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We reexamine the streaming instability in protoplanetary disks under the axisymmetric geometry. We identify the roles of all the relevant physical processes and clarify the clustering mechanism of dust, offering a new physical interpretation. Through back-reaction, the dust density fluctuations in combination with the mean relative drift velocity provide a source for the gas velocity, which in turn drives the velocity and divergence of the dust flow. In the limit where the dust-to-gas ratio , the action of the Coriolis force on the radial gradient of the azimuthal dust velocity generates the divergence of the dust. In the opposite limit , the vertical gradient of the vertical dust velocity makes the main contribution to the dust divergence. The different dust clustering mechanisms at and are referred to as Mode I and Mode II, respectively. In both cases, the dust divergence further enhances the dust density fluctuations, resulting in a positive feedback loop. In Mode I (or Mode II), the growth rate is contributed by the mean azimuthal (or radial) drag force in the gas equation of motion, while the mean radial (or azimuthal) drag tends to reduce it. The instability makes a transition from Mode I to Mode II when the coupling between the perturbed gas and dust velocities is stronger than the Coriolis force in the gas equations, which occurs at and for Stokes numbers St < 1 and St > 1, respectively. [ABSTRACT FROM AUTHOR]

We study the relative velocity of inertial particles suspended in turbulent flows and discuss implications for dust particle collisions in protoplanetary disks. We simulate a weakly compressible turbulent flow, evolving 14 particle species with friction timescale, τp, covering the entire range of scales in the flow. The particle Stokes numbers, St, measuring the ratio of τp to the Kolmogorov timescale, are in the range 0.1 ≲ St ≲ 800. Using simulation results, we show that the model by Pan & Padoan gives satisfactory predictions for the rms relative velocity between identical particles. The probability distribution function (PDF) of the relative velocity is found to be highly non-Gaussian. The PDF tails are well described by a 4/3 stretched exponential function for particles with τp ≃ 1-2 TL, where TL is the Lagrangian correlation timescale, consistent with a prediction based on PP10. The PDF approaches Gaussian only for very large particles with τp ≳ 54 TL. We split particle pairs at given distances into two types with low and high relative speeds, referred to as continuous and caustic types, respectively, and compute their contributions to the collision kernel. Although amplified by the effect of clustering, the continuous contribution vanishes in the limit of infinitesimal particle distance, where the caustic contribution dominates. The caustic kernel per unit cross section rises rapidly as St increases toward ≃ 1, reaches a maximum at τp ≃ 2 TL, and decreases as for τp ≫ TL. [ABSTRACT FROM AUTHOR]

Motivated by its important role in the collisional growth of dust particles in protoplanetary disks, we investigate the probability distribution function (PDF) of the relative velocity of inertial particles suspended in turbulent flows. Using the simulation from our previous work, we compute the relative velocity PDF as a function of the friction timescales, τp1 and τp2, of two particles of arbitrary sizes. The friction time of the particles included in the simulation ranges from 0.1τη to 54TL, where τη and TL are the Kolmogorov time and the Lagrangian correlation time of the flow, respectively. The relative velocity PDF is generically non-Gaussian, exhibiting fat tails. For a fixed value of τp1, the PDF shape is the fattest for equal-size particles (τp2 = τp1), and becomes thinner at both τp2 < τp1 and τp2 > τp1. Defining f as the friction time ratio of the smaller particle to the larger one, we find that, at a given f in (1/2) ≲ f ≲ 1, the PDF fatness first increases with the friction time τp, h of the larger particle, peaks at τp, h ≃ τη, and then decreases as τp, h increases further. For 0 ⩽ f ≲ (1/4), the PDF becomes continuously thinner with increasing τp, h. The PDF is nearly Gaussian only if τp, h is sufficiently large (≫TL). These features are successfully explained by the Pan & Padoan model. Using our simulation data and some simplifying assumptions, we estimated the fractions of collisions resulting in sticking, bouncing, and fragmentation as a function of the dust size in protoplanetary disks, and argued that accounting for non-Gaussianity of the collision velocity may help further alleviate the bouncing barrier problem. [ABSTRACT FROM AUTHOR]

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, 〈 w2〉 1/2, as a function of the friction times, τp1 and τp2, of the two particles and show that the PP10 prediction is in satisfactory agreement with the data, confirming its physical picture. For a given τp1 below the Lagrangian correlation time of the flow, TL, 〈 w2〉 1/2 as a function of τp2 shows a dip at τp2 ≃ τp1, indicating tighter velocity correlation between similar particles. Defining a ratio f ≡ τp, l/τp, h, with τp, l and τp, h the friction times of the smaller and larger particles, we find that 〈 w2〉 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 〈 w2〉 1/2 scales as for τp, h in the inertial range of the flow, stays roughly constant for TL ≲ τp, h ≲ TL/f, and finally decreases as for τp, h ≫ TL/f. The acceleration term is independent of the particle distance, r, and reduces the r dependence of 〈 w2〉 1/2 in the bidisperse case. [ABSTRACT FROM AUTHOR]