Turbulence-Induced Relative Velocity Of Dust Particles. II. The Bidisperse Case
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We extend our earlier work on turbulence-induced relative velocity between equal-size particles ( Paper I, in this series) to particles of arbitrarily different sizes. The Pan & 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) similar or equal to tau(p1), indicating tighter velocity correlation between similar particles. Defining a ratio f equivalent to tau(p,1)/tau(p,h), with tau(p,1) and tau(p,h) 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 less than or similar to 1/4. At a fixed f, our model predicts that < w(2)>(1/2) scales as tau(1/2)(p,h) tau(p,h) for in the inertial range of the flow, stays roughly constant for T-L less than or similar to tau(p,h) less than or similar to T-L/f, and finally decreases as tau(-1/2)(p,h) p, h for tau(p,h) >> T-L/f. The acceleration term is independent of the particle distance, r, and reduces the r dependence of < w(2)>(1/2) in the bidisperse case.