“…The limit a 1 /a 2 → 0, by contrast, is expected to be quite different. In this case, Kn-corrections to the tangential mobilities qualitatively change the dynamics [29]. How the phase-space portraits and the bifurcation diagrams change in this limit are open questions.…”
Section: Discussionmentioning
confidence: 96%
“…[11] we did not account for Kn-corrections to the tangential mobilities B and M . While non-continuum effects may change how the mobilities approach their values upon contact [29], it is expected that this change has a small effect on the collision dynamics [11], at least when the droplet radii are not too different. There are two reasons why the collision dynamics (3) is not smooth.…”
Saffman and Turner (1957) argued that the collision rate for droplets in turbulence increases as the turbulent strain rate increases. But the numerical simulations of in a steady straining flow show that the Saffman-Turner model is oversimplified because it neglects droplet-droplet interactions. These result in a complex dependence of the collision rate on the strain rate and on the differential settling speed. Here we show that this dependence is explained by a sequence of bifurcations in the collision dynamics. We compute the bifurcation diagram when strain is aligned with gravity, and show that it yields important insights into the collision dynamics. First, the steady-state collision rate remains non-zero in the limit Kn → 0, contrary to the common assumption that the collision rate tends to zero in this limit (Kn is a non-dimensional measure of the mean free path of air). Second, the non-monotonic dependence of the collision rate on the differential settling speed is explained by a grazing bifurcation. Third, the bifurcation analysis explains why so-called 'closed trajectories' appear and disappear. Fourth, our analysis predicts strong spatial clustering near certain saddle points, where the effects of strain and differential settling cancel.
“…The limit a 1 /a 2 → 0, by contrast, is expected to be quite different. In this case, Kn-corrections to the tangential mobilities qualitatively change the dynamics [29]. How the phase-space portraits and the bifurcation diagrams change in this limit are open questions.…”
Section: Discussionmentioning
confidence: 96%
“…[11] we did not account for Kn-corrections to the tangential mobilities B and M . While non-continuum effects may change how the mobilities approach their values upon contact [29], it is expected that this change has a small effect on the collision dynamics [11], at least when the droplet radii are not too different. There are two reasons why the collision dynamics (3) is not smooth.…”
Saffman and Turner (1957) argued that the collision rate for droplets in turbulence increases as the turbulent strain rate increases. But the numerical simulations of in a steady straining flow show that the Saffman-Turner model is oversimplified because it neglects droplet-droplet interactions. These result in a complex dependence of the collision rate on the strain rate and on the differential settling speed. Here we show that this dependence is explained by a sequence of bifurcations in the collision dynamics. We compute the bifurcation diagram when strain is aligned with gravity, and show that it yields important insights into the collision dynamics. First, the steady-state collision rate remains non-zero in the limit Kn → 0, contrary to the common assumption that the collision rate tends to zero in this limit (Kn is a non-dimensional measure of the mean free path of air). Second, the non-monotonic dependence of the collision rate on the differential settling speed is explained by a grazing bifurcation. Third, the bifurcation analysis explains why so-called 'closed trajectories' appear and disappear. Fourth, our analysis predicts strong spatial clustering near certain saddle points, where the effects of strain and differential settling cancel.
“…We also expect that the non-continuum interactions for the tangential motions will not significantly alter our collision rate calculations because axisymmetric and asymmetric relative motions of an inertialess particle pair are independent. However, one must consider non-continuum lubrication forces for asymmetric motions for calculating the collision rate of inertial particles in a gas (see Li Sing How, Koch & Collins 2021). Figure 2.( a ) Schematic of the coordinate system used in the analysis.…”
We study the gravity-induced collisions of charged spheres of dielectric materials dispersed in a gaseous medium. When the gap thickness between the surfaces of two spheres is shorter than the mean free path of the surrounding fluid medium, continuum assumptions for the hydrodynamics interactions are no longer valid, and the non-continuum lubrication interactions result in surface-to-surface contact in finite time. Two like-charged dielectric spheres attract each other at close separations for a wide range of size and charge ratio values. We use trajectory analysis to calculate the collision rate and, thus, explore the role of electrostatic interactions in the collision dynamics of a pair of like-charged dielectric spheres. We present the modifications of pair trajectories due to electrostatic forces and show how collision efficiencies vary with the non-dimensional parameter capturing the relative strength of the electrostatic force to gravity as well as the charge ratio and size ratio.
“…Similar enhancement of collisions due to particle inertia has also been seen in differentially sedimenting spheres in a quiescent fluid with van der Waals interactions (see Davis 1984). The current analysis is valid in the regime, and a detailed collision study with the arbitrary equations will be conducted in a future study where we will need to include the non-continuum tangential lubrication forces derived recently by Li Sing How, Koch & Collins (2021).…”
Section: Effects Of Weak Particle Inertia On Collisions ()mentioning
We study the collisions in a gaseous medium of a dilute bidisperse suspension of non-Brownian spherical particles sedimenting along the flow axis of a simple shear flow. Continuum lubrication forces prevent particles from coming into contact in a finite time, thus collisions can occur only due to attractive interactions such as the van der Waals force. However, in a low-pressure medium, the lubrication forces are weaker than their continuum counterparts and allow particle pairs to collide, even without any attractive forces. The Knudsen number, defined as the ratio of the mean free path of the medium to the mean radius of the interacting spheres, captures the significance of non-continuum interactions. We use uniformly valid hydrodynamic mobility functions, reflecting non-continuum lubrication at small separations and full continuum hydrodynamic interactions at moderate to larger separations. Due to the nature of the pair trajectory topology, the collision efficiency vanishes at a critical Knudsen number when simple shear flow alone drives the dynamics. Thus we perform collision calculations where particles experience the combined effects of van der Waals attraction and non-continuum hydrodynamics; van der Waals interactions enable collisions below the critical Knudsen number. Next, we calculate the collision efficiency for coupled differential sedimentation and simple shear driven motion in the presence of van der Waals interaction and non-continuum hydrodynamics. Finally, we explore the role of small particle inertia on relative trajectories and collision efficiencies in a non-continuum gas subject to a simple shear flow, ignoring the van der Waals force and gravity.
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