2.1. Dimensionless numbers

We consider two liquid drops denoted $1$ and $2$ falling under gravity in a gas and subject to thermal diffusion. The position of their centre of mass is denoted $\boldsymbol r_i$ and their radii $R_i$. The first dimensionless number in the problem is the drop radius ratio

(2.1)\begin{equation} \varGamma=\frac{R_1}{R_2}\ge 1. \end{equation}

The curvature of the gap between the drops depends on the characteristic drop size:

(2.2)\begin{equation} a \equiv R_1R_2/(R_1+R_2). \end{equation}

Here $a$ varies from $R_2/2$ when both drops have the same radius and $R_2$ when $R_1$ is much larger than $R_2$. The gas mean free path $\bar \ell$ plays an important role in the problem, as it controls the transition to the non-continuum Knudsen aerodynamical regime. A second dimensionless number is therefore

(2.3)\begin{equation} {\mathcal{A}}\equiv\frac{a}{\bar \ell} = \frac{R_1R_2}{(R_1+R_2)\bar \ell}. \end{equation}

Two further parameters compare the viscosity of the liquid $\eta _\ell$ and that of the gas $\eta _g$, and the density of the liquid $\rho _\ell$ and that of the gas $\rho _g$:

(2.4a,b)\begin{equation} {\mathcal{N}} \equiv \frac{\eta_\ell}{\eta_g}\quad {\rm and} \quad \mathcal{D} \equiv \frac{\rho_\ell}{\rho_g}. \end{equation}

We consider water drops in air at 25 $^\circ {\rm C}$ and 1 atm for which $\eta _g=18.5\times 10^{-6}$ Pa s, $\eta _\ell =8.9\times 10^{-4}$ Pa s, $\rho _g=1.2$ kg m$^{-3}$, $\rho _\ell =1000$ kg m$^{-3}$ and $\bar \ell =68$ nm (Jennings Reference Jennings1988). The dimensionless numbers are therefore ${\mathcal {N}}=48$ and ${\mathcal {D}}=830$. The influence of inertia in the drop dynamics is controlled by the dimensionless number

(2.5)\begin{equation} {\mathcal{G}}=\left( \frac{ \rho_\ell (\rho_\ell-\rho_g) g } {\eta_g^2} \right)^{1/3} a. \end{equation}

The ratio ${\mathcal {G}}/{\mathcal {A}}$ does not depend on the drop sizes and is equal to $2\times 10^{-2}$ for water drops in air. We introduce the typical radius $b$ at which ${\mathcal {G}}$ is equal to $1$:

(2.6)\begin{equation} b=\left( \frac{\eta_g^2}{ \rho_\ell (\rho_\ell-\rho_g) g } \right)^{1/3}. \end{equation}

For water drops in air, we get $b = 3.3$ $\mathrm {\mu }$m, which is the typical size of drops in clouds and fogs. The thermal noise is controlled by the dimensionless number

(2.7)\begin{equation} \mathcal{K} = \frac{3\rho_\ell kT}{4{\rm \pi}\eta_g^2 \bar\ell}. \end{equation}

At ambient temperature, for water, it is around $\mathcal {K} = 4.2\times 10^{-2}$. The surface tension $\gamma$ controls both drop deformations and van der Waals interaction. It gives the dimensionless number

(2.8)\begin{equation} {\mathcal{S}}=\frac{\rho_\ell \gamma }{\eta_g^2}a. \end{equation}

The ratio ${\mathcal {S}}/{\mathcal {A}}$ does not depend on the drop sizes and is equal to $1.4 \times 10^4$. Two electrostatic effects are taken into account. Van der Waals interactions are parametrised by the surface tension $\gamma$ and by the Hamaker constant $A$, which can be rewritten as $A=24{\rm \pi} \gamma \varsigma ^2$, where $\varsigma$ is the Israelachvili length. van der Waals interactions are therefore characterised by the dimensionless parameter

(2.9)\begin{equation} {\mathcal{T}}=\frac{\varsigma}{\bar \ell}=\frac{\sqrt{A/(24{\rm \pi} \gamma)} }{\bar \ell}. \end{equation}

Here ${\mathcal {T}}$ is equal to $1.21 \times 10^{-3}$ for water. The effect of a static electric field $E_0$ is also taken into account, which is encoded into the dimensionless parameter:

(2.10)\begin{equation} \mathcal{E}=\sqrt{12 {\rm \pi}\rho_\ell} \frac{ \varepsilon E_0 \bar \ell}{\eta_g}. \end{equation}

The typical electric field in non-precipitating warm cloud is on the order of $\mathcal {E}\simeq 3\times 10^{-4}$. The electric field for which air electrical breakdown occurs is on the order of $E_0=3\times 10^6$ V m$^{-1}$, which gives $\mathcal {E}\simeq 6$: the electric field in the atmosphere may vary over 4 orders of magnitude.

2.2. Dynamical equations

We consider that both drops are entrained by the same background fluid velocity and denote by $\boldsymbol V_i$ their velocity with respect to this background velocity. The aerodynamic interaction is decomposed into a long-range contribution due to viscous stress, computed using the Oseen approximation, and a short-range contribution due to pressure, computed in the lubrication approximation, as shown in figure 4. We neglect the gradients of velocity at the scale of $R_1$ and $R_2$. We denote by $H=|\boldsymbol r_2-\boldsymbol r_1|-R_1-R_2$ the distance between drops. We consider the Oseen approximation, valid at Reynolds number ${\mathcal {R}}_i \ll 1$. The equation of motion of drop $i$ reads

(2.11)\begin{align} \frac 43 {\rm \pi}R_i^3 \rho_\ell \frac{{\rm d} \boldsymbol V_i}{{\rm d}t} &= \frac 43 {\rm \pi}R_i^3 (\rho_\ell-\rho_g) \boldsymbol g - 6 {\rm \pi}\eta_g R_i\left(1+\frac 38 {\mathcal{R}}_i \right) {\boldsymbol V_i}+\boldsymbol F_{ji} -6 {\rm \pi}\eta_g a^2 \zeta'(H) \dot H \boldsymbol e_{ij} \nonumber\\ &\quad - f_{vdW} \boldsymbol e_{ij} + \boldsymbol F^e + \boldsymbol W_i. \end{align}

The index $j$ is equal to $2$ for $i=1$ and to $1$ for $i=2$. Inertia in the air flow surrounding the drops is controlled by the Reynolds number of each drop, with $\boldsymbol V_i(t)$ the drop velocity,

(2.12)\begin{equation} \mathcal{R}_i=\frac{\rho_g V_i R_i}{\eta_g}. \end{equation}

Here $\boldsymbol F_{ji}$ is the long-range aerodynamic force exerted by the drop $j$ on the drop $i$, $f_{vdW}$ is the van der Waals interaction and $\boldsymbol F^e$ is the electrostatic force between drops. The correction $3/8 {\mathcal {R}}_i$ to the drag on each drop arises from the Oseen approximation (Batchelor Reference Batchelor2010, p. 244). Consistently, the terminal velocity under gravity, denoted $U^t_i$, obeys

(2.13)\begin{equation} \left(1+\frac{3 \rho_g R_i U^t_i}{8 \eta_g} \right) U^t_i = \frac{ 2 (\rho_\ell-\rho_g) g R_i^{2}} {9 \eta_g}. \end{equation}

For water drops in air, the associated Reynolds number ${\mathcal {R}}_i$ reaches $1$ around a radius $R_i\simeq 56\ \mathrm {\mu }{\rm m}$. The term $6 {\rm \pi}\eta _g a^2 \zeta '(H) \dot H \boldsymbol e_{ij}$ is the generic form of the lubrication force originating from the pressure between the two drops. The function $\zeta$ is derived in § 3.2. The dimensionless parameter controlling the relative influence of inertia and viscous damping is the Stokes number, defined here as

(2.14)\begin{equation} St\equiv \frac{\rho_\ell a (U^t_1-U^t_2)}{\eta_g}. \end{equation}

Consider two drops of sizes in the same range, say $R_1=2R_2=3a$. In the viscous aerodynamical regime (Stokes drag), the Stokes number simplifies into $St = \frac 32 \mathcal {G}^3$. The dimensionless number $\mathcal {G}$ can therefore be interpreted as a Stokes number at the terminal velocity, to a power $1/3$. Here $\mathcal {G}$ characterises the influence of inertia for a drop falling around the equilibrium between gravity and viscous friction. The low-Stokes-number regime, where inertia is negligible, is referred to as the overdamped regime. Overdamped dynamics is described by (2.11) without the acceleration term on the left-hand side, i.e. assuming force balance at all times.

In the equations of motion, $\boldsymbol W_i$ is the thermal noise, delta-correlated in time. The noise is normalised using the fluctuation–dissipation theorem described in § 5. When particles are far apart, it leads to a relative diffusion of the two droplets with a diffusion coefficient

(2.15)\begin{equation} D=\frac{k_B T}{6 {\rm \pi}\eta_g a}. \end{equation}

The relative amplitude of aerodynamic effects and thermal diffusion is controlled by the Péclet number, defined as

(2.16)\begin{equation} Pe = \frac{(R_1+R_2)(U^t_1-U^t_2)}{D}. \end{equation}

The diffusive regime, where Brownian motion dominates, corresponds to the low-Péclet-number asymptotics.

2.3. Diffusive, electrostatic and inertial regimes

Figure 1(a) is adapted from the classical textbook of Pruppacher & Klett (Reference Pruppacher and Klett2010, chap. 15). It shows the collisional kernel $K$ between drops of size $R$ with one drop of size 1 $\mathrm {\mu }$m. Multiplied by the number of drops of radius $R$ per unit volume, $K$ gives the collision frequency of a 1 $\mathrm {\mu }$m drop with drops of size $R$. The figure shows a gentle cross-over between Brownian coagulation and gravitational coagulation for drops around $R\simeq 2\ \mathrm {\mu } {\rm m}$. Figure 1(b) presents our results for the same problem. The dot-dashed blue line shows the results obtained when taking into account gravity and aerodynamics only. Below $R=10\ \mathrm {\mu } {\rm m}$, the kernel is ten times smaller than that in panel (a) but above $R=10\ \mathrm {\mu } {\rm m}$, it increases much faster. The dashed red line takes into account van der Waals interaction between drops. Finally, the full model, including Brownian motion is shown in solid green line. The diffusive regime, for $R<1\ \mathrm {\mu } {\rm m}$ is similar to that in figure 1(a). However, in between $R=1\ \mathrm {\mu } {\rm m}$ and $R=10\ \mathrm {\mu } {\rm m}$, the dominant effect turns out to be van der Waals forces.

Figure 1. A change of conceptual model. (a) Graph adapted from the textbook of Pruppacher & Klett (Reference Pruppacher and Klett2010, chap. 15), showing the current vision in cloud microphysics. Comparison between two collisions modes for a spherical particle of 1 $\mathrm {\mu }$m interacting with a second particle of radius $R$, showing a cross-over between Brownian coagulation (green solid line) and gravitational coagulation (dot-dashed blue line). The calculations are based on Klett (Reference Klett1975) and Hidy (Reference Hidy1973). (b) Predictions made here, including gravity, inertia and aerodynamics (dot-dashed blue line), adding van der Waals interactions (dashed red line) and, then, Brownian diffusion (green solid line). A new regime appears between the diffusive regime and the inertial regime, where electrostatic effects become dominant.

As mentioned previously, $K$ must be multiplied by the number density of drops of size $R$ to obtain the collision frequency with drops of size 1 $\mathrm {\mu }$m. As the density of drops generally decays rapidly with $R$, figure 1 must be interpreted with caution. The quasi-plateau in the purely diffusive regime corresponds, once weighted by the density, to a decrease of Brownian coagulation rate with $R$.

In figure 2, the contributions of the dynamical mechanisms to the collision rate is analysed as a function of the two key dimensionless numbers: the Stokes number and the Péclet number. The ratio $\varGamma$ of the sizes of the large and small drops is kept constant. This measures the relative change of collision rate when one dynamical mechanism is suppressed. The dot-dashed blue line is obtained using overdamped equations (no inertia). It shows that above a Stokes number $St$ on the order of $10$ inertia is dominant. Suppressing it completely changes the collision rate. Similarly, the solid green curve is obtained by suppressing the thermal noise from the equations of motion and shows that below a Péclet number $Pe$ of unity, Brownian coagulation is dominant. Over three decades in drop size $a$, both inertia (${St}<1$) and thermal noise (${Pe}>1$) are inefficient, so that a third mechanism becomes dominant: electrostatic interactions. One observes that removing the van der Waals forces changes the collision rate by $50\,\%$ (dashed red line). The gap between the diffusive and inertial regimes constitutes the central result of this paper.

Figure 2. Effect of the various dynamical mechanisms on the collision rate for $\varGamma = R_1/R_2 = 5$. Solid green line: difference in the collision rate with and without thermal diffusion, compared with the collision rate with diffusion, as a function of the Péclet number (top axis). Dash-dotted blue line: difference in the collision rate with and without inertia, compared with the collision rate with inertia, as a function of the Stokes number (bottom axis). Inertia dominates at $St>1$ whereas thermal diffusion dominates at $Pe<1$, leaving $3$ decades in $St$ where neither inertia nor thermal diffusion are relevant. Dashed orange line: difference in the collision rate with and without van der Waals interactions, compared with the collision rate with van der Waals interactions. Lubrication is the dominant effect to reduce the collision rate between the diffusive and inertial regimes. van der Waals interactions limits this gap, but only accounts for a fraction of it.

2.4. Experimental data

Figure 3 compares the collision efficiencies modelled here with experimental data from the literature. Experimental efficiencies roughly collapse on a master curve when plotted as a function of the Stokes number ${St} = \rho _\ell a (U_1^t-U_2^t)/\eta _g$. They show a drop in efficiency when inertia becomes comparable to aerodynamic effects, as predicted here and in most numerical works since Langmuir (Reference Langmuir1948). The efficiency decreases when the smaller drop does not have enough inertia to cross the streamlines around the larger drop. Efficiencies for $\varGamma$ close to $1$ are less reliable and do not follow this trend, as the small velocity difference means that the drops interact for very long times that may not be reached experimentally. The most accurate experiment is that by Vohl et al. (Reference Vohl, Mitra, Wurzler, Diehl and Pruppacher2007), who used a single collector drop in a controlled airflow at its terminal velocity. The inset of figure 3 shows the Stokes number for which $E=0.2$ (red dotted line) as a function of $\varGamma$. These measurements overlap with our computations for $\varGamma <10$. For larger size ratios, the experimental data rather follows the critical Stokes number for head-on collisions given by (3.24). Unfortunately, efficiencies around the minimum for ${St}=1$ (i.e. 6 $\mathrm {\mu }$m) have never been measured experimentally, nor below this cross-over value. This presents its own experimental challenges, as the critical impact parameter near the minimum is at nanometre scale. Further experimental work is needed to understand the fine details of the collision in the parameter range for which the collision frequency drops, the regime most relevant to cloud microphysics (Beard & Ochs Reference Beard and Ochs1993).

Figure 3. Experimental dataset of the collision efficiency corrected for diffusion $E_d$ as a function of the Stokes number $St = \rho _\ell a (U_1^t-U_2^t)/\eta _g$. Colour indicates radius ratio $\varGamma =R_1/R_2$. The dash-dotted green line is the computed curve in the absence of an electric field for $\varGamma =5$. Inset: transitional Stokes number as a function of the radius ratio $\varGamma$. Dashed blue line: critical Stokes number $St_\varLambda$ for head-on collisions given by (3.24), with $\varLambda = 4.7$. Dotted red line: Stokes number for the full model at which the efficiency reaches $E=0.2$ in the inertial regime. Crosses: Stokes number for which $E=0.2$ obtained from fitting $\varGamma$-aggregated data from Vohl et al. (Reference Vohl, Mitra, Wurzler, Diehl and Pruppacher2007) to the full model presented here. Agreement is good for $\varGamma <10$ but degrades at larger size ratios. The simple head-on collision model (3.24) captures well the observed trend in the experimental data.

3.1. Long-range aerodynamic interactions

In first approximation, $\boldsymbol F_{ji}$ can be deduced from the effective velocity induced by the drop $j$ at the location of the drop $i$ considered. The drag force can be linearised with respect to the velocity difference between the drop and the gas. Denoting by $\boldsymbol u(\boldsymbol r)$ the velocity field induced by the drop $j$ and taking into account the Faxén correction to the drag force (Guazzelli et al. Reference Guazzelli, Morris and Pic2012), we get

(3.1)\begin{equation} \boldsymbol F_{ji}=6{\rm \pi} \eta_g R_i \left(1+\frac 34 {\mathcal{R}}_i \right) \left(\boldsymbol u+\frac 16 R_i^2 {\boldsymbol \nabla}^2 \boldsymbol u\right)_{\boldsymbol r_i}. \end{equation}

The Oseen solution is obtained by linearising the equations around the mean flow. The polar coordinate system is centred on the drop $j$ inducing the field. Here $\theta =0$ is the direction of the velocity vector $\boldsymbol V_j$. The radial velocity $u_r$ and the tangential velocity $u_\theta$ are given by

(3.2a,b)\begin{equation} u_{r} =\frac{1}{r^{2} \sin \theta} \frac{\partial \varPsi}{\partial \theta} \quad {\rm and} \quad u_{\theta} ={-}\frac{1}{r \sin \theta} \frac{\partial \varPsi}{\partial r}, \end{equation}

where the stream function reads (Lamb Reference Lamb1911)

(3.3)\begin{align} \varPsi&={-}V_j R_j^{2} \sin ^{2} \theta \frac{R_j}{4 r}+\frac{3 V_j R_j^{2}}{2{\mathcal{R}}_j}(1-\cos \theta)(1-\phi_j ), \nonumber\\ &\qquad\mathrm{with}\ \phi_j\equiv \exp \left(-\frac{r {\mathcal{R}}_j}{2 R_j}(1+\cos \theta)\right). \end{align}

The velocity field reads

(3.4)\begin{gather} \frac{u_r}{V_j} ={-}\frac{R_j^{3} \cos \theta}{2 r^{3}}+\frac{3 R_j^{2}}{2 r^{2} {\mathcal{R}}_j}(1-\phi_j ) -\frac{3 R_j(1-\cos \theta)}{4 r} \phi_j, \end{gather}

(3.5)\begin{gather}\frac{u_\theta}{V_j} ={-}\frac{R_j^{3} \sin \theta}{4 r^{3}}-\frac{3 R_j \sin \theta}{4 r} \phi_j. \end{gather}

The solution presents an intermediate asymptotics which coincides with the Stokes solution, the velocity field decaying as $r^{-1}$. However, at distances much larger than $R_j/{\mathcal {R}}_j$, the solution decays much faster, as $r^{-2}$. The Faxén correction requires the evaluation of the Laplacian:

(3.6)\begin{gather} \nabla^2 \boldsymbol u |_r ={-}\frac{3 (2R_j+r {\mathcal{R}}_j) (r {\mathcal{R}}_j \sin ^2\theta+4 R_j \cos \theta)}{8R_j r^3} \phi_j, \end{gather}

(3.7)\begin{gather}\nabla^2 \boldsymbol u |_\theta ={-}\frac{3 \sin \theta (4R_j^2+ r {\mathcal{R}}_j (2R_j+r {\mathcal{R}}_j)(1+\cos \theta))}{8 R_j r^3} \phi_j. \end{gather}

3.2. Lubrication force

For rigid spheres, the gap $h(r)$ between the drops can be locally expanded as

(3.8)\begin{equation} h(r) = H+R_1+R_2-\sqrt{R_1^2-r^2}-\sqrt{R_2^2-r^2}\simeq H+ \frac{r^2}{2a} \end{equation}

as shown in figure 4(b). In order to regularise the lubrication force, we first introduce the slip length, which is approximately equal to the mean free path $\bar \ell$ in a gas. Slip at the interface is taken into account using the Navier slip boundary conditions $u_r = \bar \ell \,{\rm d} u_r/{\rm d}z$ at $z = 0$ and $u_r = -\bar \ell \,{\rm d} u_r/{\rm d}z$ at $z = h$. Using cylindrical coordinates, the velocity profile, in the lubrication approximation, reads

(3.9)\begin{equation} u_r=\frac{1}{2 \tilde \eta_g} \partial_{r} P (z^{2}-(z+\bar \ell) h), \end{equation}

where $\tilde \eta _g$ is an effective viscosity which is equal to the viscosity $\eta _g$ at large $H/\bar \ell$, but which gets smaller in the Knudsen regime $H/\bar \ell <1$ (figure 4c). Following Sundararajakumar & Koch (Reference Sundararajakumar and Koch1996), a good approximate expression of $\tilde \eta _g$ is

(3.10)\begin{equation} \tilde \eta_g=\frac{\eta_g}{1-\dfrac{2}{\rm \pi} \ln\left(\dfrac{3H}{3H+\bar \ell}\right)} \simeq \frac{\eta_g}{\upsilon}, \quad\mathrm{with}\ \upsilon=1+\frac{2}{\rm \pi} \ln\left(1+\frac{\bar \ell}{3H}\right). \end{equation}

The continuity equation integrates into $\int _0^h u_r \,{\rm d}z=- r \dot H /2$. Integrating a second time, one obtains the pressure field, which reads

(3.11)\begin{equation} P=3\eta_g a \zeta''(h)\dot H, \quad\mathrm{with}\ \zeta''(h)={-}\frac{1}{ 3 \upsilon \bar \ell} \left(\frac{1}{ h}+\frac{1}{6 \bar \ell}\log \left(\frac{h}{6 \bar \ell+h}\right) \right). \end{equation}

The lubrication force is obtained by integrating the pressure over the surface:

(3.12)\begin{align} F&\simeq \int_{0}^{\infty} 2{\rm \pi} r P(r) \,{\rm d}r={-}6 {\rm \pi}\eta_g a^2 \zeta'(H) \dot H,\nonumber\\ &\qquad\mathrm{with}\ \zeta'(H)=\frac{1}{3 \upsilon \bar \ell} \left[ \left(1+\frac{H}{6 \bar \ell}\right) \log \left(1+\frac{6 \bar \ell}{H}\right)-1\right]. \end{align}

A second dynamical mechanism can lead to a regularisation of the lubrication force: the induction of a motion inside the liquid (figure 4b). Davis et al. (Reference Davis, Schonberg and Rallison1989) have solved this problem for non-deformable drops, as a function of the fluid to gas viscosity ratio ${\mathcal {N}}$. The integration of the equations giving the pressure profile and the force, taking both the flow inside the drop and the mean free path into account can only be performed numerically. Here, we make use of exact asymptotic results to derive an approximate analytical formula.

Figure 4. Hydrodynamic mechanisms involved in drop collisions. (a) Long-range aerodynamic interaction. (b) Lubrication in the air film near contact. The shear of the squeezing air flow creates a flow inside the drops. (c) When the gap is comparable to the mean free path $\bar \ell$, rarefaction of the air between the drops leads to partial slip boundary conditions and a lower effective viscosity. (d) The pressure inside the lubrication air film induces a capillary flattening of the interface.

Let us consider both entrainment of the liquid inside the drop, characterised by an interfacial velocity $u_t$ and a Poiseuille contribution:

(3.13)\begin{equation} u_r=u_t+\frac{1}{2 \tilde \eta_g} \partial_{r} P (z^{2}-(z+\bar \ell) h). \end{equation}

In the mass conservation equation, the flux now reads

(3.14)\begin{equation} \int_0^h u_r \,{\rm d}z={-}\frac{r}{2} \dot H=u_t h-\frac{\upsilon}{12 \eta_g} h^2 (h+6 \bar \ell) \partial_{r} P. \end{equation}

Using the Green function formalism, the tangential stress $\sigma _t$ can be related to the tangential velocity $u_t$ by a non-local relationship. Dimensionally, one obtains the scaling law:

(3.15)\begin{equation} \sigma_t=\frac{h}{2}\partial_r P \sim {\mathcal{N}} \eta_g \frac{u_t}{\sqrt{ah}}. \end{equation}

Considering this scaling law as a local relationship, the flow inside the drop would lead to a term $\sim \sqrt {ah}/{\mathcal {N}}$ added to $h+6 \bar \ell$ in (3.14). There are therefore two possible regularisation processes. Slip occurs in the Knudsen regime, below a gap $h\sim \bar \ell$. The cross-over between a dissipation taking place in the lubrication gaseous film and in the drop takes place at $h\sim a/{\mathcal {N}}^2$.

We therefore propose to modify the function $\zeta '(H)$ giving the force $F$ into

(3.16) \begin{equation} \zeta'(H)=\frac{1}{3 \upsilon \bar \ell} \Bigg[ \Bigg(1+\frac{H+s \sqrt{aH}/{\mathcal{N}}}{6 \bar \ell}\Bigg) \log \left(1+\frac{6 \bar \ell}{H+s \sqrt{aH}/{\mathcal{N}}}\right)-1\Bigg], \end{equation}

where $s$ is a constant. When ${\mathcal {N}}$ goes to infinity, one recovers equation (3.12). Letting $a$ go to infinity, one gets the intermediate asymptotics associated with a drop dominated dissipation: $\zeta '(H)={\mathcal {N}}/s\sqrt {aH}$. Identifying with the result obtained by Davis et al. (Reference Davis, Schonberg and Rallison1989) in this limit, we find $s=1.143\cdots \simeq 8/7$.

3.3. Capillary-limited drop deformation

The deformability of the drop is controlled by the liquid–vapour surface tension $\gamma$. The thermal waves have a negligible amplitude $\sim \sqrt {k_BT/\gamma }\simeq 0.2$ nm so that the relevant deformations can be predicted using aerodynamics. The drop $j$ flattens over an extension $\delta = \sqrt {2 \tilde ah}$ which modifies the curvature $a^{-1}$ into $\tilde a^{-1}$ and the gap between drops $H$ into $\tilde H$. We denote by $u_j(r)$ the disturbance to the interfacial profile of drop $j$. The volume $\sim \delta ^2 u_j$ is assumed small enough not to change the outer radius $R_j$ through the conservation of volume: the inside pressure for drop $j$ remains $2\gamma /R_j$. The gap $h(r)$ is therefore given by (figure 4d)

(3.17)\begin{equation} h(r) \simeq H+u_1(r)+u_2(r) \simeq \tilde H+ \frac{r^2}{2\tilde a}, \quad\mathrm{with} \ \tilde H=H+u_1(0)+u_2(0). \end{equation}

The pressure inside the lubrication film reads $3\eta _g \tilde a \zeta ''(h)\dot H$. Inside the drop $j$, the pressure gradient balances the inertial term associated with the acceleration of the drop. The reference pressure is controlled by the Laplace pressure for the spherical drop, $2\gamma /R_j$. The Laplace equation evaluated at $r=0$ gives the modified curvature:

(3.18)\begin{equation} \frac{1}{\tilde a}=\frac{1}{a}-3 \tilde a \zeta''(\tilde H) \frac{\eta_g \dot H}{\gamma}. \end{equation}

It involves the capillary number ${Ca}=\eta _g \dot H/\gamma$. Integrating once the Laplace equation, one obtains

(3.19)\begin{equation} r \left(1-\frac{r^2}{R_j^2}\right)^{1/2} \gamma \frac{{\rm d}u_j}{{\rm d}r} \simeq 3 \tilde a^2 \eta_g \dot H \left(\zeta'(\tilde H)\left(1-\frac{r^2}{R_j^2}\right)^{3/2}-\zeta'(h)\right). \end{equation}

To obtain $u_j(0)$, one needs to integrate once more this equation, assuming that $u_j$ vanishes far from the contact zone. The term $\zeta '(h)$ leads to a logarithmic term of the form $\zeta '(H)\ln (r)$, which balances the divergence of the first term. In the outer asymptotic $\zeta '(h)\sim 1/h$, the integration can be performed explicitly, leading to $\zeta '(\tilde H)\ln (r/\sqrt {2\tilde ah})$. Using this approximation, one obtains

(3.20)\begin{equation} \tilde H\simeq H-3 \left(\ln\left(\frac{R_1R_2}{2 \tilde a \tilde H}\right)-1\right) \tilde a^2 \frac{\eta_g}{\gamma} \zeta'(\tilde H) \dot H . \end{equation}

Equation (3.20) is an implicit equation for $\tilde H$. It must be solved alongside (2.11) in which $H$ is replaced by $\tilde H$, yielding at the same time a modified drop separation and a small drop deformation given by (3.17).

3.4. Influence of the different forces

The equations of motion governing the relative position of the two drops are integrated numerically using a Runge–Kutta scheme of order $4$, with an adaptive time step. The equations are made dimensionless using $a=1$, $\eta _g=1$ and $\rho _\ell =1$. The drops are initially at their terminal velocity, at a distance large enough to obtain results insensitive to this initial condition. In the overdamped limit, we explicitly set ${\rm d} \boldsymbol V_i/{\rm d}t = \boldsymbol 0$ in the governing equation (2.11) and integrate the resulting first-order coupled differential equations using also a Runge–Kutta scheme of order $4$. Over the range of parameters where the inertial and overdamped equations can both be integrated accurately, their results are identical when the drops are small enough.

To investigate the effect of the different forces, we first consider head-on collisions. Figure 5(a) shows trajectories in the phase space $(H, \dot H)$. The solid green line shows the reference case, where all aerodynamic forces are neglected; the relative velocity then remains equal to $U^t_1-U^t_2$. The dotted blue line shows the result of a calculation ignoring the regularisation of the lubrication force by the mean free path. Conditions are chosen to highlight the existence of a size $a$ for which the two drops collide ($H=0$) at vanishing velocity ($\dot H=0$). When the lubrication force is removed altogether (dashed orange line), one can observe the effect of long-range aerodynamic interaction at large distances, which tends to lower the impact velocity. Lubrication forces are dominant at short separations as compared with $a$ and greatly lower the impact velocity in the absence of Knudsen effects (dot-dashed blue line). When introducing a finite mean free path $\bar \ell$ (solid black line), the lubrication force is more efficiently regularised at $H<\bar \ell$, leading to a larger impact velocity. Adding capillary deformations of the drops (dotted red line) only has a very small effect on the results in this regime.

Figure 5. Effect of the different forces on head-on collisions. (a) Example of trajectory in the phase space $\delta V=-\dot H$ vs $H$, illustrating the role of the different forces. The size ratio is $\varGamma ={R_1}/{R_2}=5$. We chose $a\simeq 89.5\bar \ell$ in the critical condition for the integration performed with the regularisation by the mean free path $\bar \ell$. (b) Normal impact velocity $\delta V=-\dot H$ as a function of the rescaled size $\mathcal {G}$ for $\varGamma = 5$.

Figure 5(b) shows the (normal) velocity at the time of impact as a function of the inertial parameter $\mathcal {G}$. The impact velocity is always non-zero except when the flow inside both drops is taken into account, but not the finite mean free path $\bar \ell$ (dotted blue line). In this case ($\bar \ell =0$), the impact velocity vanishes at a critical value of $\mathcal {G}$ below which there is no collision. This critical point is further discussed further in the following. When $\bar \ell$ is taken into account, one observes two regimes: at $\mathcal {G}>1$, the effect of $\bar \ell$ is small and the curve (black line) remains close to the curve presenting a critical point ($\bar \ell =0$). The effect of capillarity is large for big drops and tends to reduce the impact velocity. It becomes negligible at small $\mathcal {G}$, both because the impact velocity is small and because the drops are less deformable.

3.5. Collision efficiency

When the drops are far from each others, they follow a linear trajectory. We define the impact parameter $\delta$ as the horizontal distance between these vertical lines. Figure 6(b) shows the normal and tangential velocity as a function of the gap $H/a$ for particular values of $\mathcal {A}$ and $\varGamma$, at the critical impact parameter $\delta _c$. As $\delta$ goes to $0$, one recovers a head-on collision, for which the drops travel in straight lines with a non-zero normal velocity at impact. As $\delta$ increases, the normal velocity decreases and crosses $0$ at this critical impact parameter $\delta _c$. Figure 6(c) shows a critical trajectory, defined by $\delta =\delta _c$, in the frame of reference of the large drop. By definition, the trajectory is tangent at the collision point ($\dot H=0$ when $H=0$). Comparing this critical trajectory to the ballistic trajectory (figure 6c) allows one to define the collision efficiency as

(3.21)\begin{equation} E = \frac{\delta_c^2}{(R_1+R_2)^2}. \end{equation}

Spherical hard particles which do not interact in their trajectory have an efficiency $E = 1$, by definition. In practice, the limit trajectory is found numerically by bracketing over $\delta _c$ until $\dot H$ vanishes. The initial distance between the drops is chosen large enough to ensure that the results become independent of the choice made; in practice, the required initial distance is around $10^2 R_1$.

Figure 6. (a) Normal impact velocity $\delta V=-\dot H$ as a function of the impact parameter $\delta$, for $\varGamma =5$ and $\mathcal {A}=75$. (b) Evolution of normal (black solid line) and tangential (red dotted line) velocities as a function of the gap $H$, for the critical value of $\delta$. Here $\varGamma =5$ and $\mathcal {A}=75$. (c) The collision efficiency $E$ is the ration between the geometric collision cross-section and the aerodynamic cross-section. Blue: trajectory for the critical impact parameter $\delta _c = \sqrt {E}(R_1+R_2)$, below which the drops always collide and above which they never do.

Figure 7 shows the collision efficiency $E$ as a function of the rescaled drop size $\mathcal {A}$, for $\varGamma = 5$. In all cases, the ballistic limit $E = 1$ is recovered for purely inertial drops ($\mathcal {G} \gg 1$). Ignoring Knudsen effects but taking into account lubrication regularised by flow inside the drops ($\bar \ell = 0$; dash-dotted blue curve), $E$ vanishes below a critical value of $\mathcal {G}$. At this rescaled scale $a$, the critical impact parameter $\delta _c$ vanishes, which corresponds to the critical head-on collision shown in figure 5. The curve provides a good approximation of the full model (solid black line and dotted red line) above $E\simeq 0.5$ and therefore captures the cross-over value of $\mathcal {G}$ below which the efficiency $E$ drops. The efficiency obtained without any lubrication force is plotted in dotted green line for Stokes long-range flow and in dashed orange line for Oseen long-range flow (figure 7a). Both approximations overestimate the efficiency at $\mathcal {G}<1$ and present a cross-over towards $E=1$ at large $\mathcal {G}$. Oseen flow reduces to Stokes flow above the small drop $2$ (downstream of it). On the opposite, below the large drop, labelled $1$, the flow velocity decays faster (as $r^{-2}$) for the Oseen approximation than for the Stokes approximation (as $r^{-1}$). As a consequence, the large drop $1$ repels less the small drop $2$ using the Oseen approximation so that the efficiency is higher. At small rescaled size $\mathcal {G}$, inertia becomes negligible, as confirmed by the overdamped curve (dash-dotted red curve). As the efficiency decreases with $\mathcal {G}$, the efficiency presents a minimum in the cross-over towards the inertial regime. This suggests decomposing $E$ as the sum of two contributions, as shown in figure 7(b): an overdamped part, which can be accurately computed at vanishing inertia; and an inertial part, defined as the difference to the full calculation. Capillarity deformation of the drops turns out to be negligible in the whole range of parameters. This is due to the fact that, by construction, the efficiency curves are determined from very particular trajectories which have a vanishing normal velocity when colliding.

Figure 7. (a) Collision efficiency for $\varGamma = 5$ as a function of the dimensionless effective radius $a/\bar \ell$ taking into account different mechanisms. Red dotted line: full solution: inertial dynamics, long-range Oseen interaction, lubrication with drop shear flow, Knudsen effects and capillarity. Dot-dashed: overdamped dynamics. Orange dashed: long-range Stokes interaction, no lubrication. Green dashed: long-range Oseen interaction, no lubrication. Blue dot-dashed: full solution at vanishing mean free path. (b) Decomposition of the collision efficiency (black curve) as the sum of the efficiency computed in the overdamped limit (red dot-dashed curve) plus a remainder, reflecting the inertial limit (blue curve).

In order to understand the origin and the value of the minimum collisional efficiency, we make use of the decomposition of $E$ into the sum of an overdamped (figure 8a) and an inertial (figure 8b) contribution to the efficiency. In the limit $\varGamma \gg 1$, the smallest drop follows the streamlines around the large drop moving at its terminal velocity. The stream function $\psi (r, \theta ) \simeq - U_1^t r^2 \sin ^2(\theta )(1/2-3R_1/4r+R_1^3/4r^3)$ is therefore approximately constant all along the trajectory, if it is small enough to be in the overdamped regime. Initially the drops are at large distance $r$ such that $r \sin (\theta )\sim \delta$ so that $\psi = - U_1^t \delta ^2/2$. The collision happens at angle ${\rm \pi} /2$ at distance $r = R_1+R_2$. Equating these two values of $\psi$ gives $E=\frac {3}{2} \varGamma ^{-2}$ at asymptotically large $\varGamma$. In figure 8(a), the product $\varGamma ^2 E$ is therefore plotted as a function of $\mathcal {A}$ for different $\varGamma$. At large $\varGamma$, the curves collapse on a single master curve which tends to $1$ (and not $3/2$) in the limit of vanishing $\mathcal {A}$. The lubrication force acts only very close to contact so that the smaller drop eventually leaves its initial streamline. This leads to the smaller prefactor observed, with a scaling still controlled by the long-range forces.

Figure 8. Decomposition of the collision efficiency $E$. (a) Overdamped contribution: $\varGamma ^2 E$ as a function of $\mathcal {A}$, for six values of $\varGamma$: $1.1$, $4$, $10$, $100$, $500$ and $1500$. At asymptotically large $\varGamma$, the curves tend to a master curve. (b) Inertial contribution, defined as the difference between the efficiency and its overdamped contribution.

The inertial contribution to the efficiency $E$ is shown in figure 8(b). For asymptotically large $\varGamma$, the drop labelled $1$ is much bigger than the second drop. A change of efficiency is expected at a change of aerodynamical regime of the large drop. As $R_1=a (1+\varGamma )$, one expects that the efficiency is asymptotically controlled by the combination of parameters $\mathcal {G} (1+\varGamma )$. Although far to be a perfect collapse, one observes in figure 8(b) much smaller variations of the inertial contribution to the efficiency, when plotted vs $\mathcal {G} (1+\varGamma )$. This inertial contribution presents similarities with the efficiency obtained in the limit $\bar \ell \to 0$, which vanishes below a threshold. It is therefore interesting to investigate the origin of this threshold. Figure 9(a) presents initial (dotted lines) and collisional velocities (solid lines) for head-on collisions for different $\varGamma$. The impact velocity, above the threshold, remains close to the velocity difference $U_1^t-U_2^t$. The lubrication term controls the dynamics of drops immediately before collision. Let us consider the head-on collision of two drops of mass $m_1$ and $m_2$ and let us neglect gravity and the long-range aerodynamic force. There is no inertia in the gas so that the force on the drops obeys the reciprocal action principle. As a consequence, the two-body problem reduces to a single-body problem with an effective mass $m_1m_2/(m_1+m_2)=\frac {4}{3}{\rm \pi} \rho _\ell \mu a^3$ (with $\mu = (R_1+R_2)^3/(R_1^3+R_2^3)$) with the full force. The dynamical equation reads

(3.22)\begin{equation} \tfrac{4}{3}{\rm \pi} \rho_\ell \mu a^3 \ddot H={-}6 {\rm \pi}\eta_g a^2 \zeta'(H) \dot H. \end{equation}

This equation integrates into

(3.23)\begin{equation} \frac{\rho_\ell a \Delta \dot H}{\eta_g}={-} \frac{9}{2\mu } \Delta \zeta(H), \end{equation}

where $\Delta$ denotes the variation between the initial and final states considered, and $\zeta (H)$ is the antiderivative of $\zeta '(H)$.

Figure 9. Head-on collisions and non-continuum effects. (a) Normal impact velocity $\delta V=-\dot H$ without non-continuum effects as a function of the rescaled drop size $\mathcal {G}$. The radius ratio is $\varGamma ={R_1}/{R_2}=5$. Solid lines: impact velocity in the limit $\bar \ell \to 0$, i.e. without non-continuum effects. Dashed lines: initial velocity. Dotted lines: difference between the impact velocity in the presence of long-range forces only, without any lubrication, and the analytical criterion (3.24), with $\varLambda = 4.7$. (b) Critical value of $\mathcal {G}$ below which the impact velocity vanishes before collision, for different long-range forces. Solid black line: Stokes long-range force (with $\bar \ell \to 0$). Solid red line: Oseen long-range force (with $\bar \ell \to 0$). Dotted green line: criterion of (3.24) with $\varLambda = 17.4$. Solid orange line: isocurve of efficiency $E = 0.2$, including non-continuum effects. Solid purple line: efficiency minimum $\mathcal {G}_{min}$, including non-continuum effects.

As a consequence, the lubrication film prevents the collision of drops with an insufficient initial velocity difference. The precise criterion in the absence of long-range forces is a threshold Stokes number

(3.24)\begin{equation} {St}_\varLambda\equiv \frac{\rho_\ell a (U^t_1-U^t_2)}{\eta_g}=\frac{9}{2\mu}\varLambda, \end{equation}

where $\varLambda = \int \zeta '(H)\,\mathrm {d} H$ is a logarithmic factor originating from the fact that, at large $H$, $\zeta '(H) \sim H^{-1}$. Here $\varLambda$ depends on the dynamical mechanism regularising the lubrication pressure. Figure 9(a) shows in dotted lines the impact velocity obtained by subtracting equation (3.24) from the impact velocity computed only with long-range interactions, in the absence of lubrication forces. The good agreement with the full calculation, with long-range forces, using a constant $\varLambda = 4.7$ shows that there is scale separation between the near-contact lubrication and the long-range interactions.

In the viscous drag regime in which (2.13) reduces to $U_i^t = 2(\rho _\ell - \rho _g)gR_i^2/9\eta _g$, the threshold given by (3.24) can be expressed as $\mathcal {G}= \mathcal {G}_\varLambda$, with

(3.25)\begin{equation} \mathcal{G}_\varLambda=\left(\frac{81}{4} \varLambda \frac{\varGamma^2 (1+(\varGamma-1) \varGamma)}{(\varGamma-1) (1+\varGamma)^5}\right)^{1/3}. \end{equation}

Surprisingly, this scaling is close to the scaling obtained with Oseen long-range forces [solid red line in figure 9b]. Stokes long-range forces (solid black line) displays completely different behaviour as the $r^{-1}$ interaction makes the impact velocity much smaller than the terminal velocity difference over the whole range of $\mathcal {G}$.

This sheds light on the behaviour of the efficiency near its fast inertial decrease and minimum. The values of $(\varGamma, \mathcal {G})$ for which $E = 0.2$ are shown as the orange line in figure 9(b). For that iso-efficiency curve, $\mathcal {G}$ follows the scaling $\mathcal {G} \propto \varGamma ^{-1}$ at large $\varGamma$. The value of the minimum of $E$, $\mathcal {G}_{min}$, decreases faster as $\varGamma ^{-2}$. Figure 10(a) provides a summary of the relationship between collisional efficiency and the size of both droplets.

Figure 10. Isocontours of the collision efficiency $E$ as a function of $\varGamma$ and $\mathcal {A}$. Dashed line: minimum value of the collision efficiency for at a given $\varGamma$. All aerodynamical effects are taken into account. Parameters are those of water droplets. For large sizes and aspect ratios, geometric behaviour is recovered. Below a certain Stokes number $\mathcal {G}$, the efficiency decreases very rapidly and reaches a minimum. (a) Without van der Waals interactions. (b) With van der Waals interactions. The efficiency rapidly decreases from the geometric value when inertia is low enough, in the same way as in the purely aerodynamic case. Van der Waals interactions increase the minimum efficiency by a factor $10$. Collisions in the overdamped regime are also more efficient, leading to a narrower valley of collision slowdown.

4.1. Electrostatic interactions: van der Waals and Coulombian forces

The average droplet charge for weakly electrified clouds is $\sim e$ (Harrison, Nicoll & Ambaum Reference Harrison, Nicoll and Ambaum2015). When the drops are far apart, they interact as two point charges $q_1$ and $q_2$ by the electrostatic potential energy $q_1 q_2/4{\rm \pi} \varepsilon r$, where $\varepsilon$ is the air permittivity. The electric force can either be repulsive or attractive depending on the relative charges. In the case of clouds, one rather expects all drops to present the same sign, hence leading to a repulsion. The electrostatic force $F_e$ becomes comparable with the lubrication force $\propto \eta _g a^2 U^t/h$ driven by $U^t = 2/9 \rho _\ell g a^2/\eta _g$ at separations $h_e \sim q^2/(4{\rm \pi} \varepsilon \rho _\ell g a^4)$. $h_e$ is comparable to the mean free path $\bar \ell$ for $a = 0.5$ $\mathrm {\mu }$m and decreases very rapidly with $a$. In disturbed weather, storm clouds develop large electric fields, the bulk of the cloud becomes negatively charged and drops carry much larger charges, up to $10^5\;e$ (Takahashi Reference Takahashi1973). Here, we restrict our analysis to electroneutral drops, which is a good approximation in the bulk of warm clouds (Pruppacher & Klett Reference Pruppacher and Klett2010, chap. 18).

Van der Waals forces between drops can be computed using the unretarded van der Waals pair potential. At very small separation, the disjoining pressure $\varPi (h)$ is given by the phenomenological expression involving the Israelashvili length $\varsigma$:

(4.1)\begin{equation} \varPi(H)={-}\frac{4 \gamma \varsigma^2}{(\varsigma+H)^3}. \end{equation}

This formula obeys the integral relation giving the surface tension: $\int _0^\infty \varPi (h) \,{\rm d}h=-2\gamma$. Moreover, at intermediate $H$ large with respect to the molecular scale, but small with respect to $a$, one recovers the decay as $\varPi (H) \sim -4 \gamma \varsigma ^2/H^3$. The force is given by integrating the disjoining pressure over the surface:

(4.2)\begin{equation} f^{vdW}=\int_{0}^{\infty} 2{\rm \pi} r \varPi(h) \,{\rm d}r={-}4{\rm \pi} \gamma a \frac{\varsigma^2}{(\varsigma+H)^2}. \end{equation}

This expression holds at gap $H$ comparable to $\varsigma$. Hamaker (Reference Hamaker1937) showed that at all separations,

(4.3)\begin{equation} \boldsymbol f_{ji}^{vdW} ={-}A \frac{32 R_1^3 R_2^3 (R_1+R_2+H)}{3(2 R_1+H)^2 (2 R_2+H)^2 (2 (R_1+R_2)+H)^2 (H+\varsigma)^2} \boldsymbol e_{ij}, \end{equation}

where $A=3.7\times 10^{-20}$ J is the Hamaker constant. It matches with (4.3) in the small $H$ limit, since $\varsigma =\sqrt {A/(24{\rm \pi} \gamma )}$. Note that the molecular cut-off length $\varsigma$ is equal to $0$ in the original work of Hamaker (Reference Hamaker1937). At large distances, the van der Waals force asymptotically tends to

(4.4)\begin{equation} f^{vdW} ={-}A\frac{32 R_1^3 R_2^3 }{3H^7}. \end{equation}

4.2. Vertical static electric field

Fair weather clouds, which do not develop their own strong electric fields, are electrified at their boundaries, with a positive charge at cloud top and a negative charge at cloud base (Harrison et al. Reference Harrison, Nicoll and Ambaum2015). This can be modelled as a vertical electric field $\boldsymbol E_0$, as shown in figure 12(a). The drops act as conducting spheres and behave as induced dipoles of dipolar moment $\boldsymbol p_i=4 {\rm \pi}\varepsilon R_i^3 \boldsymbol E_0$. When the drops are far from each others, the electric potential around the drop $j$ obeys the Poisson equation and takes the form

(4.5)\begin{equation} V(r, \theta)=E_0 r \cos \theta\left(\frac{R_j^3}{r^3}-1\right). \end{equation}

The force exerted on the other drop, labelled $i$, derives from the energy $\boldsymbol p_i \boldsymbol {\cdot } \boldsymbol \nabla V$:

(4.6)\begin{gather} F_r={-}12 {\rm \pi}\varepsilon E_0^2 (R_iR_j)^3\frac{2 \cos^2 \theta-\sin^2 \theta}{(R_1+R_2+H)^4}, \end{gather}

(4.7)\begin{gather}F_\theta={-}12 {\rm \pi}\varepsilon E_0^2 (R_iR_j)^3\frac{\sin (2 \theta)}{(R_1+R_2+H)^4}. \end{gather}

At small separations, however, the force must be corrected for the geometrical amplification of the field in the gap separating the two drops. This effect presents similarities with the lubrication discussed previously. For practical purpose, the exact solution derived by Davis (Reference Davis1964) can be approximated by the following phenomenological formula:

(4.8)\begin{gather} F^e_r={-}12 {\rm \pi}\varepsilon E_0^2 (R_iR_j)^3\left( \frac{2 \cos^2 \theta}{(c_c+H)^{3+\alpha}H^{1-\alpha}}-\frac{\sin^2 \theta}{(c_s^2+H^2)(c_s+H)^2)}\right), \end{gather}

(4.9)\begin{gather}F^e_\theta={-}12 {\rm \pi}\varepsilon E_0^2 (R_iR_j)^3\frac{\sin (2 \theta)}{(c_\theta+H)^{4-\alpha}(c_\theta^\alpha+H^\alpha)}, \end{gather}

with $\alpha \simeq 0.2$, $c_c\simeq 1.36 (R_1+R_2)$, $c_s\simeq 1.55(R_1+R_2-a)$ and $c_\theta \simeq 0.77(R_1+R_2)$ and the notation of figure 12(a). It provides a good fit to the exact force down to $H\simeq 10^{-6} a$.

5.1. Normalisation of the thermal noise

We now take into account the thermal noise in the equation of motion (2.11). Following Batchelor (Reference Batchelor1976), we consider that the separation of time scales between the time to return to thermal equilibrium and the time for the particle configuration to change is sufficient to consider the position of the particles as constant when computing the correlations of thermal noises. In the inertial regime, diffusion is negligible. Conversely, in the regime where diffusion is important, inertial effects can be neglected. As a consequence, we use the Stokes model of long-range aerodynamic interactions rather than Oseen's model. Then, the equations for the velocity component along the direction of the axis joining the two particles and those for the two perpendicular components decouple. For simplicity, we single out one such velocity component and introduce the equation governing the velocity fluctuation $u_i$ along the chosen axis. As thermal diffusion is an Ornstein–Uhlenbeck process, it is convenient to write the Langevin equation under the following form (Risken & Frank Reference Risken and Frank1989), with Einstein summation over indices:

(5.1)\begin{equation} \frac{{\rm d} u_i}{{\rm d}t}={-}S_{ij} u_j+ W_i \quad {\rm with}\ \langle W_i(t)W_j(t') \rangle = \mathcal{W}_{ij} \delta(t-t'). \end{equation}

The indices $i$ and $j$ here refer to the particle number, the three components being considered separately. It is convenient to write the relaxation rate matrix $\boldsymbol {S}$ under the form

(5.2)\begin{equation} S_{ij} = \frac{9 \eta_g}{2\rho_\ell R_i^3} \alpha_{ij}. \end{equation}

For the axis joining the centre of the drops, the matrix $\alpha _{ij}$ reads

(5.3a,b)\begin{gather} \alpha_{11}=R_1+ a^2 \zeta'(H),\quad\alpha_{22}= R_2+ a^2 \zeta'(H), \end{gather}

(5.4)\begin{gather}\alpha_{12}=\alpha_{21}={-}\frac{R_1R_2 }{2r}\left(3-\frac{R_1^{2} +R_2^{2} }{r^{2}}\right) - a^2 \zeta'(H). \end{gather}

In the plane perpendicular to the axis joining the centre of the drops, it reads

(5.5a,b)\begin{gather} \alpha_{11}= R_1,\quad \alpha_{22}= R_2, \end{gather}

(5.6)\begin{gather}\alpha_{12}=\alpha_{21}={-}\frac{ R_1R_2 }{4r}\left(3+\frac{R_1^{2} +R_2^{2} }{r^{2}}\right). \end{gather}

The Langevin equation integrates into, summed over indices,

(5.7)\begin{equation} u_i= \int_0^t G_{ij}(t-t') W_j(t') \,{\rm d}t', \end{equation}

where $G_{ij}$ denotes the matrix elements of the Green's function $\boldsymbol {G}$, which formally obeys, in matrix notation, $\boldsymbol {G}=\exp (-\boldsymbol {S} t)$. Similarly, the velocity $u_i$ can be integrated formally to give the position.

The velocity correlation function reads, with Einstein summation,

(5.8)\begin{equation} \langle u_i(t)u_j(t) \rangle=\int_0^t G_{ik}(t')G_{jl}(t') \,{\rm d}t' \mathcal{W}_{kl}. \end{equation}

Using the generalised equipartition of energy,

(5.9)\begin{equation} \langle u_i(t)u_j(t) \rangle_{t\to\infty}= \frac{3 k T}{4 {\rm \pi}\rho_\ell R_k^3} \delta_{ik}\delta_{jk}, \end{equation}

we deduce

(5.10)\begin{equation} \mathcal{W}_{ij}=\frac{3kT}{4 {\rm \pi}\rho_\ell} \left(\frac{S_{ij}}{R_j^3}+\frac{S_{ji}}{R_i^3}\right). \end{equation}

In practice, at each integration step of the Runge–Kutta of order 4 algorithm, noise terms obeying a series of correlation rules described by Ermak & Buckholz (Reference Ermak and Buckholz1980) between positions and velocities are added to the deterministic increments. The fourth-order integration scheme is recovered in the limit where the thermal diffusion is negligible. Conversely, the scheme is designed to lead to the exact diffusion result, when diffusion is dominant, provided the thermal equilibration time scale is smaller than the typical time scale of evolution of the geometrical configuration.

5.2. Redefining the collision efficiency

We have previously defined the collision efficiency as the factor encoding the influence of aerodynamics and electrostatic forces on the collision frequency of particles. The reference frequency was derived in the ballistic limit, in which the two particles of radii $R_i$ settle with a differential speed $U = U^t_1-U^t_2$, and reads: $\nu = {\rm \pi}(R_1+R_2)^2 n_0 U$. Considering now the effect of thermal noise, the effect of diffusion must be included in the reference collision frequency to which the real rate is compared to define $E$. Each particle diffuses with a diffusion constant $D_i = k_B T/(6{\rm \pi} \eta _g R_i)$. The problem is equivalent to the advection-diffusion of particles with a diffusion coefficient $D = D_1 + D_2$. The diffusion-advection equation for the particle concentration $n$ is, in spherical coordinates,

(5.11)\begin{equation} -U \cos\theta \partial_r n + U \frac{\sin\theta}{r}\partial_\theta n = D\left(\frac{1}{r^2}\partial_r (r^2 \partial_r n) + \frac{1}{r^2 \sin\theta}\partial_\theta (\sin\theta \partial_\theta n)\right). \end{equation}

The boundary conditions are $n(r\to \infty ) = n_0$, $n(r = R_1+R_2) = 0$. The combined effects of diffusive and advective transport on droplet growth are described by the particle flux, i.e. the collision rate, at the drop surface

(5.12)\begin{align} \nu_0 &= 2{\rm \pi} D (R_1+R_2)^2 \int_0^{\rm \pi} \mathrm{d} \theta \left(\frac{\partial n}{\partial r}\right)_{r = R_1+R_2} \sin\theta\nonumber\\ &= {\rm \pi}(R_1+R_2)^2 n_0 U\ q({Pe}). \end{align}

In the purely Brownian limit, one gets $\nu = 4{\rm \pi} D (R_1+R_2) n_0$ so that $q({Pe}) \sim 4/{Pe}$. In the purely ballistic limit, $q({Pe}) \sim 1$. Equation (5.11) has been solved analytically by Simons et al. (Reference Simons, Williams and Cassell1986), and $q({Pe})$ can be expressed as

(5.13)\begin{equation} q({Pe}) = \frac{4{\rm \pi}}{{Pe}^2}\sum_{n=0}^\infty ({-}1)^n (2n+1)\frac{I_{n+1/2}\left(\dfrac{Pe}{2}\right)}{K_{n+1/2}\left(\dfrac{Pe}{2}\right)}, \end{equation}

with $I_n$, $K_n$ the modified Bessel functions. Care must be taken when evaluating this series (Sajo Reference Sajo2008). Consequently, we define the collision efficiency in the presence of diffusion and all aerodynamic effects as the ratio between the collision rate $\nu$ and the reference collision rate ${\rm \pi} (R_1+R_2)^2 n_0 U q({Pe})$:

(5.14)\begin{equation} E_d = \frac{\nu}{{\rm \pi} (R_1+R_2)^2 n_0 U q({Pe})}. \end{equation}

In practice, we compute the collision frequency $\nu$ using a Monte Carlo method. We uniformly sample impact parameters over a square upstream and compute the trajectory for each sample point. The reduced collision rate $\nu /({\rm \pi} (R_1+R_2)^2 n_0 U)$ is estimated by measuring the ratio between the surface area of impact parameters leading to a collision to the area in the geometric case ${\rm \pi} (R_1+R_2)^2$.