Velocity space averages

A. Velocity Distribution Function and Velocity Space Averages. 24... 

If the velocity distribution F(5, x, t) is known, important macroscopic properties of the electrons can be calculated by appropriate velocity space averaging over the distribution. To give some examples, the density n(x, t), the density of the mean energy u (x, t) and the vectorial particle current density j x, t) of the electrons are given by the averages (Desloge, 1966)... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

In the kinetic theory of gases we shall only have to do with mean values, such as time averages, space averages, mean values over all directions, and so on. Individual values entirely elude observation. If Ua denotes the number of molecules per unit volume with a definite property a, e.g. with a velocity of definite magnitude, or with a definite ic-component of velocity, then by the mean value of a we understand the quantity d, where... 

By the reflection method, some relations between the drag force F and the sedimentation velocity U averaged with respect to various orientations of particles of equal radius in space were obtained in [179]. It was assumed that the maximum distance l between the sphere centers is much larger than their radius a. In all considered cases, the drag force is described by formula (2.9.1), where A is the correction coefficient depending on the configuration of the system of particles. In what follows, we write out the correction coefficient for some typical configurations of particles. 

Here, ijJ v)) is the mean and angle averaged value of the local radiation field, weighted with the profile function of the local absorption coefficient. The Aij and Bij are the Einstein coefficients for spontaneous and induced transitions, while denotes the probability for a collisional transition from state j —> i. Accordingly, the first row in eq. (10.20) accounts for spontaneous emission and collision of the molecule considered with H2, whereas in the second row induced emission processes are described. This system of rate equations has to be solved simultaneously with the generalized radiative transfer equation for every point in physical and velocity space. 

The primary purpose of the kinetic treatment of the electron component in anisothermal plasmas is the determination of its velocity distribution function or only its energy distribution. The various macroscopic properties of the electrons can then be obtained fi-om the velocity distribution function by appropriate averages over the velocity space of the electrons. 

In this expansion, the dependence of the velocity distribution F U, vjv, z, t) on the direction v/v is fixed by the Legendre polynomials P v./v). Thus, averages with respect to the angle space v/v over the velocity distribution and appropriate weight functions can be performed. For example, with dv = dv d(v/v), the angle space averages over the velocity distribution F and over the product of F and F, yield according to (5) and (7) the expressions... 

Figure 5.27 Effect of Fjs/Fss ratio on the axial-velocity distribution averaged over interparticle space along the angular coordinate at three different axial positions.

Internal energy per unit volume Internal energy per unit mass Internal energy associated with bead Contribution to equilibrium internal energy Averaged bead velocity Mass-average fluid velocity Species velocity Set of phase-space coordinates Contributions to relative velocity vector Tensor in dumbbell distribution function Tensors m Rouse distribution function Contributions to the a tensor Finger tensor... 

The [[ ]] quantities in these last two equations have the same arguments as the distribution functions with which they are associated. Note that Eqs. (10.5), (10.6), and (10.7) are valid for any species in a multicomponent mixture involving flexible macromolecules. Equation (10.7) is a generalization of the usual equation of continuity for P (r , Q , t) for Rouse chains (DPL, Eqs. (15.1-5)), in that It IS applicable to models of any connectivity and with bead masses and friction coefficients different from one another. One often sees the equation written without the momentum-space averages for the velocities in such instances the equation contains an inappropriate mixture of statistical and deterministic quantities. 

In other words, a velocity can be defined only with respect to a given point in space-time and relative to a particular observer. Since one cannot measure velocity at an infinitesimal point in time or space, any measurement of a rate or a velocity requires averaging the change in the quantity of interest at some point in space over a period of time as well as averaging it over a volume at some point in time. The period of time over which the quantity is averaged determines the time scale and the volume determines the space (or distance scales). We note that the measurement precision of the resulting rate or velocity is directly proportional to the time scale and inversely proportional to the distance scales. But since these time and distance scales may be arbitrarily determined (within certain limiting constraints), the observer can influence the measurement precision and thus influence the power required to measure... 

The space-averaged mass transport and the mass transport distribution to a flat plate electrode as a function of mean linear flow velocity [22, 35-38]. 

Rayleigh convection caused by the density gradient plays the dominant role in the gravitational direction (y direction). Therefore, the time-space averaged velocity in y direction is employed to characterize the Rayleigh convection. 

Figure 8.42 shows the time-space averaged vertical velocity avg versus Ra with different Rcg. The reason that avg is increased with increasing Rcq is due to the gas flow can renew the solute concentration of gas phase at the interface and promote the convection. Therefore, both high liquid concentration and gas flow rate can enhance the volatihzation of acetone. 

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