Particle migration and free-surface topography in inclined plane flow of a suspension

Abstract

Gravity-driven free-surface flow of a suspension of neutrally buoyant particles down an inclined plane channel of constant width has been studied experimentally and by flow modelling. A uniform suspension of spheres, sieved to radii of $a \,{=}\, 53--125 $\umu$m or 125–150$\,\umu$m, was introduced to create films of initial depth $h_o$. The flow was always at small film-depth-based Reynolds numbers. The film depth and the mixture flow profile were measured at the initial and two locations at least $200 h_o$ and $400 h_o$ downstream. The bulk particle volume fraction, $\phi_{\rm B}$, was varied in the range $0.01\,{\le}\, \phi_{\rm B} \,{\le}\, 0.5$; $h_o\,{\approx}\, 1.8$–3.2 mm and the inclination angle relative to horizontal, $0.1^{\circ} \,{<}\, \alpha\,{\le}\, 90^{\circ}$, were also varied. Analysis of the particle velocity was performed by stereoscopic imaging to determine particle location followed by particle correlation velocimetry. A two-layer Newtonian viscosity model was applied to the velocimetry results in order to infer particle concentration information. Measured velocity profiles and film depth show that film thickness decreases from $h_o$, while the velocity gradient at the wall and the mean velocity increase, as the mixture flows down the plane. The free surface, examined using direct imaging, becomes progressively more deformed as $\alpha$ and $\phi_{\rm B}$ increase, with the onset of observable deformation found at a particle-scale capillary number of $\hbox{\it Ca}_{p} \,{\sim}\, {\rho g_x a^2}/{\sigma}\,{=}\, O(10^{-4})$; $\rho$ is the mixture density, $g_x\,{=}\, g \sin \alpha$ is the axial component of gravitational acceleration and $\sigma$ is the surface tension of the suspending liquid. An existing model for suspension flow which describes phase migration as driven by normal stresses caused by the suspended particles is used to predict the flow, with satisfactory agreement for film depth and development distance for the non-uniform local solid volume fraction, $\phi$. The agreement with the detailed $\phi$ profile is less good, as the model fails to predict the observed $\phi \,{\approx}\, 0$ near the solid boundary while $\phi$ is overpredicted adjacent to the free surface.