Surface tension effects for particle settling and resuspension in viscous thin films

Abstract

We consider flow of a thin film on an incline with negatively buoyant particles. We derive a onedimensional lubrication model, including the effect of surface tension, which is a nontrivial extension of a previous model (Murisic et. al [J. Fluid Mech. 2013]). We show that the surface tension, in the form of high order derivatives, not only regularizes the previous model as a high order diffusion, but also modifies the fluxes. As a result, it leads to a different stratification in the particle concentration along the direction perpendicular to the motion of the fluid mixture. The resulting equations are of mixed hyperbolic-parabolic type and different from the well-known lubrication theory for a clear fluid or fluid with surfactant. To study the system numerically, we formulate a semi-implicit scheme that is able to preserve the particle maximum packing fraction. We show extensive numerical results for this model including a qualitative comparison with two-dimensional laboratory experiments.

Existence of weak solutions for particle-laden flow with surface tension

“…Lubrication models of such flows reduce to coupled systems for the film height and a quantity tracking the second phase, whose complex dynamics have been the subject of considerable interest in recent research [11]. Here we consider one such model for gravity-driven suspension flow in one dimension that accounts for the non-uniform distribution of particles within the bulk of the fluid, proposed in [16] and recently extended to include surface tension in [15]. The model equations [15] for the film height h(x, t) and depth-integrated particle density ψ(x, t) have the form…”

“…Here we consider one such model for gravity-driven suspension flow in one dimension that accounts for the non-uniform distribution of particles within the bulk of the fluid, proposed in [16] and recently extended to include surface tension in [15]. The model equations [15] for the film height h(x, t) and depth-integrated particle density ψ(x, t) have the form…”

“…for constants [15,19]. These fluxes arise from depth-integrating the suspension (f ) and particle (g) volume fluxes, which depend on the distribution of particles in the fluid depth.…”

“…These fluxes arise from depth-integrating the suspension (f ) and particle (g) volume fluxes, which depend on the distribution of particles in the fluid depth. The exponents of φ in the dilute limit are a consequence of the particle accumulation towards the substrate of the fluid [15]. The suspension evolves as a settled layer of particles with a clear fluid layer above; the height of this layer can be shown to scale with φ 1/2 as φ → 0, so the fluxes for the particle transport in the lubrication model, which scale with the cube of the height, gain a factor of φ 3/2 .…”

“…Here h 0,δ and ψ 0,ε are smooth enough approximation functions, i. e. h 0,δ ∈ H 1 (Ω) and ψ 0,ε ∈ L 2 (Ω). Integrating (15) and (16) in Q T by 17, we get the mass conservation…”