Calculating the rotational friction coefficient of fractal aerosol particles in the transition regime using extended Kirkwood-Riseman theory

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 ($\lesssim 2)$ 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.

Figure 1

Representations of the fractal aggregates used in this study.

Figure 2

Calculated rotational slip correction factor [defined by Eq. (42)] for a dimer, linear hexamer, and octahedral hexamer. For the dimer, the continuum value in the slip correction is the exact solution from Happel and Brenner [11]; for the hexamers, the continuum values are the shell method values (SHM) from Table 2 in Carrasco and Garcıa de la Torre [20]. The free-molecule limit for each case (dashed lines) is calculated using our Monte Carlo algorithm.

Figure 3

Calculated rotational slip correction factor [defined by Eq. (42)] for four fractal aggregates. The continuum rotational friction coefficient that appears in the slip correction is calculated using the 3RD method, and the Knudsen-number-dependent friction coefficient is calculated using our EKR method. The free-molecular limit for each aggregate (dashed lines) is calculated using our Monte Carlo algorithm.

Figure 4

Rotational slip correction factor plotted versus an aggregate Knudsen number. The aggregate Knudsen number is the ratio of the friction coefficient calculated as if the aggregate is in continuum flow to the friction coefficient calculated as if the aggregate is in free-molecule flow. Continuum friction coefficients are calculated using the same reference method used to calculate the slip correction factor $C_r$, while the free-molecule coefficients are calculated using our Monte Carlo algorithm.