Local Maxwellian

It was shown, in Eqs. (1-73), (1-74), (1-75), that a = 1, afy r> => 0, a = 0. As the zeroth approximation we shall assume that A mid /a are zero (their effects are negligibly small) if Eqs. (1-86) and (1-87) are multiplied by /a and A, respectively, we obtain the condition that og0 and -oSi are zero higher order equations would show that all the coefficients are zero. Thus, the coefficients are proportional to some power of /a (or A). The zero-order approximation to the distribution function is just the local maxwellian distribution... 

We can do this simply only by making the approximation that P(c,z) P(c). This will be valid only when the streaming velocity V is much less than a, the most probable speed. (F , and the integrations become rather difficult. In such a case, however, when the plate is moving with molecular velocities, the use of a local Maxwellian velocity is probably very bad and the entire treatment breaks down. 

It is important to note that this distribution function (2.225), defined so that it resembles (2.224) but with the constant values of n, v and T in (2.224) replaced by the corresponding functions of r and t, remains a solution to (2.223). This distribution function, which is called the local Maxwellian, makes the kinetic theory much more general and practically relevant. 

Both the absolute- and local Maxwellians are termed equilibrium distributions. This result relates to the local and instantaneous equilibrium assumption in continuum mechanics as discussed in chap. 1, showing that the assumption has a probabilistic fundament. It also follows directly from the local equilibrium assumption that the pressure tensor is related to the thermodynamic pressure, as mentioned in sect. 2.3.3. 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

Applying the local Maxwellian distribution function (2.225), explicit expressions can be obtained for the heat flux, q and the pressure tensor P. 

Note that the transport term on the left-hand side of Eq. (6.1) can be larger or smaller in magnitude than the collision term. For cases in which the collision term is much more important than the transport term, the solution to Eq. (6.1) with the Boltzmann collision model is a local Maxwellian wherein ap. Up, and p depend on space and time but / is well approximated by Eq. (6.10). In this limit, the particles behave as an ideal gas and the mean velocity obeys the Euler equation. 

Note that the local values of the number density n(t, r), bulk velocity u(t, r), and temperature T(t, r) being a part of the local Maxwellian / are unknown. They are calculated via the... 

Here, and are, respectively, the nondimensionalized radial and transverse components of velocity, f is the distribution function for the species being described, and f is the local Maxwellian distribution. Charging of spherical aerosol particles of radius a by gaseous ions is modeled by solving the equation with the surface of the sphere placed at r = a and the force appropriately chosen. The distribution function must then be integrated over r = a to give the ion flux. 

The fact that a local equilibrium assumption is an essential part of the hydrodynamic description of fluid flow suggests that we look for solutions of the Boltzmann equation where the distribution function is close to a local Maxwellian distribution /ie(r, v, t) given by... 

The local equilibrium distribution function // (r, u) can be described by the local Maxwellian with respect to the non-stationary convective flow vo(r, f), that is... 

Note, the local values of the number density n t,r), bulk velocity u (t, r) and temperature T(t, r) being a part of the local Maxwellian are unknown. They are calculated via the function f(t, r, v) in accordance with the definitions (3)-(5). Thus, Eq. (8) together with (11) is non-linear integro-differential equation. The quantity v is the collision frequency assumed to be independent of the molecular velocity. The expression... 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

Of course, the equations are simpler, since they arbitrarily discard that very large class of Maxwellian systems in local thermodynamic c/Asequilibrium with the active vacuum They arbitrarily discard all those Maxwellian systems that are capable of producing electrical circuits and power systems with a coefficient of performance (COP) > 1.0. 

Despite the observational identification of a possible site of dust formation, however, the dust-driven wind could not be applied to stars without dust envelope, as noted in Sect.II. Then, a more interesting possibility is a turbulence-driven wind,in which the high turbulent pressure of the transition layer(or cool corona) pushes the gas out of star,just as the high thermal pressure in corona does in solar-type stars. In fact, if the turbulent zone is extended to about 10 stellar radii, the local escape velocity there may already be small enough to be comparable with the observed flow velocities. Thus, the Maxwellian tail of the turbulent motion in the quasi-static molecular formation zone can directly lead to stellar mass-loss in all... 

Textbooks frequently cite this work as strong empirical evidence for the existence of photons as quanta of electromagnetic energy localized in space and time. However, it has been shown that [8] a complete account of the photoelectric effect can be obtained by treating the electromagnetic field as a classical Maxwellian field and the detector is treated according to the laws of quantum mechanics. 

The principal method of introducing metal impurities into early pinch discharges was considered to be arcing. The externally applied voltages, although low, still permit the occurrence of unipolar arcing between the plasma and the wall when driven by the sheath potential. Local electron emission from a cathode spot is balanced by a uniform flow back to the surface of energetic electrons in the tail of the Maxwellian distribution. 

The distribution function/(v) is Maxwellian at local equilibrium, and is defined by... 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

The initial conditions for the nonzero moments in the Riemann shock problem are shown in Figure 8.9. On the left half of the domain the normalized number density is Moo = 1, and on the right half it is 0.1. Initially the RMS velocity is 1 on both sides of the domain and the mean velocity is null. The moments are initialized as Maxwellian, so that the initial conditions are at local equilibrium. As time increases, a shock wave starting at x = 0 moves to the right and a deflation wave moves to the left. In the limit of t = 0, the solution is the same as for the Euler equation of gas dynamics. Sample results for the moments with T = 100 at time t = 0.5 are shown in Figure 8.10. For this case, the collisions are very weak, and thus there is little transfer of kinetic energy from the M-component to the... 

As might be expected, the model leads to a great simplification over the calculations required for molecules with a continuous potential energy function, as it enables the analysis to be confined to binary collisions and permits the definition of a collision frequency. Because there is no molecular interaction between collisions, the velocity distributions of two colliding molecules may be assumed to be re-established by the time a second collision occurs between them. Thus a Maxwellian distribution around the local mass velocity may be postulated for the calculation of the mean frequency of collision and the average momentum and energy transported per collision in the nonuniform state of the liquid. 

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