2.1. Trajectory equations for zero particle inertia ($St=0$)

We consider a dilute suspension of non-Brownian spherical particles in a non-continuum gas subject to a simple shear flow, sedimenting along the flow axis. For dilute suspensions, the probability of a third particle influencing the relative motion of two interacting particles is negligible, so we restrict our analysis to binary interactions of particles with radii $a_1$ and $a_2$, as shown in figure 2. The particles are sufficiently small to neglect the role of fluid inertia; thus we assume that the Stokes equations govern this creeping flow. Though we focus on a simple shear flow in the present study, the following formulation will apply to any arbitrary linear flow. A linear flow field ${\boldsymbol {U}}^\infty (\boldsymbol {x})$ can be characterized by a spatially constant strain rate tensor

(2.1)\begin{equation} {\boldsymbol{E}}^{\infty} = \tfrac{1}{2} \left[(\boldsymbol{\nabla} {\boldsymbol{U}}^{\infty}) + (\boldsymbol{\nabla} {\boldsymbol{U}}^{\infty})^{\rm T} \right], \end{equation}

and a rigid body rotation with angular velocity,

(2.2)\begin{equation} \boldsymbol{\varOmega}^{\infty} = \tfrac{1}{2}{\boldsymbol{\nabla}} \times {\boldsymbol{U}}^{\infty}. \end{equation}

The linearity of the Stokes equations enables us to write the resultant relative velocity between two particles by vector summing the relative velocities caused by the motion of force-free and torque-free particles in the background flow, attractive van der Waals force, and gravity. Moreover, because of the axisymmetric geometry of the problem, we can resolve the relative velocity along and normal to the line joining the centres of the two spheres (see Batchelor & Green Reference Batchelor and Green1972a; Batchelor Reference Batchelor1976, Reference Batchelor1982; Wang et al. Reference Wang, Zinchenko and Davis1994; Batchelor & Wen Reference Batchelor and Wen1982):

(2.3)\begin{align} \hat{\boldsymbol{v}}_{12}(\hat{\boldsymbol{r}}) &= \boldsymbol{\varOmega}^{\infty}\times \hat{\boldsymbol{r}} + {\boldsymbol{E}}^{\infty}\boldsymbol{\cdot}\hat{\boldsymbol{r}}-\left[A\, \frac{\hat{\boldsymbol{r}}\hat{\boldsymbol{r}}}{\hat{r}^2}+B\left({\boldsymbol{I}}-\frac{\hat{\boldsymbol{r}}\hat{\boldsymbol{r}}}{\hat{r}^2}\right)\right]\boldsymbol{\cdot} \left({\boldsymbol{E}}^{\infty}\boldsymbol{\cdot}\hat{\boldsymbol{r}}\right) \nonumber\\ &\quad -\frac{1}{6{\rm \pi} \mu_f }\left(\frac{1}{a_1} + \frac{1}{a_2}\right)\left[ G\,\frac{\hat{\boldsymbol{r}}\hat{\boldsymbol{r}}}{\hat{r}^2}+H\left({\boldsymbol{I}}-\frac{\hat{\boldsymbol{r}}\hat{\boldsymbol{r}}}{\hat{r}^2}\right)\right]\boldsymbol{\cdot}\boldsymbol{\nabla}(\hat{\varPhi}_{12}) \nonumber\\ &\quad -\frac{2 \rho_p \left(a^2_1-a^2_2\right) \boldsymbol{g}}{9 \mu_f}\boldsymbol{\cdot}\left[ L\,\frac{\hat{\boldsymbol{r}}\hat{\boldsymbol{r}}}{\hat{r}^2} + M\left({\boldsymbol{I}}-\frac{\hat{\boldsymbol{r}}\hat{\boldsymbol{r}}}{\hat{r}^2}\right)\right], \end{align}

where $\hat {\boldsymbol {r}}$ is the vector from the centre of particle 2 to the centre of particle 1, and ${\boldsymbol {I}}$ is the unit second-order tensor. Here, $A$ and $B$ are the mobility functions for two hydrodynamically interacting spherical particles in a linear flow field, $L$ and $M$ are the mobility functions for two unequal-sized spherical particles settling under gravity through a quiescent fluid, and $G$ and $H$ are the mobility functions for two spherical particles interacting hydrodynamically and moving because of a central potential. Also, $A$, $L$, $G$ are axisymmetric mobilities (i.e. mobility functions responsible for the relative motion along the line-of-centres), and $B$, $M$, $H$ are asymmetric mobilities (i.e. mobility functions responsible for the relative motion normal to the line-of-centres). These mobility functions depend on the size ratio $\kappa = a_2/a_1$ and dimensionless centre-to-centre distance $r=2|\hat {\boldsymbol {r}}|/(a_1+a_2)$. Dhanasekaran et al. (Reference Dhanasekaran, Roy and Koch2021a) calculated the modifications of the axisymmetric mobilities due to non-continuum lubrication interactions, where they considered continuum hydrodynamic interactions when $\xi > O(Kn)$, and non-continuum lubrication interactions when $\xi \leq O(Kn)$. In this study, we use the uniformly valid solution of axisymmetric mobilities developed by them.

We are interested in uncharged drops or particles in a gas, thus the potential $\hat {\varPhi }_{12} = \hat {\varPhi }_{{vdW}}$ is due solely to van der Waals attraction. Assuming pairwise additivity of the intermolecular attractions, Hamaker (Reference Hamaker1937) calculated the van der Waals force between two isolated particles. The force potential as a function of dimensionless centre-to-centre distance for two unequal-sized particles without retardation is then

(2.4)$$\begin{gather} \hat{\varPhi}_{{vdW}}={-}\frac{A_H}{6}\left[\frac{8\kappa}{(r^2-4)(1+\kappa)^2}+\frac{8\kappa}{r^2(1+\kappa)^2-4(1-\kappa)^2} \right. \nonumber\\ +\left.\ln\left\{\frac{(r^2-4)(1+\kappa)^2}{r^2(1+\kappa)^2-4(1-\kappa)^2}\right\}\right], \end{gather}$$

where $A_H$ is the Hamaker constant for the materials composing the two spheres. Typically, $A_H$ is of order $10^{-19}\unicode{x2013}10^{-21}$J (see Russel et al. Reference Russel, Saville and Schowalter1991).

The induced-dipole/induced-dipole interaction between the molecules results in van der Waals attraction. We can express this induced-dipole/induced-dipole interaction in the form of a summation of characteristic electromagnetic waves with finite propagation speed. The finite propagation speed of electromagnetic waves alters the induced-dipole/induced-dipole interactions when droplet separations are comparable with or larger than the London wavelength $\lambda _L$ ($\approx 0.1\,\mathrm {\mu }{\rm m}$). The effect of electromagnetic retardation was not considered in Hamaker's calculation; hence (2.4) is valid only for separations less than $\lambda _L$. Schenkel & Kitchener (Reference Schenkel and Kitchener1960) analysed the retardation effect and provided the following expressions for $\hat {\varPhi }_{{vdW}}$ when $\xi = r-2 \ll 1$:

(2.5)\begin{equation} \hat{\varPhi}_{{vdW}} = \left\{ \begin{array}{ll} \displaystyle -\left(\dfrac{\kappa}{3(1+\kappa)^2}\right)\left(\dfrac{A_H}{\xi + 0.855 N_L \xi^2}\right), & \text{for}\ \xi < 4/N_L,\\ \displaystyle -\left(\dfrac{\kappa}{\left(1+\kappa\right)^2}\right)\left(\dfrac{A_H}{\xi}\right)\left[ \dfrac{4.9}{15 N_L \xi} - \dfrac{8.68}{45 N^2_L \xi^2} + \dfrac{4.72}{105 N^3_L \xi^3} \right], & \text{for}\ \xi > 4/N_L, \end{array}\right. \end{equation}

where $N_L$ is the radius of the two particles scaled by the retardation wavelength $\lambda _L$ (i.e. $N_L = 2 {\rm \pi}(a_1+a_2)/\lambda _L = 2 {\rm \pi}a_1 (1+\kappa )/\lambda _L)$.

We choose a spherical coordinate system $(r,\theta,\phi )$ with origin at the centre of sphere 2 and $\theta = 0$ being the vorticity axis ($x_3$). To non-dimensionalize the system, we consider $a^*\ (=(a_1+a_2)/2)$, $\dot {\gamma }^{-1}$ and $\dot {\gamma }a^*$ as the characteristic length, time and velocity scales of the problem, respectively. Thus the non-dimensional radial separation between the centres of the two spheres lies in the range from $r=2$ (referred to as the collision sphere, indicated as sphere 3 in figure 2) to $r=\infty$ (where one sphere does not influence the other). The size ratio $\kappa$, which can vary in the range $(0,1]$, captures the geometry of the two-spheres system. For ${\boldsymbol {U}}^\infty (\boldsymbol {x}) = (\dot {\gamma }x_2,0,0)$, the dimensionless relative velocity $\boldsymbol {v}\ (= \hat {\boldsymbol {v}}_{12}/\dot {\gamma }a^*)$ can be written as $\boldsymbol {v} = v_r \hat {e}_r + v_{\theta } \hat {e}_{\theta } + v_{\phi } \hat {e}_{\phi }$, where

(2.6)\begin{gather} v_r ={-}L Q \sin\theta\cos\phi + (1-A)r \sin^2\theta\sin\phi\cos\phi-\frac{G}{N_F}\,\frac{{\rm d}\varPhi_{12}}{{\rm d}r}, \end{gather}

(2.7)\begin{gather}v_{\theta} ={-}M Q \cos\theta\cos\phi + (1-B)r \sin\theta\cos\theta\sin\phi\cos\phi, \end{gather}

(2.8)\begin{gather}v_{\phi} = M Q \sin\phi - r \sin\theta \left[\sin^2\phi + \frac{1}{2}\,B \left(\cos^2\phi - \sin^2\phi \right) \right]. \end{gather}

Here, $\varPhi _{12} = \hat {\varPhi }_{12}/A_H$ is the dimensionless interparticle potential. The dimensionless number $N_F$ measures the relative importance of viscous forces due to shear and van der Waals forces:

(2.9)\begin{equation} N_F = \frac{3{\rm \pi}\mu_f a_1^3\kappa(1+\kappa)\dot{\gamma}}{2A_H}. \end{equation}

The other non-dimensional quantity is $Q$, which is the terminal velocity due to gravity scaled with $\dot {\gamma }a^*$:

(2.10)\begin{equation} Q = \frac{4 \rho_p g a_1 (1-\kappa)}{9 \mu_f \dot{\gamma}}. \end{equation}

The quantity $Q N_F$, which measures the relative strength of gravitational and van der Waals forces, is independent of the fluid viscosity $\mu _f$ and the imposed shear rate $\dot {\gamma }$. We denote $Q N_F$ as $N_g$:

(2.11)\begin{equation} N_g = \frac{2 {\rm \pi}\rho_p g a_1^4 \kappa (1-\kappa^2)}{3 A_H}. \end{equation}

With the expressions for the relative velocity (2.6)–(2.8) thus obtained, we need to solve the following relative trajectory equations:

(2.12)\begin{gather} \frac{{\rm d}r}{{\rm d}t} = v_r, \end{gather}

(2.13)\begin{gather}\frac{{\rm d}\theta}{{\rm d}t} = \frac{v_{\theta}}{r}, \end{gather}

(2.14)\begin{gather}\frac{{\rm d}\phi}{{\rm d}t} = \frac{v_{\phi}}{r \sin\theta}, \end{gather}

where $t$ is the dimensionless time.

2.2. Expressions for the particle collision rate and efficiency

The collision rate is defined as the rate at which particles of radii $a_1$ and $a_2$ with number densities $n_1$ and $n_2$ collide with each other per unit volume. Mathematically, the collision rate $K_{12}$ is equal to the flux of pairs into the collision sphere of radius $r = 2$, thus we can express it in terms of the pair distribution function $P(r)$ and the particle relative velocity $\boldsymbol {v}$. In this problem, the collision rate over the collision sphere scales as $n_1n_2\dot {\gamma }(a_1+a_2)^3$, and therefore we can write

(2.15)\begin{equation} \frac{K_{12}}{n_1n_2\dot{\gamma}(a_1+a_2)^3}={-} \frac{1}{8} \int_{(r=2)\&(\boldsymbol{v}\boldsymbol{\cdot}{\boldsymbol{n}}<0)} (\boldsymbol{v}\boldsymbol{\cdot}{\boldsymbol{n}}) P\,{\rm d}A, \end{equation}

where ${\boldsymbol {n}}$ is the outward unit normal at the spherical contact surface. For a dilute suspension, the pair distribution function is governed by the quasi-steady Fokker–Planck equation for regions of space outside the contact surface,

(2.16)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}(P\boldsymbol{v}) = 0, \end{equation}

with the upstream boundary condition $P \rightarrow 1$ as $r \rightarrow \infty$. For our numerical calculations we consider $P=1$ at $r=r_{\infty }$, a large but finite radial location.

The deterministic background flow and high particle Péclet number (negligible Brownian diffusion) make the relative motion deterministic. Therefore, we can find the collision rate using trajectory analysis. Using (2.16) and the divergence theorem, the integral in (2.15) can be taken over the surface that encloses the volume occupied by all trajectories that originate at $r=r_{\infty }$ (far upstream) and terminate at $r=2$ (collision surface). The trajectories far upstream will become parallel to the $x_1$-axis as the interparticle interactions become insignificant for large separations. Thus the collision rate is equal to the flux through a cross-section $A_c$ located far upstream and perpendicular to the flow axis:

(2.17)\begin{equation} \frac{K_{12}}{n_1n_2\dot{\gamma}(a_1+a_2)^3}={-} \frac{1}{8} \int_{A_c} (\boldsymbol{v} \boldsymbol{\cdot}{\boldsymbol{n}}')|_{r_{\infty}}\,{\rm d}A, \end{equation}

where ${\boldsymbol {n}}'$ is the unit outward normal vector at the area element of $A_c$. From here onwards, we will label this area $A_c$ as the upstream interception area. Equation (2.17) bypasses the evaluation of the pair probability $P$. The interaction between the spheres is negligible at large separation, thus we can calculate $\boldsymbol {v}$ using the far-field values of the hydrodynamic mobilities. At large separation, $\boldsymbol {v}$ becomes parallel to the flow axis. We find that $(\boldsymbol {v} \boldsymbol {\cdot }{\boldsymbol {n}}')|_{r_{\infty }} = -Q + \bar {x}_2$, where $\bar {x}_i = x_i/a^*$ ($i=1,2,3$). Therefore, (2.17) reduces to

(2.18)\begin{equation} \frac{K_{12}}{n_1n_2\dot{\gamma}(a_1+a_2)^3}= \frac{1}{8} \int_{A_c} (Q - \bar{x}_2)\,{\rm d}\kern0.06em \bar{x}_3 \, {\rm d}\kern0.06em \bar{x}_2. \end{equation}

We denote the collision rates with and without interactions as $K_{12}^{PI}$ and $K_{12}^{0}$, respectively. The collision efficiency $E_{12}$ is defined as the ratio of $K_{12}^{PI}$ to $K_{12}^{0}$:

(2.19)\begin{equation} E_{12} = \frac{K^{PI}_{12}}{K_{12}^{0}}. \end{equation}

Using trajectory analysis, we determine the upstream interception area, within which two widely separated spheres will eventually collide.

4.1. Collision efficiency without gravity and van der Waals forces ($Q=0$, $N_F^{-1}=0$)

In the absence of gravity and van der Waals forces, (2.12)–(2.14) yield the following integrals for the relative trajectories (see Batchelor & Green Reference Batchelor and Green1972a):

(4.1)\begin{gather} \bar{x}_3 = \zeta_3\,\varphi(r), \end{gather}

(4.2)\begin{gather}\bar{x}_2^2 = \varphi^2(r)\,[\zeta_2 + \varPsi(r)] , \end{gather}

where $\zeta _2$ and $\zeta _3$ are the constants specifying a particular trajectory. The expressions for $\varphi (r)$ and $\varPsi (r)$ are

(4.3)\begin{gather} \varphi(r) = \exp \left[\int_r^{\infty} \frac{A(r')-B(r')}{1-A(r')}\,\frac{{\rm d}r'}{r'}\right], \end{gather}

(4.4)\begin{gather}\varPsi(r) = \int_r^{\infty} \frac{B(r')\,r'\,{\rm d}r'}{[1-A(r')]\,\varphi^2(r')}. \end{gather}

We perform integrals in (4.3) and (4.4) using near-field and far-field asymptotic expressions (for both $A(r)$ and $B(r)$) along with an exact bispherical coordinate solution of $A(r)$ and a twin-multipole expansion solution of $B(r)$ for intermediate separations. Continuum hydrodynamic interactions between a pair of solid spheres result in two types of relative trajectories – open and closed (see Batchelor & Green Reference Batchelor and Green1972a). These trajectories are fore–aft symmetric. Figure 4(a) shows in-plane (i.e. at $\bar {x}_3 = 0$) relative trajectories for continuum hydrodynamic interactions. The separatrices (black lines in figure 4a) $\zeta _2=0$ for continuum hydrodynamic interactions separate the closed and open trajectories (see Batchelor & Green Reference Batchelor and Green1972a). We find that non-continuum lubrication interactions introduce two new kinds of relative trajectories – collision and semi-closed – but trajectories remain fore–aft symmetric. Therefore, non-continuum lubrication interactions result in four types of trajectories: (i) open trajectories that arrive from infinity and depart to infinity without reaching the collision sphere; (ii) collision trajectories that arrive from infinity and collide on the surface of the contact sphere $r=2$, or depart from the surface of the contact sphere and go to infinity; (iii) semi-closed trajectories that start from the surface of the contact sphere and return to it; and (iv) closed trajectories. Figure 4(b) shows the pattern of pair trajectories at the shearing plane (i.e. at $\theta = {\rm \pi}/2$) for non-continuum hydrodynamic interactions. Interestingly, two distinct types of separatrix exist due to non-continuum lubrication interactions. The separatrix that separates semi-closed and collision trajectories touches the $\bar {x}_1$-axis at $r=\infty$. So for this separatrix, $\bar {x}_2=0$ at $r=\infty$, which yields $\zeta _2=0$. The other separatrix, which divides regions of collision and open trajectories, is tangent to the collision sphere at $\bar {x}_2=2$, and maintains a finite gap from the $\bar {x}_1$-axis at $r=\infty$; accordingly, from (4.2), we have $\zeta _2 = (4/\varphi ^2(2)) - \varPsi (2)$. For semi-closed and closed trajectories $\zeta _2<0$, and for open trajectories $\zeta _2> (4/\varphi ^2(2))-\varPsi (2)$. The trajectories confined between these two types of separatrix are collision trajectories. These separatrices form the boundary of the collision trajectories in three dimensions. The cross-section of this boundary becomes a semicircle as $r \rightarrow \infty$ (see figure 5). From figure 5, it can be seen that out-of-plane closed trajectories are present, while no closed trajectories exist in the shearing plane.

Figure 4. A comparison is made between (a) continuum ($Kn=0$) and (b) non-continuum ($Kn=10^{-1}$) in-plane pair trajectories when $\kappa =1$. The sphere located at the centre is the test sphere, and the black circle is the collision circle in the shearing plane. Arrows on the trajectories indicate their directions, and black arrows in each quadrant indicate the flow direction. In (a), blue and green lines are open and closed trajectories, and continuous black lines are the separatrices between them. In (b), blue, continuous red and dotted green lines are open, colliding and semi-closed trajectories. Dashed red lines start from the collision sphere and depart to infinity. Continuous black lines are separatrices between regions of semi-closed and collision trajectories, and between regions of collision and open trajectories. It is important to note that the trajectories for both cases are fore–aft symmetric.

Figure 5. The sphere (with non-dimensional radius 2) represents the collision surface. The limiting collision (continuous red lines) and semi-closed (dotted green lines) trajectories, and the closed (continuous green lines) trajectories along with their projections in different planes, are plotted for $Kn=10^{-2}$ and $\kappa =0.5$. The black circle represents the upstream interception area whose radius varies depending on the values of $Kn$ and $\kappa$.

The hydrodynamic mobility functions always asymptote to a specific numerical value as the separation approaches infinity; in this case, both $A$ and $B$ approach zero as $\xi \rightarrow \infty$. The lubrication force becomes dominant for $\xi \ll 1$. The radial mobility $(1 - A)$ is $O(\xi )$ for continuum lubrication interactions and thus precludes finite time contact between the spheres. As was discussed earlier, the non-continuum lubrication interactions lead to $1-A = O(1/{\ln (\ln (Kn/\xi ))})$, which vanishes slower than its continuum counterpart. This weaker non-continuum lubrication force arises at $\xi = O(Kn)$ and is responsible for the contact between the spheres in finite time. On the shearing plane, the minimum distance of the separatrix (that separates closed and open trajectories) from the collision sphere, $d^{sep}_{min} = \bar {x}_2-2 = r-2$ at $\bar {x}_1=0$, is finite for continuum hydrodynamic interactions. To find $d^{sep}_{min}$, we put $\bar {x}_2=r$ and $\zeta _2=0$ in (4.2), then solve for $r$ numerically. For a given size ratio, $d^{sep}_{min}$ decreases gradually with increasing $Kn$, and it becomes zero when $Kn$ is greater than $Kn_c$. We call this $Kn_c$ the critical Knudsen number for a given $\kappa$. The separatrix for $Kn>Kn_c$ always maintains a finite distance from the flow axis in the far field. Zinchenko (Reference Zinchenko1984) and later Wang et al. (Reference Wang, Zinchenko and Davis1994) analysed a problem that also encounters weakened lubrication interactions, similar to the current problem. They considered collisions of small drops in linear flow fields. For drops, the radial mobility is $1-A = O(\sqrt {\xi })$ for $\xi \ll 1$, which also vanishes slower than its rigid counterpart and therefore results in collision trajectories and finite collision rates in the absence of colloidal forces. They found a critical size ratio for a given viscosity ratio necessary for a collision to occur, thus finding a phase boundary for the existence of stable emulsions in the absence of colloidal forces.

Following the works of Zinchenko (Reference Zinchenko1984) and Wang et al. (Reference Wang, Zinchenko and Davis1994), we define two regions in $\boldsymbol {r}$-space ($r>2$) as given below:

(4.5)\begin{gather} D_f: \bar{x}_2^2 < \varphi^2(r)\,\varPsi(r), \end{gather}

(4.6)\begin{gather}D_t: \bar{x}_2^2 + \bar{x}_3^2 \leq \varphi^2(r) \left[\frac{4}{\varphi^2(2)} - \varPsi(2) + \varPsi(r)\right], \end{gather}

where the domain $D_f$ consists of trajectories with a finite length (i.e. semi-closed and closed trajectories), while trajectories touching the collision sphere (i.e. semi-closed and collision trajectories) occupy the region $D_t$. According to their definitions, these two regions must have overlap regions because semi-closed trajectories have finite lengths, and at the same time, they touch the collision sphere. The same is also evident from figure 5, where projections of the separatrices in different planes overlap. Therefore, semi-closed trajectories belong to the region $D_f \cap D_t$. Collision trajectories belong to $D_t - (D_f \cap D_t)$, closed trajectories belong to $D_f - (D_f \cap D_t)$, and open trajectories do not belong to any of these regions. The boundaries of the regions $D_f$ and $D_t$ are formed by rotating the in-plane separatrices corresponding to $\zeta _2 = 0$ about the $\bar {x}_2$-axis, and $\zeta _2 = (4/\varphi ^2(2))-\varPsi (2)$ about the $\bar {x}_1$-axis. The volumes of the regions $D_f$ and $D_t$ depend on $Kn$ and $\kappa$.

Unlike the scenario with continuum interactions, the collision rate is non-zero for non-continuum lubrication interactions, even in the absence of the van der Waals force. The expression for the collision efficiency in this case is

(4.7)\begin{equation} E_{12} = \frac{1}{8}\,\zeta^{3/2}_2 = \left(\frac{1}{\varphi^2(2)} - \frac{\varPsi(2)}{4}\right)^{3/2}. \end{equation}

Figure 6(a) shows the variation of the collision efficiencies with $Kn$ for different size ratios. As expected, for a given $\kappa$, the collision efficiency decreases with decreasing $Kn$. Most importantly, the collision efficiency is zero when $Kn$ is less than some critical value $Kn_c$. In contrast, our previous study (Dhanasekaran et al. Reference Dhanasekaran, Roy and Koch2021a) observed that collisions in a uniaxial compressional flow could occur without van der Waals forces for arbitrarily small $Kn$. In simple shear flow, the driving force for relative motion is primarily tangential at the closest approach. Thus the non-continuum modification of the resistivity for normal motion is less effective in simple shear flow than in uniaxial compressional flow. The critical Knudsen number $Kn_c$ is different for different values of $\kappa$. In (4.7), the quantity $(4-\varphi ^2(2)\,\varPsi (2))$ switches sign at $Kn=Kn_c$. Figure 6(b) shows that $Kn_c$ decreases as $\kappa$ increases. The $Kn_c\unicode{x2013}\kappa$ plot shows the demarcation between the region of stable dispersion (no collision possible) and the region of unstable dispersion (collision possible). The same figure also shows the variation of $d^{sep}_{min}$ with $\kappa$, and it is evident that the trends of $Kn_c$ and $d^{sep}_{min}$ are quite similar. The region of closed trajectories is larger for particles with a smaller $\kappa$, therefore $Kn_c$ decreases with increasing $\kappa$. For a medium with the mean free path $\lambda _0$, and $\kappa =1$, the critical particle size is $a_1=\lambda _0/Kn_c$. So, using the value of $Kn_c$, we can create a stable dispersion of equal-sized drops. Another important application of $Kn_c$ might be in detecting the mean free path of a gaseous medium. For known values of $a_1$ and $\kappa$, the mean free path of the medium can be found from the expression $\lambda _0=a_1(1+\kappa )\,Kn_c/2$.

Figure 6. (a) The collision efficiency for particles in a simple shear field as a function of $Kn$ for different $\kappa$ without van der Waals forces (i.e. $N_F=\infty$). For any value of $\kappa$, the collision efficiency decreases with decreasing $Kn$. The dotted lines corresponding to each curve indicate the value of the critical Knudsen number $Kn_c$ for that value of $\kappa$. (b) Plot showing how $Kn_c$ varies with $\kappa$. The solid red circles indicate the minimum distance of the separatrices from the collision sphere for $\kappa =0.3,0.4,\ldots,1.0$ when a pair of particles interacts through continuum hydrodynamics. The lower inset in (b) shows continuum in-plane separatrices for different values of $\kappa$. The black dashed line is a separatrix that separates closed and open streamlines for a single sphere in a simple shear flow. The upper inset in (b) shows a zoomed-in view of the separatrices near $\bar {x}_1=0$ in increasing order of $\kappa$ from the top.

4.2. Collision efficiency with van der Waals forces but without gravity ($Q=0$, $N_F^{-1} \neq 0$)

The van der Waals force acts along the line joining the two spheres’ centres, thus influencing only the radial component of the relative velocity. The last term on the right-hand side of (2.6) represents the relative velocity due to van der Waals attractions, whose potentials are expressed by (2.4) and (2.5). We use a uniformly valid solution of the mobility function $G$, which captures non-continuum hydrodynamic interactions for separations $\xi \leq O(Kn)$, and continuum hydrodynamic interactions for $\xi > O(Kn)$. In the presence of van der Waals interactions, the parameter $N_F$ becomes greater than zero. The van der Waals interaction eliminates the fore–aft symmetry of the relative trajectories, thus the determination of the upstream interception area becomes non-trivial.

The dimensionless trajectory equations (2.12)–(2.14) form an autonomous system, thus we can reduce the system to two equations:

(4.8)\begin{gather} \frac{{\rm d}r}{{\rm d}\phi} ={-} \frac{r(1-A) \sin^2\theta\sin\phi\cos\phi-\dfrac{G}{N_F}\,\dfrac{{\rm d}\varPhi_{12}}{{\rm d}r}} {\left[\sin^2\phi + \tfrac{1}{2} B \left(\cos^2\phi - \sin^2\phi \right) \right]}, \end{gather}

(4.9)\begin{gather}\frac{{\rm d}\theta}{{\rm d}\phi} ={-} \frac{(1-B) \sin\theta\cos\theta\sin\phi\cos\phi}{\left[\sin^2\phi + \tfrac{1}{2} B \left(\cos^2\phi - \sin^2\phi \right) \right]}. \end{gather}

Equations (4.8) and (4.9) are solved numerically using a fourth-order Runge–Kutta method (we use the ‘ode45’ subroutine in MATLAB). The near-field and far-field expressions for ${\rm d}r/{\rm d}\phi$ and ${\rm d}\theta /{\rm d}\phi$ can be derived by using respective near-field and far-field asymptotic forms of $A(r)$, $B(r)$ and $G(r)$. In this subsection, first, we will illustrate typical in-plane and off-plane trajectories, and a typical upstream interception area in three dimensions, using only the unretarded form of the van der Waals potential. Then we will show the upstream interception areas and collision efficiencies as a function of different relevant parameter values with both unretarded and retarded van der Waals potentials.

Figure 7 shows some typical in-plane trajectories for $Kn=10^{-2}$ and $N_F=10$ when $\kappa =0.5$. We find two kinds of in-plane trajectories – open and colliding. The van der Waals force enables satellite spheres to collide on the rear side of the test sphere. This type of collision trajectory does not exist in the absence of van der Waals interactions. Some open trajectories convert into collision trajectories because of van der Waals interactions, and these converted collision trajectories terminate on the rear side of the collision sphere. Thus the separatrices that separate open and colliding trajectories asymptote with the $x_1$-axis (flow axis) in the far downstream, and maintain a finite distance from the $x_1$-axis in the far upstream. This distance is greater for non-continuum hydrodynamic interactions compared to their continuum counterparts (see figure 7). Figure 8 shows two new kinds of off-plane collision trajectories. In one case, we find that the satellite sphere undertakes multiple circulations around the test sphere before collision (see figure 8a), whereas in the other case, the satellite sphere starts from a point near the vorticity axis and spirals in towards the test sphere to collide (see figure 8b). Feke & Schowalter (Reference Feke and Schowalter1983) and Wang et al. (Reference Wang, Zinchenko and Davis1994) have reported similar collision trajectories for colloidal particles interacting through continuum hydrodynamics in a simple shear flow.

Figure 7. A comparison is made between in-plane continuum and non-continuum relative trajectories for $N_F=10$ and $\kappa =0.5$. In the presence of van der Waals interactions, pair trajectories are not fore–aft symmetric. Blue lines are open trajectories. Red continuous ($Kn=10^{-2}$) and dash-dotted ($Kn=0$) lines indicate collision trajectories starting from different downstream locations but ending at the same point on the collision circle. Collisions occur on both the front and rear sides of the test sphere; trajectories designated by $1$, $1'$, $2$, $2'$ are such examples. Black continuous and dash-dotted lines are in-plane separatrices for non-continuum and continuum hydrodynamic interactions.

Figure 8. Off-plane collision trajectories in the presence of van der Waals forces when $Kn=10^{-2}$, $N_F=10$ and $\kappa =0.5$. In (a), a trajectory coming from upstream makes multiple circulations before hitting the collision sphere. In (b), a trajectory starts near the vorticity axis and spirals in towards the collision surface. In both plots, blue and black dots are the start and end locations of the trajectories.

As discussed below, we employ the backward integration method (first used by Adler Reference Adler1981) for obtaining the upstream interception area because this method considers collision trajectories of all types. Here, the separatrices that separate colliding and non-colliding trajectories define the boundary of the upstream interception area. It is well established that the van der Waals force becomes significant only when particles are in close separation (see Zhang & Davis Reference Zhang and Davis1991; Wang et al. Reference Wang, Zinchenko and Davis1994). Therefore, the boundary of $D_f$ must merge with the separatrices in the $r\rightarrow \infty$ limit, and the far downstream part of a collision trajectory must belong to the region $D_f$. The far-field coordinates of the boundary of $D_f$ are obtained after substituting the far-field expressions for $A(r)$ and $B(r)$ into (4.2) and (4.1):

(4.10)\begin{gather} \bar{x}_3 \sim \zeta_3 \left[1 + \frac{20}{3}\,\frac{1+\kappa^3}{(1+\kappa)^3}\,\frac{1}{r^3} \right], \end{gather}

(4.11)\begin{gather}\bar{x}_2^2 \sim \left(\frac{\bar{x}_3}{\zeta_3} \right)^2\left[\zeta_2 + \frac{32}{2}\, \frac{3(1+\kappa^5)+5\kappa^2(1+\kappa)}{3(1+\kappa)^5}\,\frac{1}{r^3}\right]. \end{gather}

For determining the separatrices, we integrate (4.8) and (4.9) from far downstream to far upstream (i.e. $\phi$ varies from ${\rm \pi}$ to $0$ for $\bar {x}_2 > 0$, and from ${\rm \pi}$ to $2{\rm \pi}$ for $\bar {x}_2 < 0$). We take $r=10$ as a reasonably large separation for far-field expressions to be valid. Thus the starting positions of these separatrices are obtained by fixing $r=10$ and calculating $\theta$ and $\phi$ according to (4.10) and (4.11) with $\zeta _2=0$. Figure 9 shows one-quarter of the upstream interception area in the ($\bar {x}_1,\bar {x}_2,\bar {x}_3$) space for $Kn=10^{-2}$, $N_F=10$ and $\kappa =0.5$. This area and its mirror image about the shearing plane (for the hemisphere on the negative side of the $x_3$-axis) constitute one-half of the total upstream interception area. The other half of the upstream area lies far upstream on the positive $x_1$ side, where the flow is in the negative $x_1$ direction. It is interesting to note that the curve bounding the area approaches infinity in the vorticity direction (i.e. $\bar {x}_3 \rightarrow \infty$ as $\bar {x}_2 \rightarrow 0$) because of the attractive van der Waals force between the spheres. Quarters of the upstream interception areas for different values of $N_F$ when $Kn=10^{-2}$ and $\kappa =1.0$ are plotted in figure 10(a). As expected, the area decreases with increasing $N_F$, and the area for the unretarded van der Waals interaction is larger than that for the retarded one at the same value of $N_F$. We obtain qualitatively similar plots of upstream interception areas for different $Kn$ when $N_F=10^2$ and $\kappa =1.0$ (see figure 10b). However, in this case, the area increases as $Kn$ increases.

Figure 9. Separatrices (that separate colliding and non-colliding trajectories) calculated using the backward integration method, plotted for $Kn=10^{-2}$, $N_F=10$ and $\kappa =0.5$. The area enclosed by the red lines is one-quarter of the upstream interception area.

Figure 10. Quarters of the upstream interception areas in the $\bar {x}_2\unicode{x2013}\bar {x}_3$ plane. (a) Upstream interception areas for different values of the van der Waals (vdW) force parameter $N_F$ (both unretarded and retarded) when $Kn=10^{-2}$ and $\kappa = 1.0$. (b) Upstream interception areas for various values of $Kn$ when $N_F=10^2$ (both unretarded and retarded) and $\kappa =1.0$. The solid and dashed lines in both the plots are for the retarded ($N_L=250$) and unretarded van der Waals interactions, respectively.

Figures 11(a) and 11(b), respectively, show the variation of the collision efficiency with $Kn$ (keeping $N_F$ fixed) and $N_F$ (keeping $Kn$ fixed) for two equal-sized spheres in a simple shear flow with non-continuum lubrication interactions and attractive van der Waals forces. Here too, the collision efficiency decreases with decreasing $Kn$, but unlike the previous case, for $Kn< Kn_c$, $E_{12}$ does not approach zero because van der Waals interactions between the particles then drive the entire collision dynamics. As $Kn$ approaches zero, the problem becomes equivalent to particles interacting through continuum hydrodynamics and van der Waals forces. The collision efficiency decreases as the strength of van der Waals forces relative to shear-driven viscous forces decreases, or in other words, $N_F$ increases (see figure 11b). The collision efficiency for unretarded van der Waals forces is considerably higher than that of the retarded cases when the value of $N_F$ is not high (approximately less than $10^4$). From an experimental perspective, figure 11(a) would correspond to a set of experiments holding $a_1$, $\dot {\gamma }$ and the sphere material fixed, and varying the mean free path $\lambda _0$ by varying gas pressure; and figure 11(b) would correspond to keeping $a_1$ and $\lambda _0$ fixed, and varying the shear rate $\dot {\gamma }$. In figure 12, we plot the collision efficiency of water drops in air with $a_1=10\,\mathrm {\mu }{\rm m}$ and $a_1=50\,\mathrm {\mu }{\rm m}$ as a function of $\kappa$ at different shear rates $\dot {\gamma }$. Given the numerical values of $\kappa$ and $\dot {\gamma }$, we can calculate easily the relevant dimensionless parameters required to determine the collision efficiencies. The mean free path of air at standard temperature and pressure is ${\approx }70$ nm. With decreasing pressure, the mean free path increases, a scenario relevant in the context of droplet coalescence in warm clouds, thus $\lambda _0 \approx 0.1\,\mathrm {\mu }{\rm m}$ would be an appropriate choice (see Lamb & Verlinde Reference Lamb and Verlinde2011). The Knudsen numbers as a function of size ratio are given by $Kn = 0.02/(1+\kappa )$ for $a_1=10\,\mathrm {\mu }{\rm m}$, and $Kn = 0.004/(1+\kappa )$ for $a_1=50\,\mathrm {\mu }{\rm m}$. For water droplets in air, $A_H\approx 5\times 10^{-20}$ J and $\mu _f\approx 1.8\times 10^{-5}\,{\rm Pa}\,{\rm s}$. Thus the dimensionless numbers $N_L$ and $N_F$ vary according to the relations $N_L\approx 628(1+\kappa )$ and $N_F\approx 1.7 \kappa (1+\kappa )\dot {\gamma }$ when $a_1=10\,\mathrm {\mu }{\rm m}$; and $N_L\approx 3141(1+\kappa )$ and $N_F\approx 212 \kappa (1+\kappa )\dot {\gamma }$ when $a_1=50\,\mathrm {\mu }{\rm m}$. For any fixed value of $\dot {\gamma }$, the collision efficiency increases with increasing $\kappa$. It is important to note that $\dot {\gamma }=\infty$ is equivalent to no van der Waals interactions. Therefore, the line $\dot {\gamma }=\infty$ must set the minimum value of the collision efficiencies for a given $\kappa$. For a fixed $\kappa$, the collision efficiency increases monotonically with decreasing $\dot {\gamma }$.

Figure 11. (a) Collision efficiency as a function of $Kn$ for $\kappa =1.0$ and $N_F = 10^2,10^4$ with $N_L=250$. Straight lines indicate the values of the collision efficiency for continuum hydrodynamic interactions in the presence of van der Waals forces. Each curve approaches the corresponding straight line as $Kn$ tends to zero. (b) Collision efficiency for two equal-sized spheres as a function of $N_F$ for $Kn=10^{-1}, 10^{-2}$. The result for $Kn=0$ obtained by Wang et al. (Reference Wang, Zinchenko and Davis1994) is included to validate our calculations.

Figure 12. Collision efficiency as a function of $\kappa$ at different shear rates $\dot {\gamma }$ for $a_1=10\,\mathrm {\mu }{\rm m}$ and $a_1=50\,\mathrm {\mu }{\rm m}$. The $\dot {\gamma }=\infty$ line corresponds to collision efficiencies resulting from non-continuum hydrodynamic interactions alone.

4.3. Collision rate and efficiency with gravity and van der Waals forces

There are several scenarios where the background flow and gravity, together, play a significant role in particle collisions. As stated earlier, we assume that the background flow is a simple shear flow. Here, we encounter an additional parameter, the inclination angle between the flow axis and gravity. Dhanasekaran et al. (Reference Dhanasekaran, Roy and Koch2021a) have recently studied collisions due to the combined effect of uniaxial compressional flow and gravity, exploring a range of inclination angles. To reduce complexity, in the current study, we fix the angle between the flow axis and gravity to be zero. In this case, the collision rate will depend on $Kn$ (quantifying the strength of non-continuum hydrodynamic interactions), $\kappa$ (describing the geometry of two interacting spheres), $Q$ (capturing the relative velocity due to gravity) and $N_F$ (capturing the van der Waals interactions). We will span these parameters to obtain salient features of the collision dynamics. We consider the Knudsen numbers $Kn=10^{-2}$ (less dominant non-continuum lubrication interactions) and $Kn=10^{-1}$ (significantly dominant non-continuum lubrication interactions), and size ratios $\kappa =0.5$ (noticeably different in size) and $\kappa =0.9$ (nearly equal-sized). For any chosen $Kn$ and $\kappa$, we will span $Q$ from 0 (simple shear flow dominated regime) to 20 (differential sedimentation dominated regime).

We will now discuss the detailed procedure for determining the upstream interception area without van der Waals interactions. In the presence of gravity and shear, the determination of the upstream interception area using trajectory analysis is more complicated than in the previous cases, where gravity was absent. The path traced by a satellite sphere that starts far upstream and collides with the test sphere located at the origin is classified as a collision trajectory. A set of such collision trajectories will constitute the upstream interception area. The computational cost of trajectory evolution would be immense if we set the initial conditions on the spherical shell at $r_{\infty }$. Most trajectories starting from the surface at $r_{\infty }$ would never reach the collision sphere. Instead, we start the satellite spheres from the collision surface and evolve them according to their time-reversed motion. This is possible because of the quasi-steady nature of the relative trajectory equations. The radial relative velocity of a colliding satellite sphere must be inwards at the collision sphere. Thus we further reduce the computation by selecting only those points on the collision sphere where the sign of the radial relative velocity is negative. However, exactly at $r=2$, $v_r =0$ because even in the presence of non-continuum lubrication interactions, $A=1$, $L=0$ and $G=0$ at $r=2$. For this reason, we consider small separations ($\xi \ll 1$) from the collision sphere. We will show converged results without too much computational load at offset $\xi = 10^{-6}$. Therefore, we evaluate relevant hydrodynamic mobilities and radial relative velocity from (2.6) at $r=2+10^{-6}$. The regions where $v_r$ is negative are called influx regions, and the regions where $v_r$ is positive are called efflux regions. Only influx regions contribute to the collision rate.

Figure 13 shows the influx–efflux regions on the collision sphere for four different values of $Q$ when $Kn=10^{-2}$, $\kappa =0.5$ and $N_F=\infty$. The collision sphere is coloured blue. The regions bounded by the black or red lines are the influx regions, and the remaining regions are the efflux regions. For $Q=0$, two distinct but identical influx regions exist on the collision sphere, corresponding to the simple shear flow. In both the regions, $\theta$ varies from $0$ to ${\rm \pi}$, whereas $\phi \in [{\rm \pi} /2, {\rm \pi}]$ in one region (bounded by black lines) and $\phi \in [3{\rm \pi} /2, 2{\rm \pi} ]$ in the other (bounded by red lines). With increasing $Q$, the influx region towards the gravity direction (i.e. $\bar {x}_1>0$ side) increases, and the influx region opposite to the gravity direction (i.e. $\bar {x}_1<0$ side) decreases. The influx regions approach each other with increasing $Q$. Eventually, at high enough $Q$, they merge. We found that for the above value of $Kn$ and $\kappa$, these two regions merge when $Q>10.2$, and the entire region lies within the hemisphere in the gravity direction. The influence of the simple shear flow becomes negligible with a further increase in $Q$, and gravity alone governs the collision dynamics. The bifurcation in the pair trajectory topology with increasing $Q$ has also been observed for particles settling in a uniaxial compressional flow by Dhanasekaran et al. (Reference Dhanasekaran, Roy and Koch2021a) and Dubey et al. (Reference Dubey, Gustavsson, Bewley and Mehlig2022).

Figure 13. The influx–efflux regions on the collision sphere for $Kn=10^{-2}$, $\kappa = 0.5$ and $N_F=\infty$, for various values of $Q$. The influx regions satisfy $\boldsymbol {v}\boldsymbol {\cdot }{\boldsymbol {n}}<0$, and the efflux regions satisfy $\boldsymbol {v}\boldsymbol {\cdot }{\boldsymbol {n}}>0$. Influx regions are bounded either by black lines or by red lines. In (a), with $Q=0$ (i.e. pure simple shear flow), two distinct but identical influx regions exist on the collision sphere. One region lies on the $\bar {x}_1<0$ side (opposite to the gravity direction), and the other region lies on the $\bar {x}_1>0$ side (gravity direction). In (b), with $Q=4$, there are also two influx regions but the region on the $\bar {x}_1<0$ side decreases, and the region on the $\bar {x}_1>0$ side increases. In (c), with $Q=8$, the two influx regions are close to each other. In (d), with $Q=12$, only one influx region exist and it lies in the gravity direction.

We integrate (2.12), (2.13) and (2.14) backwards in time to obtain relative trajectories. Not all the trajectories starting from the influx region will be colliding trajectories. They may also include semi-closed trajectories that start and end on the collision sphere and do not contribute to the collision rate. We identify the colliding trajectories by applying the condition that they depart to infinity. All these collision trajectories that go to infinity become parallel to the flow axis. Thus we determine the upstream interception area formed by these collision trajectories. Figure 14 shows the boundaries of the collection of collision trajectories constituting $A_c$ when $\kappa =0.5$, $Kn=10^{-2}$ and $N_F=\infty$. Two distinct boundaries exist for smaller values of $Q$. One of them lies in the gravity direction, where $\bar {x}_1>0$ (see figure 14b), and the other lies in the direction opposite to gravity, where $\bar {x}_1<0$ (see figure 14a). For $Q=0$, the upstream interception area is a circle (a sum of two identical semicircles) corresponding to simple shear flow alone. As expected, the area on the $\bar {x}_1<0$ side becomes thinner with increasing $Q$, and it disappears for $Q>5.3$ (see figure 14a); whereas the area on the $\bar {x}_1>0$ side becomes circular as $Q$ increases, corresponding to pure differential sedimentation. It is notable from figure 14(a) that with increasing Q, the entire area stretches significantly in the vorticity direction, and narrows in the gradient direction.

Figure 14. The upstream interception areas for various values of $Q$ when $Kn=10^{-2}$, $\kappa =0.5$ and $N_F=\infty$. For $0 \leq Q<5.3$, one area lies in the direction opposite to gravity (a), and the other lies in the gravity direction (b). For $Q=0$, each of them is a semicircle. With increasing $Q$, the area in the direction opposite to gravity decreases, and it vanishes for $Q > 5.3$. On the other side, as $Q$ increases, the area becomes a circle, corresponding to pure differential sedimentation.

The collision rates are shown in figure 15(a) as functions of $Q$ for $Kn=10^{-2},10^{-1}$ and $\kappa = 0.5,0.9$ when $N_F = \infty$. The qualitative behaviours of collision rates for various combinations of $Kn$ and $\kappa$ are similar. The background shear contributes significantly to the collision rate when $Q$ is quite small, and gravity almost exclusively contributes to the collision rate when $Q$ is high. It is evident from figure 15(a) that the collision rate due to the combined effect of gravity and shear is not a linear superposition of the collision rates resulting from them individually. We find that the collision rate increases with $Q$ for given $Kn$ and $\kappa$. However, we see a slight decrease in the collision rate for $4.7< Q<5.3$ since the collisional area that lies in the direction opposite to gravity disappears around these $Q$ values. Therefore, the small bumps that appear around $Q=5.3$ in the collision rate curves are physical. Dhanasekaran et al. (Reference Dhanasekaran, Roy and Koch2021a) and Dubey et al. (Reference Dubey, Gustavsson, Bewley and Mehlig2022) reported similar trends for the collision rate of settling spheres in a uniaxial compressional flow with non-continuum hydrodynamics.

Figure 15. Collision rates and efficiencies as functions of the strength of gravity relative to simple shear flow for $\kappa = 0.5, 0.9$ and $Kn=10^{-2},10^{-1}$ in the absence of van der Waals interactions (i.e. $N_F=N_g=\infty$). For all $Kn$ and $\kappa$, the collision rate increases, and the collision efficiency monotonically decreases with increasing $Q$.

The collision rate with interactions is smaller than the ideal collision rate evaluated in § 3. The collision efficiency, which measures the extent of this retardation and manifests interesting insights into hydrodynamic interactions, is obtained by dividing the rate with interactions (from figure 15a) by the ideal rate. Figure 15(b) shows the variation of the collision efficiency with $Q$. In the absence of van der Waals forces, the collision efficiency decreases monotonically with increasing $Q$. Like the collision rate curves, the collision efficiency curves exhibit small bumps around $Q=5.3$. For $Q\rightarrow 0$, the collision efficiency curve approaches asymptotically the value corresponding to the collision efficiency due to simple shear flow alone. At large $Q$, the asymptotic behaviour of the collision efficiency indicates the sedimentation-dominated regime. The asymptotic value for $Q\rightarrow \infty$ is significantly lower than the $Q\rightarrow 0$ result for both $\kappa =0.5$ and $\kappa =0.9$.

The divergent nature of the van der Waals attraction force near contact makes the radial relative velocity negative over the entire collision surface. Therefore, the entire collision sphere becomes the influx region in the presence of van der Waals forces. We determine the upstream interception area using the numerical technique discussed earlier in this section, with the initial conditions spreading over the entire collision sphere. The parameter $N_g\ (=QN_F)$, which measures the strength of gravity relative to van der Waals forces, is finite in this case. Figure 16(a) shows how the collision efficiency varies with $Q^{-1}$ for $N_g=1, 10, 10^2, \infty$ when $Kn=10^{-2}$, $\kappa =0.5$ and $N_L = 500$. A motivation for plotting $E_{12}$ with $Q^{-1}$ is to allow comparisons with a possible experimental study where the shear rate varies while the parameters describing the particle and medium properties are kept fixed. The collision efficiency with van der Waals forces is always higher than that without, for given values of $Kn$ and $\kappa$. The collision efficiency increases monotonically with $Q^{-1}$ when only non-continuum hydrodynamics is present in the problem ($N_g=\infty$ curve in figure 16a). For a fixed $N_g$ (other than $\infty$), as $Q^{-1}$ increases, the strength of van der Waals attractions relative to shear-induced viscous forces decreases since $N_F\ (=Q^{-1}N_g)$ increases. Therefore, with increasing $Q^{-1}$ (increasing shearing rate), $E_{12}$ increases up to a maximum value and then starts decreasing, and finally asymptotes to the value corresponding to the collision efficiency of particles in a simple shear flow in the presence of van der Waals forces and non-continuum hydrodynamics. The trend of $Q^{-1}$ versus $E_{12}$ curves for different $Kn$ are similar qualitatively (see figure 16b). For a given $Kn$, $E_{12}$ is higher for a smaller $N_g$, and for a given $N_g$, $E_{12}$ is higher for a higher $Kn$. The collision efficiencies due to the combined effects of gravity, background shear flow and van der Waals attractions are shown in figure 17 for various combinations of $\dot {\gamma }$ and $\kappa$ when $a_1=10$ and $20\,\mathrm {\mu }{\rm m}$. Like the $Q=0$ result given in figure 12, the collision process becomes more efficient as $\kappa$ increases. However, contrary to the $Q=0$ result, the collision efficiency is larger for a higher shearing rate (or higher $Q^{-1}$) because the van der Waals force accelerates the collision dynamics in the parameter space considered here.

Figure 16. (a) Collision efficiencies as functions of $Q^{-1}$ for $\kappa = 0.5$ and $Kn=10^{-2}$ when $N_g=1, 10, 10^2, \infty$ and $N_L = 500$. (b) Collision efficiencies for $Kn=10^{-1}, 10^{-2}, 10^{-3}$ when $N_g=10$. It is important to note that unlike the $N_g=\infty$ case, the variation of the collision efficiency with $Q^{-1}$ is non-monotonic for small and moderate values of $N_g$.

Figure 17. Collision efficiency due to the combined effects of gravity, shear and van der Waals force (retarded) as functions of size ratio $\kappa$ and shear rate $\dot {\gamma }$ when (a) $a_1=10\,\mathrm {\mu }{\rm m}$, and (b) $a_1=20\,\mathrm {\mu }{\rm m}$. For a water droplet in air, $A_H\approx 5\times 10^{-20}$ J, and the dynamic viscosity of air is $\mu _f\approx 1.8\times 10^{-5}\,{\rm Pa}\,{\rm s}$. The values of the parameters $Kn$, $N_L$, $N_F$, $Q$ and $N_g$ for given values of $\kappa$, $\dot {\gamma }$ and $a_1$ can be calculated from their expressions stated earlier.