2.1 Two spheres in continuum flow

The instantaneous velocity of two spheres subjected to external forces $\boldsymbol{F}_{1}$ and $\boldsymbol{F}_{2}$ has been studied by a number of authors (Smoluchowski Reference Smoluchowski1912; Stimson & Jeffery Reference Stimson and Jeffery1926; Burgers Reference Burgers1995; Happel & Brenner Reference Happel and Brenner1965; Batchelor Reference Batchelor1976; Felderhof Reference Felderhof1977). Exact solutions are available for two spheres moving parallel and perpendicular to their line of centres. For arbitrary $\boldsymbol{r}$ , the problem can be solved using the method of reflections, whereby the mobility tensors are determined as expansions in powers of the inverse of interparticle separation distance $r$ . (See Happel & Brenner (Reference Happel and Brenner1965) for more information about the method of reflections, including its application to the two-sphere problem and to the problem of a particle in a tube.) Since the mobility tensors are symmetric about the line of centres connecting the spheres, we can formally write

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D648}_{ij}=A_{ij}\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}+B_{ij}\left(\unicode[STIX]{x1D644}-\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right),\end{eqnarray}$$

where the non-dimensional coefficients $A_{ij}$ and $B_{ij}$ can be written as expansions in powers of $r^{-n}$ ,

(2.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle A_{ij}=\mathop{\sum }_{n=0}^{\infty }a_{ij,n}r^{-n}, & \displaystyle\end{eqnarray}$$

(2.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle B_{ij}=\mathop{\sum }_{n=0}^{\infty }b_{ij,n}r^{-n}. & \displaystyle\end{eqnarray}$$

$a_{ij,n}$

$b_{ij,n}$

One can include successive terms in the expansions for $A_{ij}$ and $B_{ij}$ to obtain increasingly accurate results for the mobility tensors. The simplest approach is the point force approximation, where $\unicode[STIX]{x1D648}_{ii}=\unicode[STIX]{x1D701}_{i,0}^{-1}\unicode[STIX]{x1D644}$ , $\unicode[STIX]{x1D648}_{ij}$ is given by the Oseen tensor (the $r^{-1}$ term in (2.5) below), $\unicode[STIX]{x1D644}$ is the identity tensor and $\unicode[STIX]{x1D701}_{i,0}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a_{i}$ is the monomer friction coefficient for a sphere with radius $a_{i}$ in a fluid with viscosity $\unicode[STIX]{x1D707}$ . The next level of approximation yields $\unicode[STIX]{x1D648}_{ii}=\unicode[STIX]{x1D701}_{i,0}^{-1}\unicode[STIX]{x1D644}$ and

(2.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D648}_{ij} & = & \displaystyle \frac{1}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}r}\left[\left(\unicode[STIX]{x1D644}+\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)+\left(\frac{a_{i}^{2}+a_{j}^{2}}{3r^{2}}\right)\left(\unicode[STIX]{x1D644}-\frac{3\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)\right]\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}r}\left[\left(1-\frac{a_{i}^{2}+a_{j}^{2}}{3r^{2}}\right)\frac{2\boldsymbol{r}\boldsymbol{r}}{r^{2}}+\left(1+\frac{a_{i}^{2}+a_{j}^{2}}{3r^{2}}\right)\left(\unicode[STIX]{x1D644}-\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)\right]\end{eqnarray}$$

for $i\neq j$ . For $a_{i}=a_{j}$ , equation (2.5) reduces to the Rotne–Prager–Yamakawa tensor (Rotne & Prager Reference Rotne and Prager1969; Yamakawa Reference Yamakawa1970). Batchelor (Reference Batchelor1976) provides expressions for the mobility tensors accurate to order $r^{-5}$ , while Felderhof (Reference Felderhof1977) uses the method of reflections to derive expressions valid to order $r^{-7}$ for mixed slip-stick boundary conditions. Results from Felderhof’s interaction tensor are equal to available exact solutions to the two-sphere problem except for very small separation distances (i.e. nearly touching spheres), where higher-order terms become important (Felderhof Reference Felderhof1977).

Note that if sphere 2 is much smaller than sphere 1 (i.e.  $a_{2}\ll a_{1}$ ) and if $r$ is large enough that terms higher than order $r^{-3}$ are negligible, sphere 2 will simply be advected by the Stokes velocity field generated by sphere 1. This becomes clear solving equation (2.1) for $\boldsymbol{U}_{2}$ with $\boldsymbol{F}_{2}=\mathbf{0}$ , $\boldsymbol{F}_{1}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a_{1}\boldsymbol{U}_{1}$ and using equation (2.5) for $\unicode[STIX]{x1D648}_{21}$ with $a_{2}\ll a_{1}$ :

(2.6) $$\begin{eqnarray}\boldsymbol{U}_{2}=\left[\frac{3a_{1}}{4r}\left(\unicode[STIX]{x1D644}+\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)+\frac{a_{1}^{3}}{4r^{3}}\left(\unicode[STIX]{x1D644}-\frac{3\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)\right]\boldsymbol{\cdot }\boldsymbol{U}_{1}\equiv \unicode[STIX]{x1D651}(\boldsymbol{r})\boldsymbol{\cdot }\boldsymbol{U}_{1}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D651}(\boldsymbol{r})\boldsymbol{\cdot }\boldsymbol{U}_{1}$ is the Stokes velocity field around a sphere moving with velocity $\boldsymbol{U}_{1}$ . We will refer to $\unicode[STIX]{x1D651}(\boldsymbol{r})$ as the Stokes tensor.

Any of the above expressions for the mobility tensors can be used to solve the two-sphere problem for arbitrary sphere sizes $a_{1}$ and $a_{2}$ (provided the continuum approximation is still valid), centre-to-centre distance and orientation $\boldsymbol{r}$ and external forces $\boldsymbol{F}_{1}$ and $\boldsymbol{F}_{2}$ . Table 1–3 show results for the speed of each sphere as a function of separation distance for equal-sized spheres and equal force magnitude. (For $a_{1}=a_{2}$ , the speeds of the spheres are equal.) The tables include results for the exact solutions to the two-sphere problem (Stimson & Jeffery Reference Stimson and Jeffery1926; Brenner Reference Brenner1961; Goldman et al. Reference Goldman, Cox and Brenner1966), the mobility expression of Felderhof for stick boundary conditions (Felderhof Reference Felderhof1977), and our EKR method (i.e. using the Stokes tensor) for the three cases shown in figure 1. In all cases, the Stokes tensor results are in very good agreement (less than 1 % difference) with the exact results for $r/a>5$ . As expected, the error is more significant for smaller separation distances because the Stokes tensor does not include terms of order $r^{-4}$ and higher and because there is a missing factor of 2 in the $r^{-3}$  term.

Table 1.

Speed of two spheres moving parallel to their line of centres, relative to the speed of isolated spheres subjected to the same external force. The EKR or Stokes tensor solution refers to $\unicode[STIX]{x1D648}_{ii}=\unicode[STIX]{x1D701}_{i,0}^{-1}$ and $\unicode[STIX]{x1D648}_{ij}=\unicode[STIX]{x1D701}_{i,0}^{-1}\unicode[STIX]{x1D651}(\boldsymbol{r})$ , where $\unicode[STIX]{x1D651}(\boldsymbol{r})$ is defined in (2.6). The Felderhof tensors (Felderhof Reference Felderhof1977) are accurate to order $r^{-7}$ . The exact solution is given by Stimson & Jeffery (Reference Stimson and Jeffery1926).

Table 2.

Speed of two spheres moving anti-parallel to their line of centres, relative to the speed of isolated spheres subjected to the same external force. The exact solution is given by Brenner (Reference Brenner1961).

Table 3.

Speed of two spheres moving perpendicular to their line of centres, relative to the speed of isolated spheres subjected to the same external force. The exact solution is given by Goldman et al. (Reference Goldman, Cox and Brenner1966).

2.2 Aggregates in continuum flow

Experiments and simulations have shown that particles formed by diffusion-limited cluster aggregation have a fractal morphology with fractal dimension $d_{f}\approx 1.78$ and prefactor $k_{0}\approx 1.3$ (Sorensen Reference Sorensen2011). These parameters relate the radius of gyration to the number of spheres in the aggregate,

(2.7) $$\begin{eqnarray}N=k_{0}\left(\frac{R_{g}}{a}\right)^{d_{f}}.\end{eqnarray}$$

Notable examples of DLCA aggregates include soot, as well as titania and other ceramic powders synthesized in aerosol flame reactors (Pratsinis Reference Pratsinis1998).

In the previous section, we reviewed how one can solve the two-sphere problem to various levels of approximation. We now describe how to use a similar approach for aggregates.

In approaches based on Kirkwood–Riseman theory (Kirkwood & Riseman Reference Kirkwood and Riseman1948), one uses any of the above expressions for the interactions between pairs of spheres to determine the force on a macromolecule or particle consisting of $N$ spherical subunits. The resulting system of equations is analogous to (2.1):

(2.8) $$\begin{eqnarray}-\boldsymbol{U}_{i}=\mathop{\sum }_{j=1}\unicode[STIX]{x1D648}_{ij}\boldsymbol{\cdot }\boldsymbol{F}_{j}.\end{eqnarray}$$

Here, $\boldsymbol{F}_{j}$ is the force exerted by the fluid on the $j$ th primary sphere in an aggregate, which is unknown, whereas in (2.1) $\boldsymbol{F}_{j}$ represents the known external force on the $j$ th particle in a two-particle system. (The negative sign appearing in (2.8) is a consequence of the shift from the force exerted by the particle – as in (2.1) – to the force exerted on the $i$ th primary sphere by the fluid.) To determine the force on a macromolecule or particle whose subunits have velocities $\boldsymbol{U}_{i}$ , one solves the above linear system for the force on each sphere, then sums over all spheres to get the total force. Alternatively, if the total external force on the particle is given (e.g. gravity or the force on charged particle in an electric field), one can solve (2.8) iteratively to get the particle velocity such that the calculated force distribution is consistent with the total force.

KR-based treatments rely on the underlying linearity of the Stokes flow equations. In their original formulation, Kirkwood & Riseman (Reference Kirkwood and Riseman1948) set $\unicode[STIX]{x1D648}_{ij}=\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D701}_{i,0}^{-1}+(1-\unicode[STIX]{x1D6FF}_{ij})\unicode[STIX]{x1D64F}_{ij}$ , where $\unicode[STIX]{x1D6FF}_{ij}$ is the Kronecker delta and $\unicode[STIX]{x1D64F}_{ij}$ is the Oseen tensor. Later applications of KR theory (García de la Torre & Rodes Reference García de la Torre and Rodes1983; Chen, Deutch & Meakin Reference Chen, Deutch and Meakin1984; Lattuada, Wu & Morbidelli Reference Lattuada, Wu and Morbidelli2003) replace the Oseen tensor with the Rotne–Prager–Yamakawa tensor. One can use higher-order approximations for the mobility tensors (e.g. the Felderhof expressions); one can also account for multi-body effects using mobility tensors derived by Mazur & Van Saarloos (Reference Mazur and Van Saarloos1982). However, Carrasco & García de la Torre (Reference Carrasco and García de la Torre1999) have shown that using these more rigorous methods to calculate particle friction and diffusion tensors offers little improvement over using KR-theory with the Rotne–Prager–Yamakawa tensor.

One can use the same KR-based methods to determine the forces between two aggregates: one again solves the system of equations given by (2.8), but now one sums over the spheres in each aggregate to determine the total hydrodynamic force. Again, one can perform this calculation for aggregates with specified velocities or specified external forces. By comparing this result to the force on an isolated aggregate, one can determine the effect of the second aggregate on the mobility of the first.

2.3 Extended Kirkwood–Riseman theory

The previous subsections have focused on situations where the continuum approach is valid. For particles consisting of spheres with diameters of the order of tens of nanometres moving through air at standard temperature and pressure ( $\unicode[STIX]{x1D706}\approx 67$ nm), one cannot use the expressions for interactions between particles in the continuum, but one can use the same general (2.8) relating the velocity of particles in the transition regime to the drag forces exerted on the particles by the fluid. One only needs to determine the mobility tensors as a function of the primary sphere Knudsen numbers.

Noting that the Stokes velocity tensor provides a reasonable approximation for the mobility tensor between two spheres in the continuum (see tables 1–3), we (Corson et al. Reference Corson, Mulholland and Zachariah2017c ) proposed replacing the Stokes velocity tensor with a Knudsen-number-dependent velocity tensor obtained by solving the Bhatnagar–Gross–Krook (BGK) equation (Bhatnagar, Gross & Krook Reference Bhatnagar, Gross and Krook1954) for flow around a sphere. (The BGK equation is a simplified, linearized version of the Boltzmann transport equation that is valid for near-equilibrium situations (Kogan Reference Kogan1958). Velocities calculated using the BGK equation compare well to solutions of the linearized Boltzmann equation (Loyalka Reference Loyalka1992; Takata, Sone & Aoki Reference Takata, Sone and Aoki1993).) The resulting set of equations relating the sphere velocities to the drag is

(2.9) $$\begin{eqnarray}\boldsymbol{F}_{i}=-\unicode[STIX]{x1D701}_{i,0}(Kn_{i})\boldsymbol{U}_{i}-\unicode[STIX]{x1D701}_{i,0}(Kn_{i})\mathop{\sum }_{i\neq j}^{N}\unicode[STIX]{x1D701}_{j,0}^{-1}(Kn_{j})\unicode[STIX]{x1D651}_{ij}(Kn_{j})\boldsymbol{\cdot }\boldsymbol{F}_{j}.\end{eqnarray}$$

Here, $N$ is the number of spheres in the system and $\unicode[STIX]{x1D651}_{ij}$ is the velocity tensor around a sphere with Knudsen number $Kn_{j}$ (i.e. the transition regime analogue of the Stokes tensor defined in (2.6)). The velocity tensor is a function of the vector connecting the centres of spheres $i$ and $j$ (see Corson et al. Reference Corson, Mulholland and Zachariah2017c for futher details). Note that the force on sphere $i$ depends on the friction coefficient for sphere $i$ and the velocities around the other $j$ spheres.

Our previous work (Corson, Mulholland & Zachariah Reference Corson, Mulholland and Zachariah2017a ; Corson et al. Reference Corson, Mulholland and Zachariah2017c ) has demonstrated that using the velocity and drag results from the BGK equation and solving equation (2.9) yields accurate results for the translational friction coefficient of fractal aggregates across the entire Knudsen regime. This suggests that the EKR method can be used to determine the hydrodynamic forces between particles in the kinetic regime.

2.4 Point force approach

If one is only concerned about interactions between widely separated particles, the analysis can be greatly simplified by treating the particles as point forces. In the continuum, the flow field generated by the point force is often called the Stokeslet and can be written as

(2.10) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{r})=\frac{1}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}r}\left(\unicode[STIX]{x1D644}+\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)\boldsymbol{\cdot }\boldsymbol{F},\end{eqnarray}$$

where the term multiplying the point force is the Oseen tensor. If the object is a sphere, $\boldsymbol{F}$ is given by Stokes’ law, and (2.10) gives the flow field for large $r$ . Interestingly, this flow field is valid regardless of the shape of the object exerting the force on the fluid. This means that the flow field far from an object looks like the flow around a sphere exerting the same force on the fluid as the object, even when the object is highly non-spherical. This behaviour is due to viscous effects in the fluid.

Based on the preceding discussion, we can determine the interactions between the aggregates using a point force approach, such that the total hydrodynamic force on each aggregate can be obtained by simultaneously solving the following equations:

(2.11) $$\begin{eqnarray}\boldsymbol{F}_{i}=-\unicode[STIX]{x1D729}_{i,0}\boldsymbol{\cdot }\boldsymbol{U}_{i}-\unicode[STIX]{x1D729}_{i,0}\boldsymbol{\cdot }\mathop{\sum }_{i\neq j}^{N}\left[\frac{1}{8\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}r_{ij}}\left(\unicode[STIX]{x1D644}+\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)\right]\boldsymbol{\cdot }\boldsymbol{F}_{j}.\end{eqnarray}$$

Here, $N$ is the number of aggregates in the system, $\boldsymbol{r}_{ij}$ is the vector between the centre of mass of $i$ th and $j$ th aggregates and $\unicode[STIX]{x1D729}_{i,0}$ is the translational friction tensor for aggregate $i$ when it is alone in an infinite fluid. Mackaplow & Shaqfeh (Reference Mackaplow and Shaqfeh1998) demonstrated that the point force approach applied to the problem of sedimentation of fibres in a semi-dilute homogeneous suspension yields results similar to those of a more sophisticated Monte Carlo simulation of the problem.

The point force approach should also be valid in the transition regime, albeit with a Knudsen-number-dependent hydrodynamic interaction tensor in place of the Oseen tensor. In fact, the transition regime interaction tensor should have the same form as the Oseen tensor but with a different coefficient in place of $1/8$ . This is because the fluid velocity far from a sphere has the general form

(2.12) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{r})=-\frac{c_{1}(Kn)}{2}\frac{a}{r}\left(\unicode[STIX]{x1D644}+\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)\boldsymbol{\cdot }\boldsymbol{U}=-\frac{c_{1}(Kn)}{2\unicode[STIX]{x1D701}_{i,0}(Kn)}\frac{a}{r}\left(\unicode[STIX]{x1D644}+\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)\boldsymbol{\cdot }\boldsymbol{F},\end{eqnarray}$$

where $\boldsymbol{U}$ is the sphere velocity, $\boldsymbol{F}$ is the force exerted by the sphere on the fluid and $c_{1}(Kn)$ is a coefficient that can be obtained by solving the BGK or linearized Boltzmann equation for flow around a sphere (Takata et al. Reference Takata, Sone and Aoki1993; Corson et al. Reference Corson, Mulholland and Zachariah2017c ). In the continuum, $c_{1}=-3/2$ , and (2.12) reduces to (2.10).

Substituting (2.12) for the Oseen tensor in (2.11), we get the following result for a system of $N$ aerosol aggregates:

(2.13) $$\begin{eqnarray}\boldsymbol{F}_{i}=-\unicode[STIX]{x1D729}_{i,0}(Kn_{i})\boldsymbol{\cdot }\boldsymbol{U}_{i}-\unicode[STIX]{x1D729}_{i,0}(Kn_{i})\boldsymbol{\cdot }\mathop{\sum }_{i\neq j}^{N}\left[-\frac{c_{1}(Kn_{j,eff})}{2\unicode[STIX]{x1D701}_{j,0}}\left(\unicode[STIX]{x1D644}+\frac{\boldsymbol{r}\boldsymbol{r}}{r^{2}}\right)\right]\boldsymbol{\cdot }\boldsymbol{F}_{j}.\end{eqnarray}$$

Here, $a_{j,eff}$ is the radius and $Kn_{j,eff}=\unicode[STIX]{x1D706}/a_{j,eff}$ the Knudsen number of a sphere with friction coefficient $\unicode[STIX]{x1D701}_{j,0}=|\unicode[STIX]{x1D729}_{j,0}(Kn_{j})\boldsymbol{\cdot }\boldsymbol{U}_{j}/\boldsymbol{U}_{j}|$ , where $\unicode[STIX]{x1D729}_{j,0}(Kn_{j})$ is the friction tensor for aggregate $j$ alone in an infinite fluid and $\boldsymbol{U}_{j}$ is the velocity of aggregate $j$ . One solves implicitly for the effective radius of the cluster,

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D701}_{j,0}(Kn_{j,eff})=\frac{6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a_{j,eff}}{C_{c}(\unicode[STIX]{x1D706}/a_{j,eff})},\end{eqnarray}$$

where $C_{c}$ is the Cunningham slip correction factor. We have explicitly stated the dependence of the aggregate friction tensors on the Knudsen number of the primary spheres in the aggregate; of course, the friction tensor also depends on the number of spheres in the aggregate and the configuration of those spheres. Note that (2.14) accounts for the orientation of each cluster through the use of the friction tensors, and by determining the effective radius based on the drag force for a particular aggregate orientation relative to the flow. However, one can further simplify (2.13) by replacing the friction tensors with the scalar friction coefficients based on orientation averages for the particles. This simplification is justified for aerosol particles due to the randomizing effects of Brownian forces.

To apply the point force approximation for a system of aggregates in the transition regime, one must first determine the friction tensor for each aggregate (e.g. by solving (2.9)). Next, one must determine the effective sphere radius for each aggregate. One obtains the coefficient $c_{1}(Kn_{j,eff})$ for each aggregate from a table of BGK or linearized Boltzmann results. With this information, one can solve (2.13) for the hydrodynamic forces on the system of aggregates. Again, one can replace the friction tensors with the scalar friction coefficient for each aggregate. One can use the adjusted sphere method (Dahneke Reference Dahneke1973; Zhang et al. Reference Zhang, Thajudeen, Larriba, Schwartzentruber and Hogan2012) or our EKR method (Corson et al. Reference Corson, Mulholland and Zachariah2017c ) to quickly determine the friction coefficient for each aggregate, which significantly reduces the complexity of the problem.

Using the point force approach is beneficial because one can significantly reduce the number of simultaneous equations that one must solve compared to considering the pairwise interactions among all of the spheres in all of the aggregates. This is especially useful if one is considering a system consisting of identical aggregates (such that one need only determine the friction tensor once) or if one is considering interactions between coagulating aerosols (such that the friction tensors for the aggregates remain constant until two particles coagulate, at which point one need only calculate the friction tensor for the new particle). While the point force approach is strictly valid only for widely separated particles, we will show in the following section that this approach provides reasonable results even when the aggregates are fairly close together.

3.1 Sphere results

Figure 2 shows the results for two spheres moving parallel, anti-parallel and perpendicular to their line of centres. Results are plotted as a function of separation distance for three different Knudsen numbers. Here, we show the ratio of the velocity of each sphere subjected to the same external force (albeit in opposite directions for the anti-parallel case) to the velocity it would have if was alone in an infinite medium subjected to the same force. Note that one can obtain the effect on the hydrodynamic force for specified velocity by taking the inverse of the results in figure 2. The figure includes results for both the EKR method (§ 2.3, solid lines) and the point force method (§ 2.4, dashed lines). We also present the velocities calculated using Felderhof’s mobility tensors (Felderhof Reference Felderhof1977) for mixed slip-stick boundary conditions for $Kn=0.1$ and the velocities in the free molecular limit. The Felderhof tensors are accurate to order $r^{-7}$ and are expected to yield nearly exact results except for very small separation distances for this near-continuum situation. We have obtained the free molecule results using an algorithm similar to the one described by Mackowski (Reference Mackowski2006).

Figure 2.

Speed of two spheres moving (a) parallel, (b) anti-parallel and (c) perpendicular to their line of centres, relative to the speed of isolated spheres subjected to the same external force. Results using the mobility tensors of Felderhof (Reference Felderhof1977) for mixed slip-stick boundary conditions are shown for comparison to the $Kn=0.1$ results. Note that for the Felderhof calculations we use a slip length of 0.9875 times the gas mean free path based on the best-estimate results of Loyalka (Reference Loyalka1990). Results in the free molecular limit are shown for comparison to the $Kn=10$ results for the parallel and perpendicular configurations. Solid lines represent our EKR results (2.9), while dashed lines represent point force results (2.13) for each Knudsen number.

The figure shows that our EKR results are in good agreement with the Felderhof mobility tensor results for near-continuum conditions (or near-stick boundary conditions, in the case of the Felderhof results). For $r/a>3$ , the two methods yield results within about 2 % of each other; this is not surprising, since the higher-order terms in the expansions representing the mobility tensors decay quickly with increasing separation. The error is still fairly low ( ${<}8\,\%$ ) for touching spheres moving parallel and perpendicular to their line of centres but is much more significant for the anti-parallel configuration. This is because our EKR method ignores lubrication forces between the particles, whereas Felderhof’s mobility tensors partially account for these forces through the higher-order terms in the expansion.

As expected, the strength of the hydrodynamic interactions between spheres decreases with increasing Knudsen number. For the perpendicular configuration, the two spheres behave almost as if they are isolated even when they are in contact at high Knudsen number; the effect is more important in the parallel and anti-parallel configurations, likely due to direct shielding of incoming gas molecules by the other sphere and molecules that collide with both spheres before colliding with other gas molecules. Our results at large Knudsen number are in good agreement with the free molecule results for the parallel configuration. The agreement is not as good for spheres with $Kn=10$ in perpendicular flow for small separation distances, though the interaction strength quickly becomes negligible as the separation distance increases.

As we saw in the previous section, the long-range hydrodynamic interactions have a $r^{-1}$ dependence in the continuum, as given by the Oseen tensor. The $r^{-1}$ dependence is also evident in our transition regime results in figure 2. At large separation distances, the point force results are nearly identical to the EKR results, as expected.

Figure 3.

Hydrodynamic force on an aggregate as a function of the distance between its centre of mass and the centre of mass of an identical aggregate. (a) Shows results for $N=10$ , while (b) shows results for $N=1000$ . The solid lines represent the full EKR results (2.9), while the dotted lines represent the point force results (2.13). The drag force is normalized by the drag on one of the aggregates in an infinite fluid, while the separation distance is presented as the number of effective sphere radii (i.e. the radius of a sphere experiencing drag force $F_{0}$ ) between the centres of mass of the particles.

3.2 Aggregate results

Figure 3 shows the results for two aggregates moving parallel to their line of centres. Results are shown for two aggregates with 10 spheres each and for two aggregates with 1000 spheres. The results are presented as the ratio of the drag on one of the aggregates to the drag on that aggregate when it is alone in an infinite fluid. Solid lines represent the full EKR solution (2.9), while the dotted lines represent the point force results (2.13). From the figures, the point force solution is sufficiently accurate for all but the closest separations, which greatly simplifies any analysis of the effects of hydrodynamic interactions between aggregates. Also, the aggregates exhibit more continuum-like behaviour as the number of spheres in the aggregate increases, which is analogous to the behaviour we have observed for the translational and rotational friction coefficients of DLCA aggregates (Corson et al. Reference Corson, Mulholland and Zachariah2017a ; Corson, Mulholland & Zachariah Reference Corson, Mulholland and Zachariah2018a ).

Figure 3 presents the results for aggregates with a fixed orientation. Figure 4 explores the effect of orientation on the reduction in the drag force for $N=500$ and a primary sphere Knudsen number of 2.7. Results are shown for two aggregates with a shape anisotropy $A_{31}=2.0$ and for two aggregates with $A_{31}=6.5$ , where $A_{31}$ is the ratio of the largest to smallest eigenvalues of the inertia tensor of the particle. Since each cluster in the first pair is fairly isotropic ( $A_{31}$ near unity), there is little change in the hydrodynamic force on each cluster as the particles rotate. In contrast, there is a noticeable difference in hydrodynamic force as the more anisotropic particles rotate. This is especially true for small separation distances. Note that in each case we normalize by the force on the particle for that particular orientation.

Figure 4.

Effects of orientation on the hydrodynamic force on one of two 500-sphere aggregates with primary sphere $Kn=2.7$ (sphere radius of 25 nm at room temperature), calculated using the EKR method. Results are presented as the ratio of the force on an aggregate for the specified separation distance to the force on the aggregate alone in an infinite fluid at a particular orientation. Each line represents a randomly chosen orientation for each aggregate in the two-aggregate system. (a) An aggregate with anisotropy $A_{31}=2.0$ , and (b) an aggregate with $A_{31}=6.5$ .