(163c) Modeling Inclined Plane Flow of Particle-Laden Thin Films

The flows of particle-laden liquid films are essential in numerous applications, such as geophysical (erosion, debris flow, and turbidity currents) and industrial ones (commercial oil-sand and bitumen operations, semiconductor manufacturing, pharmaceutical production, and food processing). In addition, better understanding of the physical mechanisms involved in particle-laden thin film flows will undoubtedly prove useful in dealing with various related problems involving multi-phase flows.

The motivation for this work stems from a series of experiments carried out at the UCLA Applied Mathematics Laboratory and reported in [1]. In these experiments, the focus was on suspensions of polydisperse glass beads flowing down an inclined plane. By varying both the particle concentration and the inclination angle, three distinct flow regimes were observed. At low inclination angles and concentrations, rapid sedimentation occurred, resulting in particles forming a uniform particulate layer behind the advancing front, with clear liquid film flowing over this layer and eventually undergoing the fingering instability, typical for clear liquid thin film flows (see, e.g. [2]). On the other hand, at high concentrations and inclination angles it was found that particle-rich ridge forms at the advancing front, suppressing the fingering instability. Finally, a well-mixed regime was identified for intermediate concentrations and inclination angles, characterized by occurrence of regular fingers containing particles.

A theoretical model based on lubrication approximation was derived in [1]. It incorporated basic physical mechanisms, including Stokesian settling. However, the main focus was on a reduced governing system of conservation laws, which were used to develop a shock-dynamics theory relevant to this problem. While this theory introduced a gas dynamics technique to a contact line problem, it failed to fully capture the experimentally observed flow regimes.

The main goal of this study is to better understand the mechanisms responsible for different flow regimes, by achieving both qualitative and quantitative agreement between the theory and the experimental data. To this end, we derive a novel mathematical model, based on Navier-Stokes equations for an incompressible viscous fluid and the appropriate equation for particle mass concentration. The main building blocks of this model are as follows: (i) the thin film flow is such that lubrication approximation is appropriate; (ii) surface tension and gravity effects are considered; (iii) both the shear-induced particle migration and modified Stokesian settling are included into the model for the evolution of particle mass concentration; (iv) the interaction between the liquid and the solid substrate is modeled using a disjoining pressure approach (including both van der Waals and structural components); (v) the dependence of the dynamic contact angle on the local particle concentration is included; (vi) the possibility for solutal Marangoni effect is taken into account.

The model for shear-induced particle migration is based on a theory developed in [3]. While Brownian diffusive flux is neglected (due to a large Peclet number), both the effect of spatially varying interaction frequency between the particles and the effect of spatially varying viscosity are included in this model. These effects consider pair-wise interactions between particles which lead to corresponding net fluxes, entering the governing equation for particle mass concentration. To the best of our knowledge, this is is the first time shear-induced migration has been included into a lubrication approximation framework, and employed in modeling flows of particle-laden thin films.

The interaction between the liquid and the solid substrate is modeled using a two-component disjoining pressure approach. This model is a combination of van der Waals and so-called "structural'' forces, the latter taking into account direct interaction between particles and the solid substrate. Van der Waals forces, which are dominant over scales smaller than particle diameter, are described using a two-term model (see e.g. [4]). This model includes repulsive and attractive intermolecular forces leading to a stable precursor film. On the other hand, the structural component of the disjoining pressure acts on longer scales (longer than particle diameter), it is oscillatory in nature and gives rise to particle layering. The modeling of this component is based on statistical mechanics; we include it into our theoretical framework in a manner similar to the one discussed in [5]. The dependence of the dynamic contact angle on the local particle concentration in the vicinity of the advancing front is based on Cox-Voinov law, with our approach relying on the discussion in [6].

We present the derivation process in some detail and show how it eventually leads to a pair of coupled quasi-2D partial differential equations governing the evolution of liquid thickness and particle mass fraction. We develop a numerical code based on finite differences in order to solve this coupled system. In addition, we present the results of our recent experiments, carried out as an extension of experimental work reported in [1]. We proceed by comparing our numerical results with both the experimental data from [1] and our own. Finally, we outline the plan for future work, including more precise experiments which are to be carried out at the Applied Mathematics Laboratory as part of the REU (Research Experience for Undergraduates) program at UCLA.

References:

[1] Zhou, J., Dupuy, B., Bertozzi, A.L., Hosoi, A.E., 2005, Theory for Shock Dynamics in Particle-Laden Thin Films, Phys. Rev. Lett. 94, 117803.

[2] Kondic L., 2003, Instabilities in Gravity Driven Flow of Thin Fluid Films, SIAM Review 45, 95.

[3] Phillips R.J., Armstrong R.C., Brown R.A., Graham A.L., Abbott J.R., 1992, A Constitutive Equation for Concentrated Suspensions that accounts for Shear-Induced Particle Migration, Phys. Fluids A 4, 30.

[4] Schwartz L.W., Eley R.R., 1998, Simulation of Droplet Motion on Low-Energy and Heterogeneous Surfaces, J. Colloid Interface Sci. 202, 173.

[5] Matar O.K., Craster R.V., Sefiane K., 2007, Dynamic Spreading of Droplets containing Nanoparticles, Phys. Rev. E 76, 056315.

[6] Ralston J., Popescu M., Sedev R., 2008, Dynamics of Wetting from an Experimental Point of View, Rev. Mater. Res. 38, 23.