Theory for particle settling and shear-induced migration in thin-film liquid flow

Abstract

Particles suspended in a film flow can either settle out of the flow, remain well mixed, or even advance faster than the fluid, accumulating at the moving contact line. Recent experiments by Zhou et al. [Phys. Rev. Lett. 94, 117803 (2005)] have demonstrated that these three settling behaviors can be achieved by control of the average particle concentration phi and inclination angle theta . This work presents a theory for determining the settling behavior in the Stokes regime by calculating the depth profile of phi and the depth-averaged velocities of the liquid and particle phases. It is found that shear-induced particle fluxes can lead to an inversely stratified flow, in which the particles move on average faster than the liquid. The theory is directly compared to Zhou et al.'s experimental data, and the implications of stratification for lubrication-type models are also discussed.

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

“…(9) and (14) are henceforth referred to as the 'equilibrium equations'. As shown in the literature [9,10,14], the ODE solutions constitute a family of curves parametrized by the total volume of particles present within the fluid film, φ 0 (defined below). Before proceeding, since the equilibrium equations are explicitly independent of the axial flow direction and time, it is also convenient to remove their implicit dependence [present through h(x, t)], by rescaling the normal direction, z.…”

“…The solutions were found to approach the 'ridged' regime behavior but failed to capture the separation between the particle-free and particle-rich regions observed in the 'settled' regime. In the subsequent studies of Cook [14] and Cook et al [15], it was suggested that shear-induced migration phenomena are important and should be included in the dominant physics of the problem. The lubrication models derived in the above-mentioned studies assume that transport due to shear or settling occurs rapidly in the normal direction such that these processes depend solely on the axial flow direction and time.…”

Self-similarity in particle-laden flows at constant volume

“…One possibility of additional physics to include in this analysis is shear induced migration (e.g. [19,20]) which has recently been shown to give quantitatively accurate predictions of phase transitions between settling to the substrate and settling to contact line in constant flow-rate experiments [7]. It would also be interesting to incorporate a precursor into the physical experiments.…”

“…It is shown in [7] that this assumption can be improved to a distribution that is stationary in time which balances sedimentation and shear-induced migration. We refer to [7] for a precise discussion.…”

“…It is shown in [7] that this assumption can be improved to a distribution that is stationary in time which balances sedimentation and shear-induced migration. 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.…”