Self-similarity theory of stationary coagulation

Pushkin, Dmitri O.; Aref, Hassan

doi:10.1063/1.1430440

Cited by 13 publications

(8 citation statements)

References 15 publications

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“…The grain number density distribution in the case of such a fragmentation cascade has been derived by Dohnanyi (1969) and Williams & Wetherill (1994) and was found to follow a power-law number density distribution n(m) ∝ m −α with index α = 11 6 (which is equivalent to n(a) ∝ a −3.5 ), with very weak dependence on the mechanical parameters of the fragmentation process. Tanaka et al (1996), Makino et al (1998) and Kobayashi & Tanaka (2010) showed that this result is exactly independent of the adopted collision model and that the resulting slope α is only determined by the mass-dependence of the collisional cross-section if the model of collisional outcome is self-similar (in the context of fluid dynamics, the same result was independently obtained by Hunt 1982 andPushkin &Aref 2002). The value of 11 6 agrees well with the size distributions of asteroids (see Dohnanyi 1969) and of grains in the interstellar medium (MRN distribution, see Mathis et al 1977;Pollack et al 1985) and is thus widely applied, even at the gas-rich stage of circumstellar disks.…”

Section: Introductionmentioning

confidence: 52%

Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical fits

Birnstiel

Ormel

Dullemond

2010

A&A

231

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Context. Grains in circumstellar disks are believed to grow by mutual collisions and subsequent sticking due to surface forces. Results of many fields of research involving circumstellar disks, such as radiative transfer calculations, disk chemistry, magneto-hydrodynamic simulations largely depend on the unknown grain size distribution. Aims. As detailed calculations of grain growth and fragmentation are both numerically challenging and computationally expensive, we aim to find simple recipes and analytical solutions for the grain size distribution in circumstellar disks for a scenario in which grain growth is limited by fragmentation and radial drift can be neglected. Methods. We generalize previous analytical work on self-similar steady-state grain distributions. Numerical simulations are carried out to identify under which conditions the grain size distributions can be understood in terms of a combination of power-law distributions. A physically motivated fitting formula for grain size distributions is derived using our analytical predictions and numerical simulations.Results. We find good agreement between analytical results and numerical solutions of the Smoluchowski equation for simple shapes of the kernel function. The results for more complicated and realistic cases can be fitted with a physically motivated "black box" recipe presented in this paper. Our results show that the shape of the dust distribution is mostly dominated by the gas surface density (not the dust-to-gas ratio), the turbulence strength and the temperature and does not obey an MRN type distribution.

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Section: Introductionmentioning

confidence: 52%

Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical fits

Birnstiel

Ormel

Dullemond

2010

A&A

231

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“…the formation of a particle of infinite mass at finite time [12,13,14]. Similarities between the SCE and turbulence theory have been discussed previously [15], with an explicit discussion of locality of interactions. In this paper, we use results from an interacting particle system to highlight the importance of this assumption of locality when deriving scaling solutions.…”

Section: Introductionmentioning

confidence: 58%

Nonlocal interactions in coagulating particle systems

Stein,

Nazarenko

2008

Preprint

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We consider a three dimensional system consisting of a large number of small spherical particles, which move due to gravity or with laminar shear and which merge when they cross. A size ratio criterion may be applied to restrict merging to similar sized particles (locality of interactions) or particles dissimilar in size (nonlocality). We perform direct numerical simulations (DNS) of this particle system and study the resulting mass spectra. In mean field approximation, these systems can be described by the Smoluchowski coagulation equation (SCE). DNS of the particle system with locality enforced show the scaling solutions or Kolmogorov-Zakharov spectra for the SCE, signifying a constant mass flux. DNS without a size ratio criterion show −4/3 scaling for large particles in a system with gravity, signifying a constant flux in number of particles, which we also find analytically by assuming nonlocality of interactions in the SCE. For laminar shear, this nonlocality is only marginal, and our DNS show that a correction to the scaling solution is required.

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“…By applying a dimensional analysis, Hunt (1982) showed the steady state floc size distribution follows a power law. Pushkin and Aref (2002) later developed a more rigorous self-similarity theory of stationary coagulation and showed the floc size distribution follows a power law in the coagulating system. In these studies, the system is forced with particle injection at small sizes, and breakup is not considered.…”

Section: Introductionmentioning

confidence: 99%

Floc Size Distributions of Cohesive Sediment in Homogeneous Isotropic Turbulence

Balachandar

et al. 2022

Front. Earth Sci.

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Floc size distribution is one of the key parameters to characterize flocculating cohesive sediment. An Eulerian–Lagrangian framework has been implemented to study the flocculation dynamics of cohesive sediments in homogeneous isotropic turbulent flows. Fine cohesive sediment particles are modeled as the dispersed phase by the discrete element method, which tracks the motion of individual particles. An adhesive contact model with rolling friction is applied to simulate the particle–particle interactions. By varying the physicochemical properties (i.e., stickiness and stiffness) of the primary particles, the dependence of the mathematical form of the floc size distribution on sediment properties is investigated. At the equilibrium state, the aggregation and breakup processes reach a dynamic equilibrium, in which construction by aggregation is balanced with destruction by breakup, and construction by breakup is balanced with destruction by aggregation. When the primary particles are less sticky, floc size distribution fits better with the lognormal distribution. When the primary particles are very sticky, both the aggregation of smaller flocs and breakup from larger flocs play an equally important role in the construction of the intermediate-sized flocs, and the equilibrium floc size distribution can be better fitted by the Weibull distribution. When the Weibull distribution develops, a shape parameter around 2.5 has been observed, suggesting a statistically self-similar floc size distribution at the equilibrium state.

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Self-similarity theory of stationary coagulation

Cited by 13 publications

References 15 publications

Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical fits

Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical fits

Nonlocal interactions in coagulating particle systems

Floc Size Distributions of Cohesive Sediment in Homogeneous Isotropic Turbulence

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