Direct numerical simulation of gas-solid-liquid flows with capillary effects: An application to liquid bridge forces between spherical particles

In this study, a numerical method is developed to perform the direct numerical simulation (DNS) of gas-solid-liquid flows involving capillary effects. The volume-of-fluid method employed to track the free surface and the immersed boundary method is adopted for the fluid-particle coupling in three-phase flows. This numerical method is able to fully resolve the hydrodynamic force and capillary force as well as the particle motions arising from complicated gas-solid-liquid interactions. We present its application to liquid bridges among spherical particles in this paper. By using the DNS method, we obtain the static bridge force as a function of the liquid volume, contact angle, and separation distance. The results from the DNS are compared with theoretical equations and other solutions to examine its validity and suitability for modeling capillary bridges. Particularly, the nontrivial liquid bridges formed in triangular and tetrahedral particle clusters are calculated and some preliminary results are reported. We also perform dynamic simulations of liquid bridge ruptures subject to axial stretching and particle motions driven by liquid bridge action, for which accurate predictions are obtained with respect to the critical rupture distance and the equilibrium particle position, respectively. As shown through the simulations, the strength of the present method is the ability to predict the liquid bridge problem under general conditions, from which models of liquid bridge actions may be constructed without limitations. Therefore, it is believed that this DNS method can be a useful tool to improve the understanding and modeling of liquid bridges formed in complex gas-solid-liquid flows.

Figure 1

Contact angle boundary condition at gas-solid-liquid three-phase point.

Figure 2

DNS grid for gas-solid-liquid flows. Calculation is based on a Cartesian mesh, for which the grid size must be finer than the characteristic length of the particle or meniscus.

Figure 3

Liquid bridges between two spheres: typical hydrophobic case (left) and hydrophilic case (right).

Figure 4

Spreading of a droplet on spherical particle surface. Reference data are obtained from Mitra et al. [90].

Figure 5

Convergence of liquid bridge force calculation toward reference solution. The line of first-order rate is drawn as a guide to the eye.

Figure 6

Forces of a small-volume liquid bridge with different contact angles: (a) $\theta ={15}^{\circ }$, (b) $\theta ={30}^{\circ }$, (c) $\theta ={60}^{\circ }$.

Figure 7

Comparison of forces by an asymmetric liquid bridge. The reference plots of experiment and simulation are measured from Lambert et al. [22].

Figure 8

Shapes for liquid bridges at zero separation distance $H/R=0.0$ with different volume: (a) $V_{LB}/V_P=0.1$ and (b) $V_{LB}/V_P=0.2$. Rows from top to bottom correspond to contact angle $\theta ={15}^{\circ },{45}^{\circ },{90}^{\circ }$, and ${120}^{\circ }$.

Figure 9

Comparison of sectional profiles for liquid bridges with different contact angles and zero separation for $V_{LB}/V_P=0.1$ and 0.2. Black lines, particle surfaces; red points, DNS results; green lines, surface evolver results.

Figure 10

Comparison of sectional profiles for liquid bridges with $V_{LB}/V_P=0.1$. The lines indicate separation distances: solid lines, $H/R=0.1$; dotted lines, $H/R=0.2$; dashed lines, $H/R=0.4$. The colors indicate different surfaces: black lines, particle surfaces; red lines, DNS results; green lines, surface evolver results.

Figure 11

Liquid bridge force vs separation distance. (a,c) Comparison of total liquid bridge force by the DNS, surface evolver, and analytical Models A and C. (b,d) Contributions by the tension term (solid lines) and the Laplace term (dashed lines), and the symbols are as follows: circles, $\theta ={15}^{\circ }$; squares, $\theta ={45}^{\circ }$; triangles, $\theta ={90}^{\circ }$; crosses, $\theta ={120}^{\circ }$. In (a,b), $V_{LB}/V_P=0.1$; in (c,d), $V_{LB}/V_P=0.2$.

Figure 12

Liquid bridges in triangular particle configurations at different distances: (a) $H/R=0.2$, (b) $H/R=0.4$, (c) $H/R=0.6$.

Figure 13

Variation of liquid bridge force and energy with respect to separation distance for the triangular configuration. (a) Dimensionless force. (b) Dimensionless contact area or interfacial energy.

Figure 14

Liquid bridges in tetrahedral particle configurations at different distances: (a) $H/R=0.2$, (b) $H/R=0.4$, (c) $H/R=0.6$.

Figure 15

Variation of liquid bridge force and energy with respect to separation distance for the tetrahedral configuration. (a) Dimensionless force. (b) Dimensionless contact area or interfacial energy.

Figure 16

Simulated liquid bridge $(V/R^3=0.2)$ under stretching at different time instants: (a) $t=12.5\mathrm{ms}$, (b) $t=125\mathrm{ms}$, (c) $t=170\mathrm{ms}$, (d) $t=175\mathrm{ms}$.

Figure 17

Liquid bridge force on the upper particle during stretching. Colors and symbols indicate liquid volumes: red circles, $V_{LB}/R^3=0.10$; yellow squares, $V_{LB}/R^3=0.15$; green diamonds, $V_{LB}/R^3=0.20$; purple triangles, $V_{LB}/R^3=0.25$. The black markers show the corresponding static solutions without gravity obtained by the surface evolver.

Figure 18

Rupture distance of liquid bridge under stretching.

Figure 19

Initial configuration of liquid bridge. The black bold line shows the circumference of the particle, and the red line shows the surface of the liquid bridge initialized as a cylindrical shape.

Figure 20

Temporal evolution of particle connected by liquid bridge with contact angle $\theta ={36}^{\circ }$. The velocity profiles (blue arrows) are shown for different configurations of the particle position (black circles) and liquid bridge shape (red line). (a) $t=1\mathrm{ms}$, initial wetting. (b) $t=5\mathrm{ms}$, particle acceleration under attraction. (c) $t=15\mathrm{ms}$, approaching the symmetry plane. (d) $t=24\mathrm{ms}$, just after the first rebound.

Figure 21

Plot of particle distance and liquid bridge force for $\theta ={36}^{\circ }$. (a) Plot against time. (b) Distance-force plot.

Figure 22

Temporal evolution of particle connected by liquid bridge with contact angle $\theta ={90}^{\circ }$. The velocity profiles (blue arrows) are shown for different configurations of the particle position (black circles) and liquid bridge shape (red line). (a) $t=1\mathrm{ms}$, initial wetting. (b) $t=20\mathrm{ms}$, undergoing attraction. (c) $t=40\mathrm{ms}$, the bridge is now near its maximum compression. (d) $t=60\mathrm{ms}$, receding under repulsion.

Figure 23

Plot of particle distance and liquid bridge force for $\theta ={90}^{\circ }$. (a) Plot against time. (b) Distance-force plot.