We apply our extended Kirkwood-Riseman theory to compute the translation, rotation, and coupling friction tensors and the scalar rotational friction coefficient for an aerosol fractal aggregate in the transition flow regime. The method can be used for particles consisting of spheres in contact. Our approach considers only the linear velocity of the primary spheres in a rotating aggregate and ignores rotational and coupling interactions between spheres. We show that this simplified approach is within approximately 40% of the true value for any particle for Knudsen numbers between 0.01 and 100. The method is especially accurate (i.e., within about 5%) near the free-molecule regime, where there is little interaction between the particle and the flow field, and for particles with low fractal dimension (≲2) consisting of many spheres, where the average distance between spheres is large and translational interaction effects dominate. Our results suggest that there is a universal relationship between the rotational friction coefficient and an aggregate Knudsen number, defined as the ratio of continuum to free-molecule rotational friction coefficients.

We develop an approach for computing the hydrodynamic friction tensor and scalar friction coefficient for an aerosol fractal aggregate in the transition regime. Our approach involves solving the Bhatnagar-Gross-Krook equation for the velocity field around a sphere and using the velocity field to calculate the force on each primary sphere in the aggregate due to the presence of the other spheres. It is essentially an extension of Kirkwood-Riseman theory from the continuum flow regime to the entire Knudsen range (Knudsen number from 0.01 to 100 based on the primary sphere radius). Our results compare well to published direct simulation Monte Carlo results, and they converge to the correct continuum and free molecule limits. Our calculations for clusters with up to 100 spheres support the theory that aggregate slip correction factors collapse to a single curve when plotted as a function of an appropriate aggregate Knudsen number. This self-consistent-field approach calculates the friction coefficient very quickly, so the approach is well-suited for testing existing scaling laws in the field of aerosol science and technology, as we demonstrate for the adjusted sphere scaling method.

We present a method for calculating the hydrodynamic interactions between particles in the kinetic (or transition regime), characterized by non-negligible particle Knudsen numbers. Such particles are often present in aerosol systems. The method is based on our extended Kirkwood–Riseman theory (Corson et al., Phys. Rev. E, vol. 95 (1), 2017c, 013103), which accounts for interactions between spheres using the velocity field around a translating sphere as a function of Knudsen number. Results for the two-sphere problem at small Knudsen numbers are in good agreement with those obtained using Felderhof’s interaction actions for mixed slip-stick boundary conditions, which are accurate to order $r^{-7}$ (Felderhof, Physica A, vol. 89 (2), 1977, pp. 373–384). The strength of the interactions decreases with increasing Knudsen number. Results for two fractal aggregates demonstrate that one can apply a point force approach for interactions between particles in the transition regime; the interaction tensor is similar to the Oseen tensor for continuum flow. Using this point force approach, we present an analysis for the settling of an unbounded cloud of particles. Our analysis shows that for sufficiently high volume fractions and cloud radii, the cloud behaves as a gas droplet in continuum flow even when the individual particles are small relative to the mean free path of the gas. The method presented here can be applied in a Brownian dynamics simulation analogous to Stokesian dynamics to study the behaviour of a dense aerosol system.

We study the effects of electric field strength on the mobility of soot-like fractal aggregates (fractal dimension of 1.78). The probability distribution for the particle orientation is governed by the ratio of the interaction energy between the electric field and the induced dipole in the particle to the energy associated with Brownian forces in the surrounding medium. We use our extended Kirkwood-Riseman method to calculate the friction tensor for aggregates of up to 2000 spheres, with primary sphere sizes in the transition and near-free molecule regimes. Our results for electrical mobility versus field strength are in good agreement with published experimental data for soot, which show an increase in mobility on the order of 8% from random to aligned orientations. Our calculations show that particles become aligned at decreasing field strength as particle size increases because particle polarizability increases with volume. Large aggregates are at least partially aligned at field strengths below 1000 V/cm, though a small change in mobility means that alignment is not an issue in many practical applications. However, improved differential mobility analyzers would be required to take advantage of small changes in mobility to provide shape characterization.

We apply a self-consistent field method (Corson et al. 2017c) to calculate the rotational friction coefficient for fractal aerosol particles in the transition flow regime. Our method considers hydrodynamic interactions between spheres in a rotating aggregate due to the linear velocities of the spheres. Results are consistent with electro-optical measurements of soot alignment. Calculated rotational friction coefficients are also in good agreement with continuum and free molecule results in the limits of small (Kn D 0.01) and large (Kn D 100) primary sphere Knudsen numbers. As we previously demonstrated (Corson et al. 2017b) for the translational friction coefficient, the rotational friction coefficient approaches the continuum limit as either the primary sphere size and the number of primary spheres increases. We apply our results to develop an analytical expression Equation (26) for the rotational friction coefficient as a function of the primary sphere size and number of primary spheres. One important finding is that the ratio of the translation to rotational diffusion times is nearly independent of cluster size. We include an extension of previous scaling analysis for aerosol aggregates to include rotational motion. EDITORYannis Drossinos tion coefficient, as a function of primary sphere size and the number of spheres in the aggregate. $ t , J $ O;r , and J $O;c are the translational, rotational, and translation-rotation coupling friction tensors. The translational, rotational, and coupling friction tensors relate the particle translational velocity to the force on the particle, the particle angular velocity to the torque on the particle, and the particle translational or angular velocity to the torque or force on the particle, respectively. Note that the subscript O indicates that the property is described relative to the particle's center of mass, while the dagger symbol represents the transpose of a tensor. These equations apply for creeping flow in the continuum, free molecule, and transition regimes, characterized by very small, very large, and intermediate Knudsen numbers, respectively. For spheres, the Knudsen number is defined as Kn D λ=a, where λ is the gas mean free path and a is the sphere radius.Brenner (1967) demonstrated that a particle's friction and diffusion tensors are connected by a generalized Stokes-Einstein relationship, D $ O D kTM $ O ¡ 1 [4] where the grand mobility and diffusion tensors M $ O and D $ O are defined as M $ O D J $ t J $y O;c J $ O;c J $ O;r 2 4 3 5 [5] D $ O D D $ O;t