Relevance of capillary interfaces simulation with the Shan–Chen multiphase LB model

Abstract

This paper presents a numerical study of capillary interfaces using the Single Component Multi-Phase Shan–Chen model, which is based on the lattice Boltzmann method. Despite the simplicity of the model, it has been shown to be effective and the present study aims to test its ability to correctly reproduce the physics of multiphase systems. To this end, several benchmark simulations were carried out in the configurations of a drop on a flat wall and then on a spherical surface to characterize the wetting behavior and relate explicitly the contact angle to model parameters. In addition, the capillary forces induced by a liquid bridge between two spherical particles were numerically calculated. We show that the results obtained are in agreement with experimental and theoretical results from the literature. The model is thus accurate in addressing the wetting behavior and capillary interfaces in unsaturated granular soils despite the fact that surface tension and contact angles are not explicit parameters of the model. To this respect, explicit relationships with Shan–Chen parameters are provided.

Graphic abstract

Appendix: Conversion between physical and lattice units

The LBM simulations with Shan–Chen model involve four dimensional quantities: time, length, mass and temperature. The conversion between lattice units and physical units is done with use of the reduced properties concept [4, 41] for the fluid properties (namely density $\rho$, pressure P, and temperature T) and with one additional conversion factors for length.

$$\begin{aligned} \rho _R= \frac{\rho }{\rho _c}, \quad P_R= \frac{P}{P_c}, \quad T_R=\frac{T}{T_c}, \end{aligned}$$

(37)

where the subscript “R” and “c” are the reduced and critical properties, respectively. According to this concept, the reduced properties in lattice and physical units should be equal. As an example, we have $\rho _R^{lu}=\rho _R^{phy}$, leading to $\rho ^{phy}=\rho ^{lu}\rho ^{phy}_c/\rho ^{lu}_c$. The other properties can be obtained in a similar manner.

The conversion factors for length, time and density are respectively $C_l=\Delta x_{phy}/\Delta x_{lu}$, $C_t=\Delta t_{phy}/\Delta t_{lu}$, and $C_{\rho }=\rho _{phy}/\rho _{lu}=\rho ^{phy}_c/\rho ^{phy}_c$. The conversion factor for force can the be deduced as $C_f=C_{\rho } C_l^4/C_t^2$. To get the value in physical units, the corresponding value in lattice units is multiplied by the conversion factor that has the same physical units. For example, for length we have $L_{phy}=L_{lu}\times C_l$.

For capillary interfaces with no flow, where the Reynolds number is no longer relevant, the time scale is not given by viscosity.⁸ The conversion between surface tension $\gamma$ in lattice and physical units reads

$$\begin{aligned} \gamma ^{phy} = \gamma ^{lu} C_{\rho } \frac{C_{l}^3}{C_t^2} \end{aligned}$$

(38)

In the presence of gravity, the conversion between gravity in lattice and physical units is given as

$$\begin{aligned} g^{phy} = g^{lu} \frac{C_{l}}{C_t^2} \end{aligned}$$

(39)

Alternatively, physical units can be related to lattice units through dimensionless numbers (e.g. the Bond number (Bo)) instead of using the conversion factors. As an example, the dimensionless Bond number is defined as

$$\begin{aligned} Bo = \frac{(\rho _l-\rho _g)gr^2}{\gamma } \end{aligned}$$

(40)

where g is the gravity, $\gamma$ is the surface tension and r is the length scale (e.g. drop radius in the case studied in Sect. 3.2). By setting $Bo_{lattice}=Bo_{physical}$, the conversion leads to

$$\begin{aligned} \frac{(\rho _l^{phy}-\rho _g^{phy})g^{phy}(r^{phy})^2}{\gamma ^{phy}} = \frac{(\rho _{l}^{lu}-\rho _g^{lu})g^{lu}(r^{lu})^2}{\gamma ^{lu}}. \end{aligned}$$

(41)

Expression (41) enables to fix the gravity in lattice units $g^{lu}$ that can be used in LBM simulation when taking it into account.