Diffuse interface model to simulate the rise of a fluid droplet across a cloud of particles

A large variety of industrial and natural systems involve the adsorption of solid particles to the fluidic interface of droplets in motion. A diffuse interface model is here suggested to directly simulate the three-dimensional dynamics of a fluid droplet rising across a cloud of large particles. In this three-phase model the two solid-fluid boundaries and the fluidic boundary are replaced with smoothly spreading interfaces. The capillary effects and the three-phase flow hydrodynamics are fully resolved. A special treatment is adopted for the interparticle collisions. The effect of the particle concentration on the terminal velocity of a rising fluid droplet is then investigated. It is found that, at low Reynolds number, the terminal velocity of a rising fluid droplet decreases exponentially with the particle concentration. This exponential decay is confirmed by a simple rheological model.

Figure 1

Schematic of the reference ternary system representing a fluid droplet B rising in the host fluid A. The particles of the solid constituent S adsorb at the fluidic interface of the binary fluid.

Figure 2

Bulk component of the free energy density $f_b(\psi ,{\varphi }_{\mathrm{S}})$ inside and outside the solid constituent.

Figure 3

Depletion layers adjacent to the boundaries of two solid particles immersed in the host consistent A (a). The two particles are pinned. Upon a close encounter, the depletion layers deform, and the two particles attract each other because of a short-range capillary force ${\mathbf{F}}_{\mathrm{cap}}^{\mathrm{sr}}$ (b).

Figure 4

Error in the terminal velocity of a rising spherical bubble in the absence of particles. Periodicity is enforced on all side boundaries of the domain.

Figure 5

Short-range capillary force ${\mathbf{F}}_{\mathrm{cap}}^{\mathrm{sr}}$ calculated as a function of the normalized particle separation distance $r_{ij}$. The two particles are pinned and immersed in the fluid constituent A. The repulsive force $-\mathbf{\nabla }{U}_{ij}$ is also shown.

Figure 6

Effect of the corrected collision force in a multiparticle system. In panel (a) clustering occurs as opposed to panel (b). The solid lines correspond to the reference particle radius $r_s$. In panel (c) the ratio of the interfacial thickness to the particle radius is decreased. The simulations are performed with $N_{\mathrm{S}}=23$ particles.

Figure 7

Snapshot of the three-dimensional rising bubble in a multiparticle system: (a) low particle concentration; (b) high particle concentration.

Figure 8

Effect of the solid concentration in the host fluid on the terminal velocity of a rising bubble in the Stokes regime. The standard deviation of the mean bubble velocity is about as large as the symbols. The semiempirical expression, obtained with ${\lambda }_0=4$ and ${\lambda }_1=12$, is extrapolated beyond its range of validity.

Figure 9

Effect of the interfacial-length-to-particle-radius ${\xi }_{\mathrm{AB}}/r_s$ on the terminal velocity of a rising droplet. The two-dimensional convergence study was performed with $N_{\mathrm{S}}=22$ particles.