A comparative study of dynamic models for gravity-driven particle-laden flows

Abstract

The dynamics of viscous thin-film particle-laden flows down inclined surfaces are commonly modeled with one of two approaches: a diffusive flux model or a suspension balance model. The diffusive flux model assumes that the particles migrate via a diffusive flux induced by gradients in both the particle concentration and the effective suspension viscosity. The suspension balance model introduces non-Newtonian bulk stress with shear-induced normal stresses, the gradients of which cause particle migration. Both models have appeared in the literature of particle-laden flow with virtually no comparison between the two models. For particle-laden viscous flow on an incline, in a thin-film geometry, one can use lubrication theory to derive a compact dynamic model in the form of a $2\times 2$ system of conservation laws. We can then directly compare the two theories side by side by looking at similarities and differences in the flux functions for the conservation laws, and in exact and numerical simulations of the equations. We compare the flux profiles over a range of parameters, showing fairly good agreement between the models, with the biggest difference involving the behavior at the free surface. We also consider less dense suspensions at lower inclination angles where the dynamics involve two shock waves that can be clearly measured in experiments. In this context the solutions differ by no more than about 10%, suggesting that either model could be used for this configuration.

Figures (4)

• Figure 1: A schematic of a settled particle-laden suspension on an inclined flat plane. The channel is inclined at an angle $\alpha$ and the flow is driven by gravity indicated by $\mathbf{g}$.
• Figure 2: Particle volume fraction $\tilde{\phi}$ (\ref{['fig:phi_dfm_50']}, \ref{['fig:phi_sbm_50']}) and fluid velocity $\tilde{u}$ (\ref{['fig:u_dfm_50']}, \ref{['fig:u_sbm_50']}) profiles versus the non-dimensional height $s$ in the fluid layer for the diffusive flux and suspension balance models. The plots are generated at inclination angle $\alpha=50^\circ$ and maximum packing fraction $\phi_m=0.61$ to demonstrate different regimes. The families of solutions correspond to different $\phi_0$ values indicate by the labels on the plotted lines. The black lines correspond to the ridged regime, the red lines correspond to the well-mixed regime, and the blue lines correspond to the settled regime. Note that, for this particular value of $\alpha$ and parameter values given in Table \ref{['tab:param_vals']}, we find $B_d=2.307$ and $B_s=5.320$ for systems \ref{['eqn:dfm_ode']} and \ref{['eqn:corrected_sbm_ode']}.
• Figure 3: (\ref{['fig:phi_crit_61']}) Critical volume fraction $\phi_{crit}$ for the two models when $\phi_m=0.61$. (\ref{['fig:height_1']}) & (\ref{['fig:height_2']}) Height profiles $h$ (colored red) and depth-averaged particle concentration $n$ (colored blue) for the two models at two different times. The parameters we use are $\phi_0=0.25$ and $\alpha=10^\circ$.
• Figure 4: Liquid and particle front profiles with respect to time using parameters $\phi_0=0.25$ and $\alpha=10^\circ$ in \ref{['fig:exp_cff']} and \ref{['fig:exp_pf']}, and parameters $\phi_0=0.3$ and $\alpha=20^\circ$ in \ref{['fig:alpha=20_ff']} and \ref{['fig:alpha=20_pf']}. In each plot, numerical results are shown in black solid curves, which represent the DFM, and gray dashed curves for the SBM. Colored symbols indicate experimental results, with each color corresponding to a different experimental trial. Note that the numerical results include a transient regime as in murisic2013dynamics.