vertical axis of the helix, leaving a region of clear fluid.

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Hyperbolic systems of conservation laws in gravity-driven, particle-laden thin-film flows

Cited by 23 publications

References 22 publications

A conservation law model for bidensity suspensions on an incline

A conservation law model for bidensity suspensions on an incline

Experimental investigation of bidensity slurries on an incline

Behavior of a particle-laden flow in a spiral channel

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