Velocity distribution spatially dependent

A gas is not in equilibrium when its distribution function differs from the Maxwell-Boltzman distribution. On the other hand, it can also be shown that if a system possesses a slight spatial nonuniformity and is not in equilibrium, then the distribution function will monotonically relax in velocity space to a local Maxwell-Boltzman distribution, or to a distribution where p = N/V, v and temperature T all show a spatial dependence [bal75]. 

The above derivation assumes straight streamlines and a monotonic velocity profile that depends on only one spatial variable, r. These assumptions substantially ease the derivation but are not necessary. Anal5dical expressions for the residence time distributions have been derived for noncircular ducts,... 

Velocity-encoding 2D NMR imaging methods characterize general patterns of spatial velocity distributions and directly visualize different characteristics of flow behavior depending on the properties of the materials and operating param-... 

Consider now how the solution obtained in the quasi-static approximation changes if the second-order group velocity dispersion is taken into account. It is seen from Fig. 29 that in a nonlinear waveguide of the structure A with the MD and the SS effects is first observed that spatial and temporal parameters of the field vary similarly to the case of the quasi-static approximation. Then, at a given power, spatiotemporal distribution varies depending on the value and sign of the dispersion coefficient 2- The pulse duration decreases in the case of anomalous GVD (k2 < 0) and increases in the case of normal GVD ( 2 > 0). 

However, before embarking on this analysis, a brief excursion is of interest [285]. Let us specifically ignore the spatial dependence of the distribution of the N particles. It could have been calculated from the deterministic Newtonian equations of motion. Now considering, in particular, the motion of particles 1 and 2 as above, average the distribution over all position of the N particles to give the velocity distribution function... 

In conclusion, droplet size measurements in the range 10 to 100 m can be performed, also in hostile environments, from the visibility of individual scattered signal. Advantages of this method are simultaneous measurement of particle size, concentration and velocity no calibration is necessary good spatial resolution up to less than 1 mm-3 the visibility is independent on particle trajectory. Limitations are individual scattered signal can be obtained only with moderate particle concentration it is difficult to automatically process scattered signals to extract the visibility value and to check validation conditions it seems very difficult to extend the technique to cover the entire spray distribution the lower limit in the small particle end of the distribution curve depends upon experimental sensitivities and V(d) curve flatness... 

Note that this assumption simply transforms the problem of modeling the pair correlation function into the new problem of modeling o-The usual model for go assumes that the radial distribution function depends neither explicitly on the collision angle (i.e. on X12) nor explicitly on x. The former amounts to assuming that the particle with velocity V2 has no preferential spatial direction relative to the particle with velocity vi. The radial distribution function can then be modeled as a function of the disperse-phase volume fraction. For example, a typical model is (Carnahan Starling, 1969)... 

On the basis of the space-dependent two-term approximation, including elastic and conservative inelastic electron collision processes, substantial aspects of the inhomogeneous electron kinetics, such as the spatial relaxation behavior in uniform electric fields and the response of the electron component to spatially limited pulselike field disturbances, have been demonstrated and the complex mechanism of spatial electron relaxation has been briefly explained. In these cases, starting from a specific choice of the boundary condition for the velocity distribution, the succeeding spatial evolution of the electrons in the field acceleration direction up to their establishment of a steady state has been studied. 

Furthermore, the space-dependent two-term solution approach could be extended to a multiterm solution approach (Petrov and Winkler, 1997) to the space-dependent kinetic equation. By this extension, it becomes possible to accurately describe the spatial evolution of the electron component under conditions of larger anisotropy in the velocity distribution of the electrons. 

IV. Spatially dependent velocity distributions. When the spectrum is independent of position, the central problem is the determination of the energy-transfer cross sections. The calculation of the spectrum once these cross sections are known is a straightforward procedure. The cross-section aspect of the problem is both more difficult from a physics point of view, and more time consuming from the point of view of machine computation. This situation is reversed when we come to consider the spatial dependence of the slow neutron spectrum. The cross sections needed are the same ones that already have been computed for the infinite medium spectrum problem. The transport equation must now be solved in at least two variables, and in a form for which the existing approximate techniques are not very well adapted. The focus of the problem therefore shifts to the development of appropriate techniques for solution of the transport equation when the energy and position variables are coupled in such a way that neutrons can both gain and lose energy in a collision. 

Determination of the spatial dependence of the (one-point joint velocity-composition) micro-PDF, f, or its moments is often difficult, and in many cases only the distribution of properties (pA in the entire volume, that is, at the reactor scale, is desired. The volume integral or macro-average of (12.4.1-11) for statistically stationary flow is ... 

Here, n is the number density of the gas atoms and /m is the Maxwellian velocity distribution function, Eq. (2.152). We notice that the spatial dependence of the contribution from the arriving atoms follows the spatial variation of the external field, whereas that is not the case for the contribution of the scattered atoms because of the exponential term in the curly brackets. In other words, the optical response of the gas near the surface is nonlocal. 

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