Collision rate coefficient for charged dust grains in the presence of linear shear

Like and oppositely charged particles or dust grains in linear shear flows are often driven to collide with one another by fluid and/or electrostatic forces, which can strongly influence particle-size distribution evolution. In gaseous media, collisions in shear are further complicated because particle inertia can influence differential motion. Expressions for the collision rate coefficient have not been developed previously which simultaneously account for the influences of linear shear, particle inertia, and electrostatic interactions. Here, we determine the collision rate coefficient accounting for the aforementioned effects by determining the collision area, i.e., the area of the plane perpendicular to the shear flow defining the relative initial locations of particles which will collide with one another. Integration of the particle flux over this area yields the collision rate. Collision rate calculations are parametrized as an enhancement factor, i.e., the ratio of the collision rate considering potential interactions and inertia to the traditional collision rate considering laminar shear only. For particles of constant surface charge density, the enhancement factor is found dependent only on the Stokes number (quantifying particle inertia), the electrostatic energy to shear energy ratio, and the ratio of colliding particle radii. Enhancement factors are determined for Stokes numbers in the 0–10 range and energy ratios up to 5. Calculations show that the influences of both electrostatic interactions and inertia are significant; for inertialess ($\mathrm{St}=0$) equal-sized and oppositely charged particles, we find that even at energy ratios as low as 0.2, enhancement factors are in excess of 2. For the same situation but like-charged particles, enhancement factors fall below 0.5. Increasing the Stokes number acts to mitigate the influence of electrostatic potentials for both like and oppositely charged particles; i.e., inertia reduces the enhancement factor for oppositely charged particles and increases it for like-charged particles. Uniquely, at elevated Stokes numbers with attractive potentials we find collisionless “pockets” within the collision area, which are regions completely bounded by the collision area but within which collisions do not occur. Regression equations to results are provided, enabling calculation of the enhancement factor as a function of energy ratio and Stokes number. In total, this study both leads to insight into the collision dynamics of finite-inertia, charged particles in shear flows, and provides a means to simply calculate the particle-particle collision rate coefficient.

Figure 1

(a) Example results for trajectory calculations, wherein particle $j$ is placed in the center of a linear shear field, and particle $i$ is placed at a specific location at a large “$z^{*}$” distance from particle $j$. The trajectories displayed correspond to different $Q$ and St values, as well as different initial positions, with the upper set corresponding to a collision, and the lower corresponding to a noncollision. Particle $j$ only moves in response to the shear field after electrostatic forces brought about by the close approach of particle $i$ move it from its original $y^{*}$ position. (b) A depiction of the dimensionless collision area formed considering attractive potentials on the $y^{*}-x^{*}$ plane.

Figure 2

The calculated enhancement factor in the presence of laminar or linear shear considering (a) attractive and (b) repulsive Coulomb interactions between particles in the $\mathrm{St}\to \infty$ limit. Open symbols: trajectory calculation results. Lines: Eqs. (17a) and (17b) predictions.

Figure 3

(a) Selected relative particle trajectories (i.e., the motion of particle $i$ from the perspective of particle $j$) for particles initiated on the $y^{*}, z^{*}$ ($x^{*}=0$) plane. Black lines denote collision, while red lines denote noncolliding trajectories. The blue line denotes the bound ${\left(y_i^{*}-y_j^{*}\right)}^2+{\left(z_i^{*}-z_j^{*}\right)}^2=1$; hence it is the collision boundary. (b) Points corresponding to initial relative release points leading to collision. Collisionless pockets are evident at $\mathrm{St}=5$ and $\mathrm{St}=10$. For all instances, $Q=1.56$ and ${\theta }_{\mathrm{r}}=1.0$, with the full electrostatic potential considered.

Figure 4

The enhancement factor considering attractive potential interactions for $\mathrm{St}=0$. The black dashed line denotes Eq. (18) predictions.

Figure 5

The enhancement factor considering attractive potential interactions for $\mathrm{St}=0.5-10$. The legend provided in Fig. 4 applies to all displayed curves.

Figure 6

The enhancement factor as a function of $-Q$ for repulsive interactions.

Figure 7

The ratio of the laminar shear collision rate coefficient to the Brownian motion collision rate coefficient for oppositely charged particles, considering the Coulomb potential with $\mathrm{St}=0$.