Particle momentum

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Finer particles ( < 3 pm), termed respirable particles, pass beyond the ex-trathoracic airways and enter the tracheobronchial tree. Impaction plays a significant role near the tracheal jet, but sedimentation predominates as the effects of rapid conduit expansion dampen in the distal trachea and beyond. Sedimentation occurs when gravitational forces exerted on a particle equal drag forces, i.e., when particle velocity falls to u . As mean inspiratory air-stream velocity gradually declines along the tracheobronchial tree, particle momentum diminishes and 0.5-3 pm MMAD particles settle out of the airflow and onto mucosal surfaces. 

Mean airflow velocities approach zero as the inspired airstream enters the lung parenchyma, so particle momentum also approaches zero. Most of the particles reaching the parenchyma, however, are extremely fine (< 0.5 pm MMAD), and particle buoyancy counteracts gravitational forces. Temperature gradients do not exist between the airstream and airway wall because the inspired airstream has been warmed to body temperature and fully saturated before reaching the parenchyma. Consequently, diffusion driven by Brownian motion is the only deposition mechanism remaining for airborne particles. Diffusivity, can be described under these conditions by... 

How might the interaction between two discrete particles be described by a finite-information based physics Unlike classical mechanics, in which a collision redistributes the particles momentum, or quantum mechanics, which effectively distributes their probability amplitudes, finite physics presumably distributes the two particles information content. How can we make sense of the process A scatters J5, if B s momentum information is dispersed halfway across the galaxy [minsky82]. Minsky s answer is that the universe must do some careful bookkeeping, ... 

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field. 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

B-particle momentum, multiparticle collision dynamics, single-particle friction and diffusion, 115-118... 

The principle of a lattice gas is to reproduce macroscopic behavior by modeling the underlying microscopic dynamics. In order to successfully predict the macro-level behavior of a fluid from micro-level rules, three requirements must be satisfied. First, the number of particles must be conserved and, in most cases, so is the particle momentum. States of all the cells in the neighborhood depend on the states of all the others, but neighborhoods do not overlap. This makes application of conservation laws simple because if they apply to one neighborhood they apply to the whole lattice. 

The Hamiltonian function for a system of bound harmonic oscillators is, in the most general form, a sum of two positively definite quadratic forms composed of the particle momentum vectors and the Cartesian projections of particle displacements about equilibrium positions ... 

For matter waves hk is the particle momentum and the uncertainty relation AxAp > h/2, known as the Heisenberg uncertainty principle. 

For matter waves wavelength and frequency are assumed to relate to particle momentum and energy according to A = h/p and 1/ = E/h. The proportionality factor is Planck s constant h. Hence... 

The many-particle momentum space wave function, P2, P3,..., P/v) is... 

To motivate our next step recall that the LOFF spectrum can be view as a dipole [oc Pi (a )] perturbation of the spherically symmetrical BCS spectrum, where Pi(x) are the Legendre polynomials, and x is the cosine of the angle between the particle momentum and the total momentum of the Cooper pair. The l = 1 term in the expansion about the spherically symmetric form of Fermi surface corresponds to a translation of the whole system, therefore it preserves the spherical shapes of the Fermi surfaces. We now relax the assumption that the Fermi surfaces are spherical and describe their deformations by expanding the spectrum in spherical harmonics [17, 18]... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

The particles momentum is equal to its mass times the group velocity v. 

From Eq. (8.8) it is evident that particle momentum depending on mass and velocity of the particles is important to control the delivery. The particle acceleration and impact velocity are defined by particle properties such as size, density, and morphology and device properties such as pressure of the compressed gas source, nozzle geometry, and others. 

The simplest theory of impact, known as stereomechanics, deals with the impact between rigid bodies using the impulse-momentum law. This approach yields a quick estimation of the velocity after collision and the corresponding kinetic energy loss. However, it does not yield transient stresses, collisional forces, impact duration, or collisional deformation of the colliding objects. Because of its simplicity, the stereomechanical impact theory has been extensively used in the treatment of collisional contributions in the particle momentum equations and in the particle velocity boundary conditions in connection with the computation of gas-solid flows. 

Consider a cubic unit volume containing n particles in a Cartesian coordinate system. On average, about n/6 particles move in the +y-direction, and the same number of particles move in the other five directions. Each particle stream has the same averaged velocity (v). Since particle collision is responsible for the momentum transport, the averaged x-component of the particle momentum transported in the y-direction may be reasonably estimated by... 

Synthesis of the three observations led to Bohr s proposal of a planetary atom consisting of a heavy small stationary heavy nucleus and a number of orbiting electrons. Each electron, like a planet, had its own stable orbit centred at the atomic nucleus. The simplest atom, that of hydrogen, with atomic number 1 could therefore be described as a single electron orbiting a proton at a fixed, relatively large, distance. The mechanical requirement to stabilize the orbit is a balance between electrostatic and mechanical forces, expressed in simple electrostatic units, and particle momentum p = mv, as ... 

Multiplying the wavevector k by Planck s constant h and dividing by 2n yields the crystal momentum p = hk. This crystal momentum of the electron includes a lattice component and is therefore not a true free-particle momentum, as it was in free-electron theory. 

The net difference of the forces acting on the particle is equal to the rate of change of particle momentum. Thus... 

Equation 9.12 indicates that the diffusion coefficient of an aerosol particle is independent of particle density and hence is independent of particle mass. But is this really so Since particle mass is so much greater than molecular mass and the particles are continually undergoing bombardment by the molecules, one would expect changes in the direction of the particle to be gradual, compared to the rapid changes in direction with molecular diffusion. But if this is true, then particle momentum (mass) should be considered in the particle diffusion coefficient equation. 

A necessary condition for the two-term expansion of the distribution function of equation (2) to be valid is that the electron collision frequency for momentum transfer must be larger than the total electron collision frequency for excitation for all values of electron energy. Under these conditions electron-heavy particle momentum-transfer collisions are of major importance in reducing the asymmetry in the distribution function. In many cases as pointed out by Phelps in ref. 34, this condition is not met in the analysis of N2, CO, and C02 transport data primarily because of large vibrational excitation cross sections. The effect on the accuracy of the determination of distribution functions as a result is a factor still remaining to be assessed. 

With these specifications, and with the appropriate neutral particle-plasma collision terms put into the combined set of neutral and plasma equations, internal consistency within the system of equations is achieved. Overall particle, momentum and energy conservation properties in the combined model result from the symmetry properties of the transition probabilities W indices of pre-collision states may be permuted, as well as indices of postcollision states. For elastic collisions even pre- and post collision states may be exchanged in W. 

Internal consistency in the data for a1, I i and f2s, l is a crucial prerequisite for particle, momentum and energy conservation in such models. The consistency amongst the rates is ensured by employing the above mentioned recursive relations. This can be achieved either by fitting expressions which can be differentiated with respect to Ta and vn, or e.g., by employing data tables only for the lowest rate Iq and B-splines with sufficient smoothness to permit evaluation of the derivatives. 

The particle momentum is restricted to p= VS. Since the wavefunction is single-valued it follows that R must be a single-valued function of position. However, the phase function is not uniquely fixed and two S functions like S and S, where... 

Particle momentum DP can be measured by a pitot tube (Bader et al, 1988), as well as by the momentum probe described earlier. 

The quantity mu is the momentum of the particle (momentum is the product of mass and velocity), and the expression F = A(mu)/At means that force is the change in momentum per unit of time. When a particle hits a wall perpendicular to the x axis, as shown in Fig. 5.13, an elastic collision occurs, resulting in an exact reversal of the x component of velocity. That is, the sign, or direction, of ux reverses when the particle collides with one of the walls perpendicular to the x axis. Thus the final momentum is the negative, or opposite, of the initial momentum. Remember that an elastic collision means tjiat there is no change in the magnitude of the velocity. The change in momentum in the x direction is ... 

System Collision Energy (eV) Oi Percent of Particle Momentum TVansferred Momentum TVansfer Angle, 0 Hard Cube Model (Mass = 1.5 X surface atom) Binary Collision Model... 

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