On the courant-friedrichs-lewy condition for numerical solvers of the coagulation equation

Evolving the size distribution of solid aggregates challenges simulations of young stellar objects. Among other difficulties, generic formulae for stability conditions of explicit solvers provide severe constrains when integrating the coagulation equation for astrophysical objects. Recent numerical experiments have recently reported that these generic conditions may be much too stringent. By analysing the coagulation equation in the Laplace space, we explain why this is indeed the case and provide a novel stability condition which avoids time over-sampling.

Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

Asymptotic analysis of time dependent solutions for the coagulation equation with source and efflux

This article provides mathematical proof of the existence of stationary solutions for the coagulation equation including source and efflux terms. We demonstrate the convergence of time dependent solutions to these stationary solutions and highlight the exponential rate of convergence. These properties are analyzed for affine linear coagulation kernels, non-negative source terms and positive efflux rates. Numerical examples are included to demonstrate the predicted convergence behaviour.

Mathematical modeling of dispersed media flows in the presence of nucleation, coagulation and phase transitions

A model of motion of a gas-dispersed medium in the presence of processes of nucleation, coagulation and phase transitions has been constructed. A homogeneous nucleation model is used to describe the nucleation process. It is believed that the process of cluster coagulation occurs due to their Brownian motion. The analysis of the solution of the coagulation equation in the particular case of monodisperse clusters in the presence of a source and sink of particles is carried out. To determine the rate of phase transitions the Hertz-KnudsenLangmuir formula is used. The calculations were carried out on the basis of a quasi-one-dimensional model within the equilibrium approximation (when the velocities and temperatures of the phases coincide). As a result of the study the main properties of the flow of a two-phase mixture in a channel in the presence of nucleation, coagulation, and phase transformations have been established. It is shown that the vapor temperature increases along the channel and reaches the saturation temperature at some distance from the channel entrance. Calculations have shown that the coagulation process has a rather strong effect on the distribution of cluster sizes along the channel.

Uniqueness of fat-tailed self-similar profiles to Smoluchowski’s coagulation equation for a perturbation of the constant kernel

This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski’s coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel K K which can be written as K = 2 + ε W K=2+\varepsilon W . The perturbation is assumed to have homogeneity zero and might also be singular both at zero and at infinity. Under further regularity assumptions on W W , we will show that for sufficiently small ε \varepsilon there exists, up to normalisation of the tail behaviour at infinity, at most one self-similar profile. Establishing uniqueness of self-similar profiles for Smoluchowski’s coagulation equation is generally considered to be a difficult problem which is still essentially open. Concerning fat-tailed self-similar profiles this article actually gives the first uniqueness statement for a non-solvable kernel.

PLUME-MoM-TSM 1.0.0: a volcanic column and umbrella cloud spreading model

In this paper, we present a new version of PLUME-MoM, a 1-D integral volcanic plume model based on the method of moments for the description of the polydispersity in solid particles. The model describes the steady-state dynamics of a plume in a 3-D coordinate system, and a modification of the two-size moment (TSM) method is adopted to describe changes in grain size distribution along the plume, associated with particle loss from plume margins and with particle aggregation. For this reason, the new version is named PLUME-MoM-TSM. For the first time in a plume model, the full Smoluchowski coagulation equation is solved, allowing us to quantify the formation of aggregates during the rise of the plume. In addition, PLUME-MOM-TSM allows us to model the phase change of water, which can be either magmatic, added at the vent as liquid from external sources, or incorporated through ingestion of moist atmospheric air. Finally, the code includes the possibility to simulate the initial spreading of the umbrella cloud intruding from the volcanic column into the atmosphere. A transient shallow-water system of equations models the intrusive gravity current, allowing computation of the upwind spreading. The new model is applied first to the eruption of the Calbuco volcano in southern Chile in April 2015 and then to a sensitivity analysis of the upwind spreading of the umbrella cloud to mass flow rate and meteorological conditions (wind speed and humidity). This analysis provides an analytical relationship between the upwind spreading and some observable characteristic of the volcanic column (height of the neutral buoyancy level and plume bending), which can be used to better link plume models and volcanic-ash transport and dispersion models.

Particle Size Distribution of Smoluchowski Coagulation Equation for Brownian Motion at Equilibrium

The information entropy for Smoluchowski coagulation equation is proposed based on statistical mechanics. And the normalized particle size distribution is a lognormal function at equilibrium from the principle of maximum entropy and moment constraint. The geometric mean volume and standard deviation in the distribution function are determined as simple constant. The results reveal that the assumption that algebraic mean volume be unit in self-preserving hypothesis is reasonable in some sense. Based on the present definition of information entropy, the Cercignani’s conjecture holds naturally for Smoluchowski coagulation equation. Together with the proof that the conjecture is also true for Boltzmann equation, Cercignani’s conjecture will holds for any two-body collision systems, which will benefit the understanding of Brownian motion and molecule kinematic theory, such as the stability of the dissipative system, and the mathematical theory of convergence to thermodynamic equilibrium.