Particles dynamics

Solid particles are suspended within the gas of a protoplanetary disk and experience motions as a result of their interactions with the gas. These interactions depend on such factors as the particle size and local properties of the gas (pressure, density, and their respective gradients). An important measure of how a solid particle is affected by the gas is its stopping time, which is given by  

The dynamical evolution of solids in protoplanetary disks is controlled by the large-scale flows that develop due to disk evolution, diffusion associated with the turbulence that is related to disk evolution or shear instabilities, gas-drag-induced motions due to different orbital velocities of the gas and solids, and settling towards the mid-plane due to the vertical component of the central star s gravity. As the large-scale flows have already been discussed, we now discuss the other sources of particle motions. 

The resulting transport of gas in an evolving protoplanetary disk is unlikely to have been perfectly ordered. Instead, the release of large amounts of gravitational energy associated with such evolution would likely manifest itself in small-scale fluctuations and random motions within the gas. These random motions represent the turbulence that is often discussed in terms of the evolution of protoplanetary disks, and they give rise to a diffusivity within the gas. 

On a more local scale, the effects of turbulence are to impart random motions to particles that are then added to the motions that those particles experience owing to other effects. The details of the resulting motions are beyond the scope of this chapter (comprehensive discussion and equations are provided in Cuzzi Weidenschilling 2006 Ormel Cuzzi 2007). A characteristic velocity that describes these motions is the root-mean-square turbulent velocity, /acs which is the overturn velocity of the largest eddies, and an estimate of the maximum velocity two particles (both of St = 1) would develop with respect to one another (Cuzzi Hogan 2003). Under nominal conditions, such velocities can exceed 100 m s-1 (for a = 0.01). These velocities are important to consider when developing models for dust coagulation and planetesimal formation (see Section 3.4.1 and Chapters 7 and 10). 

Solids in a gas-free orbit around a star will follow Keplerian orbits, meaning that their orbital velocity is found by balancing the centrifugal force of their motions with the central force of gravity from the star, or  

In accord with our interest we restrict our exposition in this section to statistical treatments which contain as an element the quantum mechanical cross-section or transition probability discussed in Section IV. Such statistical approaches which have been applied to chemical reactions may be conveniently divided into three categories those based on the Pauli equation or similar considerations (Section V-A), a modified Boltzmann equation (Section V-B), or a quantum statistical formulation of the Onsager theory (Section V-C). These treatments have not had notable success in comparison with experiment, probably because of the implicit Born approximation or its equivalent. It is therefore of considerable importance to extend this type of treatment to cross-sections other than that derived with the Born approximation. The method presented in Section V-C would seem to offer the best hope in this direction. 

For the sake of clarity we mention here that in the later discussion there are two different processes through which the system seeks equilibrium. We stress, of course, the chemical reaction process leading the system to chemical equilibrium. But at the same time the reactants and products themselves seek equilibrium in their internal and translational states. For simplicity, it is sometimes assumed that the reactants are in an equilibrium distribution during the approach to chemical equilibrium. If the reactants are not in or are allowed to deviate from an initial equilibrium distribution as a result of the ensuing chemical reaction, we then refer to the nonequilibrium effect on the chemical reaction rate. All of the methods given in this section are susceptible to the inclusion of such nonequilibrium effects, and we have indicated this in Sections V-A to V-C. 

we first discuss preliminary matters regarding some of the distribution functions used later. Because our exposition in Section V-B is founded on the modification of a classical equation, we include the classical distribution functions as well. For both classical and quantum mechanical considerations, we take a system to be a macroscopic object (preferably in gaseous form). A hypothetical collection of such similar systems we call an ensemble. 

For a classical mechanical system of / degrees of freedom we may define a y phase space. This is a Euclidian space of 2/ dimensions, one for each configuration coordinate (configuration space). . . /, and one for each momentum coordinate (momentum space) Px - pf The state of each system in the ensemble would be given by a representative point in the y phase space. The state of the ensemble as a whole would then be a doud of points in the y phase space. We may also define a phase space as that of one molecule in the system. If we have N molecules in the system, then the state of the system is determined by one point in y space or a doud of iV points in space. For identical molecules, a cloud of JV points in fi space represents the same physical situation with interchange of the N points. And since there are iV different but equivalent arrangements, there are N points in y space that correspond to equivalent clouds in fi space. 

If a large enough number of systems is present in the ensemble, the state of the ensemble may then be specified by a density (distribution function) / of representative points in the same y phase space  

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A fiirther theme is the development of teclmiques to bridge the length and time scales between truly molecular-scale simulations and more coarse-grained descriptions. Typical examples are dissipative particle dynamics [226] and the lattice-Boltzmaim method [227]. Part of the motivation for this is the recognition that... 

Greet R D and Warren P B 1997 Dissipative particle dynamics bridging the gap between atomistic and mesoscopic simulation J. Chem. Phys. 107 4423-35... 

Espanol P and Warren P 1995 Statistical mechanics of dissipative particles dynamics Euro. Phys. Lett. 30 191... 

Espanol P 1996 Dissipative particle dynamics for a harmonic chain a first-principles derivation Phys. Rev. B 53 1572... 

Espanol P and P B Warren 1995 Statistical Mechanics of Dissipative Particle Dynamics. Europhysl Letters 30 191-196. 

Groot R D and P B Warren 1997. Dissipative Particle Dynamics Bridging the Gap Between Atomist and Mesoscopic Simulation. Journal of Chemical Physics 107 4423-4435. 

Dissipative particle dynamics (DPD) is a technique for simulating the motion of mesoscale beads. The technique is superficially similar to a Brownian dynamics simulation in that it incorporates equations of motion, a dissipative (random) force, and a viscous drag between moving beads. However, the simulation uses a modified velocity Verlet algorithm to ensure that total momentum and force symmetries are conserved. This results in a simulation that obeys the Navier-Stokes equations and can thus predict flow. In order to set up these equations, there must be parameters to describe the interaction between beads, dissipative force, and drag. 

J. C. Shillcock, DLs.sipative Particle Dynamics U.ser Guide Molecular Simulations Ltd., Cambridge (1998). 

DPD (dissipative particle dynamics) a mesoscale algorithm DREIDING a molecular mechanics force field... 

Thermodynamics Hendrick C. Van Ness, Michael M. Abbott Heat and Mass Transfer James G. Knudsen, Hoyt C. Hottel, Adel F. Sarofim, Phillip C. Wankat, Kent S. Knaebel Fluid and Particle Dynamics Janies N. Tilton Reaction Kinetics Stanley M. Walas... 

James N. Tilton, Ph.D., P.E., Senior Consultant, Process Engineering, E. I. duPont de Nemours Co. Member, American Institute of Chemical Engineers Registered Professional Engineer (Delaware) (Section 6, Fluid and Particle Dynamics)... 

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