1. Introduction
Collisions between cloud particles lead to precipitation and affect the radiation properties of clouds. The collision rate of cloud drops is typically calculated by assuming each collision is between two spheres falling vertically at different terminal velocities (gravitational coagulation). The probability of a collision between two spherical drops of radius r_{1} and r_{2} in calm air is proportional to the swept volume. This in turn is the product of their geometrical cross section π(r_{1} + r_{2})^{2} and their relative velocity υ_{2} − υ_{1} (Pruppacher and Klett 1997). The relative velocity between droplets depends on their mass (radius) and has been investigated in depth. The definition of the swept volume in turbulent flows is different from that in calm air, because the relative velocity between approaching droplets has a random value. Moreover, droplets in turbulent flow can collide not only in the vertical direction, but also at a certain angle (Pinsky et al. 2000). As a result, collision kernels in turbulent flows might be substantially larger (up to a factor 10) than in calm air, as was shown in previous studies (Khain and Pinsky 1995; Pinsky et al. 2000, 2007, 2008; Ayala et al. 2007, 2008; Wang et al. 2008).
The increase in the collision kernel in turbulent flows leads to a significant acceleration of raindrop formation (Benmoshe et al. 2012; Seifert et al. 2010), and to a decrease in first radar echo height. Additionally, turbulence increases precipitation in large mesoscale convective systems (Lee and Baik 2016).
One of the main gaps in cloud microphysics knowledge is related to ice–ice and ice–water collisions. Collisions between ice crystals give rise to aggregate formation (snow) and lead to charge separation and lightning formation (Hashino and Tripoli 2007; Formenton et al. 2013). Aggregates dominate stratiform clouds that cover large parts of the globe and determine winter precipitation (Hosler et al. 1957; Kajikawa and Heymsfield 1989). Collisions between ice crystals and cloud droplets determine the formation of graupel and hail embryos. In turn, graupel and hail contribute to the precipitation rate in deep convective clouds (Mitchell 1988; Khain and Pinsky 2018). The high sensitivity of cloud microphysics and precipitation to the calculated rate of collisions, among ice crystal and between ice crystals and drops, was shown by Benmoshe and Khain (2014).
Observations show that small ice crystals begin producing aggregates quite rapidly in stratiform clouds, convective clouds anvils and stratiform zones of squall lines (Field and Heymsfield 2003). Nevertheless, the maximum aggregation rate occurs in the dendritic area (around T = −10° to −15°C), which could be related to these branchtype crystals ability to aggregate. Connolly et al. (2012) and Phillips et al. (2015) showed that the peak in sticking efficiency is near −15°C, when ice particles branches interlock. However, even above the dendritic growth zone, radar reflectivity from stratiform clouds could reach 15–20 dBZ (Fan et al. 2015), which indicates that aggregation between columnar or platelike crystals could also be efficient. In Arctic clouds aggregates might form at temperatures lower than the typical dendrite formation temperature (Fan et al. 2011). Moreover, the collisions between ice crystals might create the first aggregates that trigger fast aggregation in the dendritic region of the deep stratiform cloud layers forming behind squall lines (Shpund et al. 2019).
Despite the importance of these collisions in the formation of microphysical cloud structures, the collision rates between ice crystals, used in cloud models, have significant inaccuracies because of the many complications associated with the nonspherical shape of ice crystals. Fall velocity of nonspherical particles is closely related to their orientation. Nonspherical particles fall in numerous orientations, and not necessarily parallel to the gravitational acceleration vector. The particles can rotate during their fall, especially in the presence of wind shear. Therefore, like spherical particles in a turbulent flow, they may collide due to differences in both their vertical and horizontal velocities.
These factors make the calculation of swept volumes of nonspherical particles a challenging mathematical problem. Calculations of collision rates among ice crystals and between ice crystals and water droplets in current cloud models are greatly simplified to sidestep this difficulty. Typically, ice crystals are treated as “equivalent volume spheres” so that the collision rate can be calculated as if it were between spherical droplets (Hall 1980). Some studies have attempted to take the nonsphericity of ice crystals into account. In these cases, the ice crystals orientation was assumed to be with the maximum cross section perpendicular to the gravity and independent of time (Khain and Pinsky 1995; Wang 2002; Khain and Pinsky 2018).
A more accurate approach is to take into account the nonsphericity of ice crystals. The spheroidal approximation is commonly used to approximate ice particle shape, because it allows for the geometric representation of two particle length scales instead of one. Spheroids have been used in the development of numerical models of ice crystal growth processes (Chen and Lamb 1999; Hashino and Tripoli 2007; Harrington et al. 2013) and they have been used to represent snow aggregates (Stein et al. 2015; Hogan et al. 2012; Sulia et al. 2021) though aggregate shapes may be better approximated by ellipsoids (Dunnavan et al. 2019; Dunnavan 2021). Jiang et al. (2019) tested the assumption that ice particle aggregates (snowflakes) are well represented by oblate spheroids by applying ellipsoid fits to aggregate images. The resulting ellipsoids had shapes closer to prolate spheroids rather than to oblate spheroids. It was found that the most probable orientation of the maximum dimension of the retrieved ellipsoids was not in the horizontal plane.
Siewert et al. (2014a), performing direct numerical simulations (DNS), tracked the orientations of spheroids of several aspect ratios in several turbulence intensities. In the absence of gravity the orientation distribution function was uniform. However, preferential orientations were found if gravity was applied. Both prolate and oblate spheroids tended to align with their longer axis parallel to gravity. This alignment increased with increased deviation of the spheroid’s shape from spherical. Increasing the kinetic energy dissipation rate led to a more uniform distribution. Siewert et al. (2014b) investigated the collision rate and the motion of ellipsoids of different aspect ratios, settling due to gravity, in direct numerical simulated isotropic turbulence. The simulated collision rates were much higher than those of spherical particles of the same mass and volume. The authors argued that these higher rates were due to the fact that the drag force depends on the orientation. Consequently the settling velocity differences were higher and could remain so until contact if particle inertia was high enough. Jucha et al. (2018) and Naso et al. (2018) simulated the collision of small oblate spheroids and of spheroids with droplets in isotropic turbulence generated by DNS. They investigated the relative effects of turbulence intensity and gravitational settling on the orientations and on the collision rate.
As was mentioned above, ice particle orientation plays an important role in their motion and interaction. Polarimetric measurements showed that the polarimetric signatures of relatively small ice crystals could be obtained when assuming that the standard deviation of their orientation could reach up to 40° (Ryzhkov et al. 2011). Garrett et al. (2015) analyzed photographs of nearly 73 000 snowflakes in free fall and determined the aspect ratios and orientations of aggregates, moderately rimed particles, and graupel. (The aspect ratio was defined as the ratio of the shortest to longest dimension.) The analysis of Garrett et al. (2015) indicates that the range of orientation angles is much broader than has been indicated by previous observational and theoretical studies.
Despite these observational and DNS findings, the microphysical cloud models use a simplified assumptions about the orientations and fall velocities of ice crystals. Abraham (1970) developed a theory for the drag on spheres extending from the viscous regime of low particle Reynolds numbers (Re) to the regime of high Reynolds number (but still laminar). Böhm (1989, 1992a) extended the theory of Abraham (1970) to nonspherical particles, to porous particles, and to aggregates. Böhm (1992b,c) applied this theory to the study of the collision kernels and of the riming and aggregation of drops on nonspherical ice particles. He assumed that a spheroid moves in the direction of maximum drag. This implies that in calm air the large axis of the spheroid is perpendicular to gravity. Consequently, the fall velocity is always parallel to gravity. Wang and Ji (2000) also assumed a horizontal fixed orientation of the falling ice crystals, (see their previous paper; Wang and Ji 1997). The underlying assumption in the computation of the drag was that it is in the direction of the flow, or, equivalently, that particles in the gravitational field settle in the direction of gravity. Besides, in these studies, the orientations of the colliding particles were prescribed initially. Böhm (1992a) obtained very good agreement with experiments for particles falling parallel to gravity. Jensen and Harrington (2015) calculate riming growth of ice particles approximated by spheroids assuming that crystals fall in the vertical direction perpendicular to the maximum ice particle cross section. Such an assumption is valid for comparatively large crystals with Reynolds number Re > 1 (Klett 1995), but is certainly not justified for particles falling with small Reynolds numbers.
In our study we calculate the swept volume of small spheroids with Re < 1. We assume that the spheroids may have arbitrary orientations, an assumption that is valid for crystals with Re < 1. In this case the swept volume may be considerably higher due to side drift of the particles (Siewert et al. 2014a,b). Theoretically the orientations of Stokesian spheroids of small inertia, embedded in general linear shear flows, converge to either fixed points or to limit cycles. In the special case of simple shear they converge to periodic solutions named Jeffery orbits (Gavze et al. 2012, 2016a,b). The convergence time scale may sometimes be on the order of the shear time scale but may sometimes be much longer: it depends 1) on the relation between the vorticity and the eigenvectors of the strain tensor and 2) on the spheroid’s shape. The rate of convergence to these attractors is important to consideration of the orientations in time–space varying flow fields. This rate is equally relevant to the understanding of orientation distributions of spheroidal particles, smaller than Kolmogorov scale and of small inertia, in turbulent flows.
In the present paper we develop a theoretical framework for calculating the mean swept volume of an ensemble of pairs of nonspherical particles. We further develop a method to calculate the swept volume of two spheroidal particles of arbitrary orientations and aspect ratios. To make the problem tractable we focus on small spheroidal particles falling in a quiescent air. Such particles could serve as adequate models for columnar and platelike ice crystals.
2. Definition of the swept volume and the normal plane
In the following sections we deal with the collisions of small nonspherical Stokesian particles in calm air. The nonspherical particles are represented by spheroids that possess three symmetry axes, the rotation axis and two equal equatorial axes, thus forming a rotational symmetry with respect to the rotation axis. We designate the half length of the rotation axis by C and the half lengths of the two equatorial axes by A. We assume that the following conditions hold:

During the collision process the colliding particles’ orientations remain unchanged.^{1}

The relative velocities between the colliding particles are determined by their terminal velocities v^{∞}. They are not necessarily oriented in the vertical direction since the terminal velocities of nonspherical particles may have different orientations.

The particle ensemble is large enough for the probability distribution function (pdf) of each particle type’s rotation axis orientation to be defined.
The collision rate is calculated with the aid of the socalled swept volume of a collecting particle, which contains all the collected particles that collide with the collecting particle in a unit time. The product of the collected particles concentration and the swept volume yields the number (or mass) of the particles collected during a unit time. The swept volume is the product of the relative velocity between two particles and the cross section on a plane normal to the relative velocity, through which the collecting particle collides with the collected particle [defined mathematically below in Eq. (2)]. We shall call this plane the normal plane and the area of this cross section the collecting area. The definition of the normal plane is illustrated in Fig. 1.
In the following we shall make use of three coordinate systems. The first is the space coordinate system, defined by the orthogonal unit vectors
List of symbols.
3. A local coordinate system relative to the normal plane
The coordinate systems and related vectors.
4. Computation of the collecting area on the normal plane
Note that according to Eq. (9) the swept volume depends on the vector w only. It follows from Eqs. (A3) and (A5) in appendix A, and Fig. 5, that w does not dependent on which of the two colliding particles is considered the collector. Similarly, the swept volume has the same property.
The swept volume SV(θ_{1}, θ_{2}, ϕ_{1}, ϕ_{2}) can now be computed by substituting the collecting area from Eq. (9) into Eq. (2). The azimuth angle average
5. Terminal velocity of Stokesian spheroidal particles
a. The translation tensor of Stokesian spheroids
The elements of the translation tensor K.
b. Fall velocities parallel to gravity
Asymptotic values of terminal velocities when the symmetry axes are parallel to gravity and the volume is kept constant.
It is observed that the nondimensional fall velocity of prolate spheroids decreases to zero faster than that of oblate spheroids. The ratio between the asymptotic fall velocities of vertically and horizontally oriented particles is 1.5–2 for both prolate and oblate spheroids. Most spheroids fall slower than the equivalent volume spheres (corresponding to Φ = 1), except for prolate spheroids oriented parallel to gravity whose aspect ratio Φ is no larger than about five.
The aspect ratio effect on the relative velocity is better illustrated in Fig. 8, which shows cross sections from Fig. 7. Shown are normalized fall velocities differences between a sphere, Φ_{2} = 1, and spheroids whose aspect ratios change from Φ_{1} = 10^{−2} to 10^{2} (Fig. 8a) and similarly for a sphere Φ_{1} = 1 and spheroids (Fig. 8b). The elevation angles are θ_{1} = θ_{2} = 0 (i.e., the rotation axis is parallel to gravity) and the volume ratios are η = V_{1}/V_{2} = 1.02, 2, and 5. Figure 8 shows in detail the magnitude and the correct sign of the velocity difference in Fig. 7. (Note that it is immaterial, which is the collector and which is the collected spheroid.) Figure 8a shows the velocity difference between particle 1 (spheroid), the larger, and particle 2 (sphere). At aspect ratio Φ_{1} = 1 the two particles are spheres. The first sphere falls faster because its mass is larger (relative velocity is positive). If the aspect ratio is large (prolate) or small (oblate), the spheroidal particle falls slower than the sphere. Accordingly, the positive velocity difference is replaced by a negative one. Figure 8b shows the velocity difference when the particle with the larger volume (particle 1) is a sphere and the particle with the smaller volume is a spheroid. In this case the sphere falls almost always faster, and the velocity difference is actually positive. It is apparent from the η = 1.02 curve that spheres fall faster than spheroids of the same volume, except for a very small range of prolate aspect ratios close to spherical, in accord with Fig. 6. In the first column of Fig. 7 (Figs. 7a,d,g,j), which correspond to η = 1.02, the velocity difference in zones 1 and 3 is negative while it is positive in zones 2 and 4. The negative values in zones 1 and 3 are apparent in Fig. 8a (θ_{1} = θ_{2} = 0), where negative values are seen almost everywhere, as discussed above. The strictly positive values in zones 2 and 4 are in regions where the spheroid shape, of a slightly higher volume, is close to spherical. In Figs. 7b, 7e, 7h, and 7k (the second column of Fig. 7), which correspond to η = 2, the negative velocity differences in zones 1 and 3 have decreased while the positive velocity differences in zones 2 and 4 have expanded: this is because particles with twice the volume fall faster. In the right column (Figs. 7c,f,i,l), which correspond to η = 5, i.e., to a large volume difference, the negative velocity difference zones have decreased even further. Zone 1 has almost disappeared in Figs. 7c and 7f and so has zone 3 in Figs. 7i and 7l. This high volume ratio leads to a very homogeneous velocity difference in most of the aspect ratios range, irrespective of the orientations.
c. Fall velocities of arbitrarily oriented spheroids
6. Swept volumes of spheroids of different aspect ratios
a. Design of the computational tests
The actual swept volumes are obtained by multiplying the normalized swept volumes by the swept volumes of the equivalent spheres [Eq. (25)]. Since these quantities are independent of the aspect ratio, figures showing the actual swept volumes will look the same except for a multiplicative constant, which depends on the volume fraction η and on the radius r_{2} of the collected particle. Taking, for example, that radius to be r_{2} = 50 μm, then, in order to convert the normalized swept volumes, the multiplicative constants for η = (1.02, 2, 5) are SV_{eq}(η) = (1.3 × 10^{−4}, 7.1 × 10^{−3}, 3.4 × 10^{−2}) cm^{3}.
b. Relative velocity parallel to gravity
1) Nearly equal volumes: η = 1.02
Figure 12 presents the azimuthal average of the normalized swept volumes for spheroid pairs, of volume ratio η = 1.02. The aspect ratios range between 0.01 and 100. The swept volumes are normalized by the swept volume of equivalent volume spheres [Eq. (25)]. As seen in Fig. 12, the normalized mean swept volumes can exceed values of 100. These high values are obtained because the swept volume of a pair of equivalent volume spheres of nearly equal volumes, used for the normalization, is very close to zero.
The swept volume of a spheroid pair is the outcome of the collecting area and the fall velocity joint effects. The structure of the swept volume fields in Fig. 12 resembles that of the relative normalized fall velocities in Fig. 7a, 7d, 7g, and 7j. It is characterized by small narrow strips of low values close to the diagonals that result from very low relative velocities. These strips define four separate zones of high values designated as zones 1, 2, 3, and 4 on the plots. In zone 1 spheroid 1 is prolate, in zone 2 spheroid 2 is prolate, in zone 3 spheroid 1 is oblate, and in zone 4 spheroid 2 is oblate. The other spheroid in each zone assumes all aspect ratios. It may be deduced from Fig. 8 (for θ_{1} = θ_{2} = 0) that zones 1 and 3 (Fig. 8a) correspond to zones of negative relative velocities whereas zones 2 and 4 (Fig. 8b) correspond to positive relative velocities. Unlike the relative velocity, these swept volume zones lack the symmetry of the fall velocities and may roughly be divided into two pairs of high and low values thus demonstrating the effect of the collecting area. The highest values in the different orientations are obtained in the following aspect ratio ranges:

In Fig. 12a, zones 3 and 4: Both spheroids are in the vertical position.

In Fig. 12b, zones 2 and 3: The first spheroid is in the vertical position while the second is in the horizontal position.

In Fig. 12c, zones 1 and 4: The first spheroid is in the horizontal position and the second in the vertical position.

In Fig. 12d, zones 1 and 2: Both spheroids are in the horizontal position.
In all four cases the highest values of the swept volume were obtained in the zones where both spheroids have their largest dimension normal to gravity, i.e., the rotation axis in the case of prolate spheroids and both radial axes in the case of oblate spheroids. Such a configuration results in the largest collecting area. In these zones the highest swept volume were obtained between a sphere and a spheroid, either extremely prolate or extremely oblate, in which case the relative velocity is high. As already mentioned above, the aspect ratios of pristine crystals typically range from 0.1 to 10 (Pruppacher and Klett 1997) but may even be as small as 0.02 (Naso et al. 2018; Connolly et al. 2012; Wang 2002). Note in this connection that the increase in the swept volume is large even at these aspect ratios and that many ice crystals formed by secondary nucleation, like breakup by collisions or ice splinter formation by drop explosion due to freezing, can have larger aspect ratios than the regular primary ice crystals, (Rangno 2008).
In all plots the counter diagonal (bottomleft to the upperright corner) represents interactions of spheroids having the same aspect ratio. Therefore, if the orientations and volumes of the spheroids were the same it would serve as an axis of reflection. This is nearly the case in Figs. 12a and 12d where the orientations are the same and the volumes are only slightly different. For the same reason, Figs. 12b and 12c are an almost exact reflection of each other about this axis.
2) Moderate volume ratio: η = 2
Figure 13 presents the azimuthal average of the normalized swept volumes of spheroid pairs of volume ratio η = 2. The aspect ratios range between 0.01 and 100. The normalized swept volumes values are lower than in the case of η = 1.02 because the swept volumes of the equivalent volume spheres, used for the normalization, are much larger. The roles of the relative velocities and of the collecting area in this case are similar to those when η = 1.02. The structure of Fig. 13 resembles that of the relative fall velocities in Figs. 7b, 7e, 7h, and 7k. Zones 1 and 3 are surrounded by arcs of low values, which correspond to the arcs of zero relative fall velocities in Fig. 7. These zones correspond to negative relative velocities, whereas zones 2 and 4 correspond to positive relative velocities (see Fig. 8). As in the case of η = 1.02, the collecting area effect is apparent in the largest swept volume values. These values are obtained in zones where the largest dimension of the spheroid is perpendicular to gravity so that the collecting area is the largest.
3) Large volume ratio: η = 5
Figure 14 presents the azimuthal average of the normalized swept volumes of spheroid pairs, of volume ratio η = 5. The aspect ratios range between 0.01 and 100. The normalized swept volume values are lower than when η = 2 because the normalizing equivalent sphere swept volumes are larger. A dual, but somewhat more complicated, effect of the relative velocity and the collecting area is observed. As already stated the relative velocity dominated by the large spheroid is much more uniform and barely sensitive to the orientation, except for when Φ_{1} ≈ 10^{2} (zone 1) and Φ_{1} ≈ 10^{−2} (zone 3), in the right column of Fig. 7, where it approaches zero. This approach to zero has the largest effect on the swept volume in zone 3 of Figs. 14a and 14b and zone 1 of Figs. 14c and 14d. The reduced relative velocity acts against the collecting area tendency to increase the swept volume in these zones when η = 1.02 and 2 (see Figs. 12 and 13). In general the swept volume structure is relatively uniform, similar to the structure of the relative velocity.
c. Relative velocity not parallel to gravity
Here we present azimuthal averages of the normalized swept volumes of pairs of spheroids with a relative velocity that is not parallel to gravity. Here too the swept volumes are normalized by the swept volume of equivalent volume spheres [Eq. (25)]. Since equivalent spheres have no horizontal relative velocity component, we observe an increase in the normalized swept volumes, especially for small volume ratios. The effect is illustrated in Figs. 15–17.
1) Nearly equal volumes: η = 1.02
Figure 15 presents the azimuthal mean of the normalized swept volumes of spheroid pairs of volume ratio η = 1.02. The aspect ratios range between 0.01 and 100. A comparison with the Fig. 12 shows a significant increase in the normalized swept volumes for colliding spheroids of nearly the same volumes. Most values, except for the central zone where the shapes of the spheroids are close to spherical, are close to 1000. As was mentioned above this effect is related to the fact that the swept volume of effective spheres of similar volumes is close to zero.
2) Moderate volume ratio: η = 2
Figure 16 presents the azimuthal mean of the normalized swept volumes of spheroid pairs, of volume ratio η = 2 and aspect ratios range between 0.01 and 100. The fields of normalized volumes at η = 2 (Fig. 16) resemble those in Fig. 13. However, the minimum values do not fall to zero because of the horizontal component’s effect of the relative velocity.
3) Large volume ratio: η = 5
The azimuthal averages of the normalized swept volumes of colliding spheroids with large volume ratios of η = 5 (Fig. 17) resemble those in Fig. 14. There are, however, significant differences. First, the normalized swept volumes in the presence horizontal components of the relative velocity never fall below ≈0.5. Second, the swept volumes in the case of Δv ∦ g are more symmetric with respect to the line Φ_{2} = 1 (cf. Fig. 17c and Fig. 14c). This, again, is related to the existence of significant horizontal relative velocity of spheroids with high aspect ratios.
7. Discussion and conclusions
In this work we developed a method to compute gravityinduced swept volumes of spheroidal particles in calm air. In clouds, prolate spheroids can be associated with columnar crystals, while oblate spheroids with plates and dendrites. The analytic expressions for the swept volumes are suitable for any particle size, and the numerical realization is simple and computationally efficient. Terminal velocities and swept volumes were computed for Stokesian particles, which correspond to ice crystals up to about 100–150 μm. The azimuthal averaged swept volumes were shown to be dependent on particle mass, aspect ratios and the elevation angles of the colliding particles. To investigate the role of these parameters, the swept volumes were normalized and compared with swept volumes of equivalent volume spheres that have identical density and volumes as the corresponding spheroids.
The significant variability in the swept volumes is related to variability in the aspect ratios and orientations. Most swept volumes, normalized to the corresponding swept volumes of equivalent spheres, are larger than 1 and can be as large as 1000. The horizontal component of the relative velocity between the spheroids increases the normalized swept volumes. In this study we considered spheroidal particles of small Reynolds numbers whose fall velocity is not necessarily parallel to gravity. Note that in microphysical models it is usually assumed that ice crystals of any shape fall parallel to gravity. Such an assumption may be wrong, not only for small Stokesian particles, but also for larger particles due to nonspherical shape, oscillations performed in larger Re, and an uneven distribution of mass. An additional horizontal velocity component may increase the collision rates. The DNS computation of Siewert et al. (2014b) exhibits a significant increase in the collision rate of spheroids relative to that of spheres of equal volume. Therefore, we conclude that utilization of the effective spheres assumption when computing collisions between spheroids could lead to a substantial underestimation of the aggregation rate, particularly if ice crystals are modeled as spheres. This underestimation is especially important because of the nonlinear feedback in the collision process: the increase in the collision kernel leads to a nonlinear increase in the collision rate.
In this study we present a solution to the first problem related to collisions of nonspherical particles, namely, calculation of the swept volume of nonspherical particles in calm air. In addition to the swept volume, a complete calculation of the collision kernel requires the determination of the collision efficiency. Unlike spherical drops, collision efficiency between ice crystals and between ice crystals and drops is expected to depend on their aspect ratios and their orientation distribution. The general problem of collision of nonspherical particles requires consideration of additional two important mechanisms: hydrodynamic interaction between colliding particles and the turbulence effects on collisions. Problems of hydrodynamic interaction and turbulent effects in the case of nonspherical particles are much more complicated than those in spherical droplets because the rotation of the particles should be taken into account in addition to their translational motions.
The calculation of the swept volume, presented in this study, is the first important step toward the investigation of collisions between nonspherical particles in clouds. The next steps will include the incorporation of hydrodynamic interactions and the effects of turbulence.
Change of orientations of Stokesian particles in calm air may occur due to hydrodynamic interactions. These interactions become important when the distance separating the two particles is on the order of a few times the size of the particles. The change of the orientations may be neglected if 1) the relative velocity of the particles is large enough so that the time they spend in the vicinity of each other is short enough and 2) the relaxation times of the two particles are long enough. We are aware that this condition might be a little too stringent. See section 7 for a further consideration of this point.
For a 2D domain in the (x, y) plane, bounded by a curve L, define F = x, G = −y. Then
To gain some understanding of this asymmetry note first that the terminal velocity, when parallel to gravity, is inversely proportional to the corresponding elements of the translation tensor. For parallel and perpendicular orientations the drag increases with increasing Φ and 1/Φ thus showing a direct proportionality to the surface area (at constant volume the surface area is minimal for a sphere), which is reasonable for Stokes flow where viscosity plays the main role. On the other hand, it is seen in Fig. 6a that the drag is also proportional to the crosssection area, perpendicular to the motion, a behavior typical of pressure, which balances the viscous forces. Thus, for example, the drag on an oblate spheroid with its rotation axis parallel to the motion, is larger than that on an oblate spheroid when its rotation axis is perpendicular to the motion. An opposite behavior is observed for the prolate spheroid. It is therefore the interplay between these two mechanisms that determines the actual drag.
Acknowledgments.
The study was supported by the Israel Science Foundation (Grant 2027/17) and the U.S. Department of Energy (Grants DEFOA0001638, DEAC0576RL01830 81). We thank Kaushal Gianchandani for his help and valuable suggestions.
APPENDIX A
Computation of w—The Collecting Area Boundary
The computation order is as follows:

Compute
${\tilde{\theta}}_{i}$ ,${\tilde{\varphi}}_{i}$ and$\mathrm{\Delta}\tilde{\varphi}$ from Eq. (6). 
Compute a_{i} and b_{i} from Eq. (8).

Choose a point u on the boundary of the collecting particle projection on the normal plane.

Compute N from Eq. (A7).

Compute M from Eq. (A10).

Compute L from Eq. (A6).

Compute w from Eq. (A14).
APPENDIX B
Computation of the Collecting Area
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