Shock Solutions for Particle-Laden Thin Films

Abstract

Abstract. We derive a lubrication model describing gravity-driven thin film flow of a suspension of heavy particles in viscous fluid. The main features of this continuum model are an effective mixture viscosity and a particle settling velocity, both depending on particle concentration. The resulting equations form a 2 × 2 system of conservation laws in the film thickness h(x, t) and in φh, where φ(x, t) is the particle volume fraction. We study flows in one dimension under the constant flux boundary condition, which corresponds to the classical Riemann problem, and we find the system can have either double-shock or singular shock solutions. We present the details of both solutions and examine the effects of the particle settling model and of the microscopic length scale b at the contact line.

Self-similarity in particle-laden flows at constant volume

“…The flow of a particle-laden fluid down an inclined plane has recently been modeled by a system of scalar conservation laws [3,6],…”

“…We refer to [7] for a precise discussion. The depth-average of the velocity of the fluid, v tot found in (1), can then be derived via standard techniques in lubrication theory for thin liquid films [6], which is valid for fluids with small Reynolds numbers and characteristic height much smaller than the characteristic length. The first-order term describes its dominant behavior as:…”

“…Cook [6], modified the function to include the maximum packing fraction of particles (φ m = 0.67 in their case) to avoid singular shocks in solutions to the Riemann problem for (1) and (2) that occur when φ m = 1. This form also ensures that sedimentation stops once the maximum concentration is reached.…”

“…We compare different settling functions in our model to analyze this effect numerically. We also note that the lubrication models with settling require a precursor; they are singular at vanishing precursor [3,4]. Thus it makes sense to compare the dynamics of the lubrication model with settling to an exact solution of the problem without settling and with precursor.…”

“…This paper compares the behavior of a constantmass particle-laden fluid flow down an incline with a pure fluid flow. We compare the analytical solutions of the pure fluid flow to both physical experiments and numerical simulations of the particle-laden fluid (PLF) flow, using a model proposed in [3,4].…”