Turbulence-Induced Relative Velocity of Dust Particles II: The   Bidisperse Case
  We extend our earlier work on turbulence-induced relative velocity between equal-size particles (Pan and Padoan, Paper I) to particles of arbitrarily different sizes. The Pan and Padoan (PP10) model shows that the relative velocity between different particles has two contributions, named the generalized shear and acceleration terms, respectively. The generalized shear term represents the particles' memory of the spatial flow velocity difference across the particle distance in the past, while the acceleration term is associated with the temporal flow velocity difference on individual particle trajectories. Using the simulation of Paper I, we compute the root-mean-square relative velocity, <w^2>^1/2, as a function of the friction times, tau_p1 and tau_p2, of the two particles, and show that the PP10 prediction is in satisfactory agreement with the data, confirming its physical picture. For a given tau_p1 below the Lagrangian correlation time of the flow, T_L, <w^2>^1/2 as a function of tau_p2 shows a dip at tau_p2~tau_p1, indicating tighter velocity correlation between similar particles. Defining a ratio f=tau_pl/tau_ph, with tau_pl and tau_ph the friction times of the smaller and larger particles, we find that <w^2>^1/2 increases with decreasing f due to the generalized acceleration contribution, which dominates at f<1/4. At a fixed f, our model predicts that <w^2>^1/2 scales as tau_ph^1/2 for tau_p,h in the inertial range of the flow, stays roughly constant for T_L <tau_ph < T_L/f, and finally decreases as tau_ph^-1/2 for tau_ph>>T_L/f. The acceleration term is independent of the particle distance, r, and thus reduces the r-dependence of <w^2>^1/2 in the bidisperse case. 
Liubin Pan, Paolo Padoan, and John Scalo
astro-ph.EP physics.flu-dyn