Lattice Boltzmann method for simulation of wettable particles at a fluid-fluid interface under gravity

A computational technique was developed to simulate wettable particles trapped at a fluid-fluid interface under gravity. The proposed technique combines the improved smoothed profile-lattice Boltzmann method (iSP-LBM) for the treatment of moving solid-fluid boundaries and the free-energy LBM for the description of isodensity immiscible two-phase flows. We considered five benchmark problems in two-dimensional systems, including a stationary drop, a wettable particle trapped at a fluid-fluid interface in the absence or presence of gravity, two freely moving particles at a fluid-fluid interface in the presence of gravity (i.e., capillary floatation forces), and two vertically constrained particles at a fluid-fluid interface (i.e., capillary immersion forces). The simulation results agreed well with theoretical estimations, demonstrating the efficacy of the proposed technique.

Figure 1

Schematic illustration of a circular rigid particle suspended in an incompressible Newtonian fluid composed of two immiscible isodensity phases A and B. The order parameter ϕ distinguishes between the particle-side domain $(\varphi =1)$ and fluid-side domain $(\varphi =0)$, while the compositional order parameter ψ represents not only the fluid of phases A $(\psi ={\psi }_{\mathrm{A}})$ and B $(\psi ={\psi }_{\mathrm{B}})$ but also the fluid-wettable particle $(\psi ={\psi }_{\mathrm{P}})$.

Figure 2

Pressure difference Δ$P$ between a circular drop of phase A with a radius of $R_{\mathrm{d}}$ and a host fluid of phase B as a function of ${R_d}^{-1}$ for system A (see Table 1). The filled circles represent the simulation results, and the solid line represents the theoretical estimations based on Laplace's law [Eq. (46)] with an interfacial tension of $\sigma =(2.99\times {10}^{-4})\mathrm{\Delta }x$.

Figure 3

(a) Schematic illustration of system B (see Table 1) in a mechanically equilibrated state, where a circular particle with a radius of $R$, mass density of ${\rho }_{\mathrm{p}} (={\rho }_{\mathrm{f}})$, and contact angle of α (< 90°) is trapped at a flat fluid-fluid interface in the absence of gravity, exhibiting a downward distance $h$ from the fluid-fluid interface to the particle center. (b) The value of cos α$(=h/R)$ as a function of the affinity parameter χ defined by Eq. (47). The filled circles represent the simulation results, and the solid line represents the theoretical estimations based on Eq. (49).

Figure 4

(a) Schematic illustration of system C (see Table 1) in a mechanically equilibrated state, where a circular particle with a radius of $R$, mass density of ${\rho }_{\mathrm{p}}(>{\rho }_{\mathrm{f}})$, and contact angle of α$(>{90}^{\circ })$ is trapped at a deformed fluid-fluid interface in the presence of gravity, exhibiting a downward distance $h$ from the fluid-fluid interface to the particle center and a slope angle of the fluid-fluid interface Ψ at the point of contact between the interface and the particle surface. The deformed fluid-fluid interface exhibits a circular arc shape with a curvature radius of $R_{\mathrm{c}}$ and central angle of Ψ. (b), (c) Values of Ψ and $h$, respectively, as a function of the Bond number, Bo, defined by Eq. (50). The filled symbols represent the simulation results for the cases of $\alpha ={45}^{\circ }$, 90°, and 135°, and the lines represent the theoretical estimations of Ψ from Eqs. (56) and (57) and those of $h$ from Eq. (58).

Figure 5

Schematic illustrations of system D (see Table 1) in a mechanically equilibrated state, where two freely moving particles with the same radius $R$ and contact angle α $(>{90}^{\circ })$ are transiently separated by a horizontal center-to-center distance of 2δ and trapped at a deformed fluid-fluid interface in the presence of gravity, experiencing a capillary floatation force in the horizontal direction: (a) two identical heavy-weight particles with a mass density of ${\rho }_{\mathrm{p}}>{\rho }_{\mathrm{f}}$, (b) two identical light-weight particles with ${\rho }_{\mathrm{p}}<{\rho }_{\mathrm{f}}$, and (c) two similar particles with different mass densities (${\rho }_{\mathrm{p},1}={\rho }_{\mathrm{f}}+\mathrm{\Delta }\rho$ and ${\rho }_{\mathrm{p},2}={\rho }_{\mathrm{f}}-\mathrm{\Delta }\rho$). The other mathematical symbols are explained in the main text.

Figure 6

Snapshots for the simulation of system D (see Fig. 5) in a mechanically equilibrated state: (a) Two identical heavy-weight particles with ${\rho }_{\mathrm{p},1}={\rho }_{\mathrm{p},2}=1.5{\rho }_{\mathrm{f}}$, (b) two identical light-weight particles with ${\rho }_{\mathrm{p},1}={\rho }_{\mathrm{p},2}=0.5{\rho }_{\mathrm{f}}$, and (c) two similar particles with different mass densities of ${\rho }_{\mathrm{p},1}=1.5{\rho }_{\mathrm{f}}$ and ${\rho }_{\mathrm{p},2}=0.5{\rho }_{\mathrm{f}}$, where a contact angle of $\alpha ={135}^{\circ }$ was employed throughout system D (see Table 1). The left- and right-hand sides of every panel display snapshots at horizontal interparticle distances of $2\delta =48\mathrm{\Delta }x (=3R)$ and $2\delta =96\mathrm{\Delta }x (=6R)$, respectively.

Figure 7

Simulation results (filled symbols) and theoretical estimations (lines) for system D, corresponding to Figs. 5 and 6 (a) the downward distance $h$ for two identical particles (left-hand-side axis) or the vertical distance $2d$ between two similar particles with different mass densities (right-hand-side axis) as a function of their center-to-center distance 2δ; (b) the capillary floatation force $F^{\mathrm{lat}}$ between the two particles in the horizontal direction as a function of 2δ. The lines in panels (a) and (b) represent the theoretical estimations from Eq. (58) or (74) and those from Eqs. (60) or (70), respectively.

Figure 8

Simulation results for system D.3 containing two similar particles with different mass densities of ${\rho }_{\mathrm{p},1}=1.5{\rho }_{\mathrm{f}}$ and ${\rho }_{\mathrm{p},2}=0.5{\rho }_{\mathrm{f}}$, which were initially placed at the same vertical position and separated by a horizontal center-to-center distance of $2\delta =40\mathrm{\Delta }x$ [e.g., Fig. 5] and then allowed to move freely in all directions toward mechanically equilibrated positions [e.g., Fig. 5]: (a) time-series snapshots and (b) the $x$ and $y$ positions (left- and right-hand-side axes) of particles 1 and 2 as a function of time.

Figure 9

Typical snapshots for the simulation of system E in a mechanically equilibrated state, corresponding to Fig. 5: (a) system E.1 with a contact angle of $\alpha ={45}^{\circ }$, (b) system E.2 with $\alpha ={90}^{\circ }$, and (c) system E.3 with $\alpha ={135}^{\circ }$, where two identical particles with a radius of $R=16\mathrm{\Delta }x$ were constrained at the same vertical position and separated by a horizontal center-to-center distance of $2\delta =64\mathrm{\Delta }x (=4R)$ throughout system E (see Table 1). For every panel, from left to right, the snapshots obtained for downward distances of $h<h_0, h\approx h_0$, and $h>h_0$ are shown, where $h_0\equiv R\cos \alpha ,$ corresponding to slope angles of $\mathrm{\Psi }<0^{\circ }, \mathrm{\Psi }\approx 0^{\circ }$, and $\mathrm{\Psi }>0^{\circ }$, respectively.

Figure 10

Simulation results (filled symbols) and theoretical estimations (lines) of system E (see Table 1) in a mechanically equilibrated state, which are similar to parts (a) and (b) of Figs. 5 and 6, where two identical particles constrained at the same vertical position and separated by a horizontal center-to-center distance of $2\delta =64\mathrm{\Delta }x$ are trapped at a deformed fluid-fluid interface in the absence of gravity, experiencing not only a capillary immersion force in the horizontal direction but also a vertical force depending on their downward distance $h$. (a) The slope angle of the fluid-fluid interface Ψ as a function of $h$, (b) capillary immersion force $F^{\mathrm{lat}}$ between the two particles as a function of $h$, and (c) vertical force $F^{\mathrm{ver}}$ as a function of $h$, where different contact angles of $\alpha ={45}^{\circ }$, 90°, and 135° were considered. The lines in panels (a)–(c) represent the theoretical estimations from Eqs. (78), (60), and (76), respectively.

Figure 11

Volume fractions of phase A (filled symbols) and particle domain (open symbols) as a function of vertical position, $y-y_0$, along the lines of $x=\mathrm{\Delta }x$ for system B of $\chi =–1$ (a) and $x=64\mathrm{\Delta }x$ for system B of $\chi =–1$ (b), χ = 0 (c), and χ = 1 (d) (see Table 1), where $y=y_0$ ($=61.5\mathrm{\Delta }x, 64.8\mathrm{\Delta }x$, and $67.8\mathrm{\Delta }x$ for $\chi =–1,0$, and 1, respectively) represents the level of a fluid-fluid interface. Every solid line is to guide the eyes.