Drag force on spherical particles trapped at a liquid interface

The dynamics of particles attached to an interface separating two immiscible fluids are encountered in a wide variety of applications. Here we present a combined asymptotic and numerical investigation of the fluid motion past spherical particles attached to a deformable interface undergoing uniform creeping flows in the limit of small Capillary number and small deviation of the contact angle from ${90}^{\circ }$. Under the assumption of a constant three-phase contact angle, we calculate the interfacial deformation around an isolated particle and a particle pair. Applying the Lorentz reciprocal theorem to the zeroth-order approximation corresponding to spherical particles at a flat interface and the first correction in Capillary number and correction contact angle allows us to obtain explicit analytical expressions for the hydrodynamic drag in terms of the zeroth-order approximations and the correction deformations. The drag coefficients are computed as a function of the three-phase contact angle, the viscosity ratio of the two fluids, the Bond number, and the separation distance between the particles. In addition, the capillary force acting on the particles due to the interfacial deformation is calculated.

Figure 1

Illustrations of (a) a spherical particle on a flat interface between two immiscible viscous fluids and (b) the cross-section view of a spherical particle at a deformable interface with contact angle ${\theta }_s$ and immersion depth $b$.

Figure 2

The $y\text{−}z$ cross-section views of (a) the static deformation $\delta h^{(0,1)}$ and (b) the flow-induced deformation $\text{Ca}{h}^{(1,0)}$ with $\text{Ca}=\delta ={\tilde{\theta }}_s=1,\text{Bo}=1,\lambda =2$, and ${\tilde{\theta }}_s=1,\tilde{b}=0.$

Figure 3

The $y\text{−}z$ cross-section views of the interfacial deformation $h=\text{Ca}{h}^{(1,0)}+\delta h^{(0,1)}$ with $\text{Ca}=\delta ={\tilde{\theta }}_s=1,\lambda =2$, and varying values of Bo.

Figure 4

Geometry for the Lorentz reciprocal theorem.

Figure 5

The drag coefficient $f^{\left(1\right)}$ and its approximations $f_n^{\left(1\right)}$ for large Bond number plotted as functions of $\sqrt{\text{Bo}}$.

Figure 6

Comparison of the normalized drag [$F_D^{*}$: Eq. (52)] with numerical results from Loudet et al. (2020), for contact angles ${\theta }_s={75}^{\circ }$ and ${110}^{\circ },\text{Bo}\approx 0.2,\tilde{b}=0.$

Figure 7

Comparison of the normalized drag [$F_D^{*}$: Eq. (52)] with numerical results from Loudet et al. (2020) for viscosity ratio $\lambda =0.75,\text{Bo}\approx 0.2,$ and $\tilde{b}=0$.

Figure 8

Illustration of two spherical particles straddling a fluid interface between two immiscible viscous fluids, where the background uniform flow is perpendicular to the line of centers of the two particles.

Figure 9

Sketch of the fluid interface near two spherical particles.

Figure 10

The numerically calculated interfacial deformation, $h=\text{Ca}{h}_{\perp }^{(1,0)}+\delta h_{\perp }^{(0,1)}$, around two spherical particles with $\text{Ca}=\delta ={\tilde{\theta }}_s=1$, Bo $=1,\lambda =2$ and $L=6$: (a) 3D visualization, where the color map shows the interfacial height near the particle; (b) $x\text{−}z$ cross-section view for $y=0$; (c) $y\text{−}z$ cross-section view for $x=L/2$.

Figure 11

The drag coefficient plotted as functions of (a) Bond number Bo and (b) separation $L$.

Figure 12

Illustration of two spherical particles straddling a fluid interface between two immiscible viscous fluids, where the background uniform flow is parallel to the line of centers of the two particles.

Figure 13

The drag coefficient plotted as functions of (a) Bond number Bo and (b) separation $L$.

Figure 14

Comparison of the drag coefficients, $f_{\perp }^{\left(1\right)}$ and $f_{\parallel }^{\left(1\right)}$, and their corresponding asymptotic approximations, $f_{\perp ,\text{approx}}^{\left(1\right)}$ and $f_{\parallel ,\text{approx}}^{\left(1\right)}$, for large Bond number with $L=4,$ where $C_1\approx 14.31$ and $C_2\approx -25.97$ for $f_{\perp }^{\left(1\right)}$ and $C_1\approx 11.45$ and $C_2\approx -19.02$ for $f_{\parallel }^{\left(1\right)}$.

Figure 15

Comparison of the normalized drag coefficients, $f_{\perp }^{*\left(1\right)}$ and $f_{\parallel }^{*\left(1\right)}$, and their corresponding asymptotic approximations, $f_{\perp }^{*\left(1\right)}$ and $f_{\parallel }^{*\left(1\right)}$, for large $L$ with Bo =1, where $D\approx 0.95$ for $f_{\perp }^{*\left(1\right)}$ and $D\approx 1.91$ for $f_{\parallel }^{*\left(1\right)}$.

Figure 16

Sketch of the $x\text{−}y$ cross-section view of two particles at a fluid interface undergoing arbitrarily oriented uniform flow in the $x\text{−}y$ plane.

Figure 17

The magnitude of the drag force plotted as a function of the orientation angle $\mathrm{\Theta }$ with $\delta ={\theta }_s=1,\lambda =1/2$, and varying values of (a) separation $L$ and (b) Bond number Bo.

Figure 18

The dimensionless capillary force plotted as a function of the Bond number Bo. The green curve shows the single-particle capillary force; the red, black, and blue curves are the capillary forces exerted on one of the two particles due to the two-particle deformations in perpendicular background flows, with separation distance $L=4,8$, and 12, respectively. Parameter values: $\text{Ca}=1,\delta ={\tilde{\theta }}_s=1,\tilde{b}=0$.

Figure 19

The dimensionless capillary force exerted on particle I (${\mathrm{\Sigma }}_P^{\text{I}}$) due to the parallel flow-induced deformations plotted as a function of the Bond number Bo, with separation $L=4,8,$ and 12 (Ca $=1,\lambda =0$).

Figure 20

The dimensionless capillary force due to the total deformations near two particles undergoing parallel background flows. Parameter values: $\text{Ca}=1,\delta ={\tilde{\theta }}_s=1,\lambda =0.$

Figure 21

Comparison of the normalized drag computed from $F_D$ [Eq. (50)] with experimental data from Petkov et al. [25] and theoretical predictions from the literature [7, 8, 9].