摘要:

The precise mathematical relation that the Hilbert and Chapman‐Enskog expansions bear to the manifold of solutions of the Boltzmann equation is described. These expansions yield inherently imprecise descriptions of a gas in terms of macroscopic fluid variables instead of a molecular distribution function. It is shown that these expansions are asymptotic to a very special class of solutions of the Boltzmann equation for sufficiently small mean free path. Next, a generalization of the Hilbert and Chapman‐Enskog expansions is described in terms of extended sets of macroscopic state variables, each governed by partial differential equations similar to those found in fluid dynamics, but sufficiently general to approximate an arbitrary distribution function. The generalized expansions are shown to be asymptotic to quite arbitrary solutions of the Boltzmann equation. It is then shown that the ordinary Hilbert and Chapman‐Enskog expansions can also be made asymptotic to very general solutions of the Boltzmann equation by reinterpreting the variables that enter these expansions as certain well‐defined replacements for the actual fluid state of the gas. In this way the scope of the Euler, Navier‐Stokes, Burnett equations, etc., is greatly extended by interpreting them as governing the artificial variables. Not only are general solutions of the Boltzmann equation shown to be approximated by fluid dynamics (in the limit of small mean free path), but the rapid decay of an arbitrary initial distribution function to a special Hilbert distribution function is also governed by sets of partial differential equations similar to those found in fluid dynamics.

ISSN:0031-9171    

DOI:10.1063/1.1706716

出版商:AIP    

年代:1963

数据来源: AIP

摘要:

Attention is called to several mechanisms which effectively couple translational and vibrational motions through the intermediary of chemical reactions. It is proposed that these processes lead to shorter relaxation times than those expected from direct energy exchange via inelastic collisions. Quantitative estimates are made, and relaxation times for these different processes are compared as a function of the reciprocal temperatures. At high temperatures, it is proposed that the transfer of energy from rotations to vibrations at approximately constant total energy can be induced by distant collisions. These serve primarily to balance the change in rotational angular momentum. Such events provide alternate paths for rapid vibrational relaxation, particularly for the upper vibrational levels, and may account for the fact that the observed rates of dissociation are considerably higher than the rates calculated under the assumption that these are limited by vibrational equilibration via inelastic collisions.

ISSN:0031-9171    

DOI:10.1063/1.1706717

出版商:AIP    

年代:1963

数据来源: AIP

摘要:

Manometric measurements of the electrohydrostatic pressure distribution in dielectric liquids permit determination of space charge and field distribution, and of carrier generation rate.

ISSN:0031-9171    

DOI:10.1063/1.1706718

出版商:AIP    

年代:1963

数据来源: AIP

摘要:

Small perturbations of plane Couette flow in stably and unstably stratified fluid are considered. It is found that the system is more unstable when it is bounded both above and below than when its depth is infinite, but a finite negative Richardson numberJis required to maintain the perturbation for both cases. For the former case, this limiting Richardson number is−3k12/4(k12+ k22),while for the latter it is−2k12/(k12+ k12),wherek1andk2are wavenumbers in the mean flow and the transversal direction. These results show that in an unstably stratified layer of Couette flow, the preferred mode of motion is roll‐type convection, with the dimension in the wind direction much larger than in the transversal direction. The amplification factor &sgr; for the perturbations has been determined as a function of the modified Richardson numberJ¯ = gSzUz−2(1 + k1−2k22),and the dimensionless wavenumber&agr; = h(k12+ k12)1/2.Four different regimes have been found, each corresponding to a different type of perturbation. An application of the theory is made to the formation of longitudinal cloud rolls observed in the earth's atmosphere and in certain laboratory experiments.

ISSN:0031-9171    

DOI:10.1063/1.1706719

出版商:AIP    

年代:1963

数据来源: AIP

摘要:

An approximate solution of the equation which governs the inviscid modes of instability of steady flow between concentric rotating cylinders is derived. The problem is treated in the narrow‐gap approximation, and the solution is applicable over the entire range of opposite rotation rates, but is limited to the treatment of disturbances which are nearly axially symmetric. Numerical results are given for the limiting cases where the ratio of angular velocities of the cylinders approaches zero and minus infinity. It is found that the growth rates of unstable modes which are very nearly symmetric are always less than the growth rate of the corresponding axisymmetric mode.

ISSN:0031-9171    

DOI:10.1063/1.1706720

出版商:AIP    

年代:1963

数据来源: AIP

摘要:

Formal solutions are found to the linearized Boltzmann equation for the initial‐value problem. These are decomposed into an infinity of modes, which are orthogonal under a suitable inner product. All but five of these modes exhibit an exponential time decay. These five remaining modes form a generalization of hydrodynamics. Under two different asymptotic assumptions one finds quantitative solutions. If the characteristic wavelength of the initial disturbance is large compared to the mean free path, the solution appears as an expansion in their ratio. If the elapsed time is large compared to the mean free time, the solution may be represented as an expansion in their ratio. As a specific example of the theory, the fundamental solution of the one‐dimensional shear‐free initial value problem is computed. This appears as an infinity of diffusing modes, a subclass of which also propagate.

ISSN:0031-9171    

DOI:10.1063/1.1706721

出版商:AIP    

年代:1963

数据来源: AIP

摘要:

A method for studying the behavior of fields by splitting their behavior into independent field modes is presented. The method is used to explore the characteristics of steady, two‐dimensional, linearized magnetohydrodynamic fields with finite viscosity and resistivity and arbitrary orientation of the magnetic vector relative to the velocity vector.It is shown that in general boundary layers and wakes cease to exist in magnetohydrodynamics. Their place is taken by diffusing waves which, in reality, are the fields of a set of viscous‐resistive sources, vortices, poles and currents whose field lines are strongly oriented along the characteristic wave directions. When the viscosity and resistivity are equal, these waves diffuse in a simple and independent way, but when these quantities are not equal, the diffusing waves generate a new kind of wake which is located, veil‐like, in the fan‐shaped region between the two wave directions. These wakes are fed from the differential diffusion of the primary waves. In the special case for which the resistivity is much greater than the viscosity, a new type of pseudo boundary layer is shown to exist in the velocity field. When the viscosity is much greater than the resistivity, this pseudo boundary layer occurs in the magnetic field.

ISSN:0031-9171    

DOI:10.1063/1.1706722

出版商:AIP    

年代:1963

数据来源: AIP

摘要:

A collisionless plasma in equilibrium with a slightly nonuniform magnetic field is studied to determine its stability against growing low‐frequency oscillations whose propagation vector is perpendicular to the magnetic field. The method employs the collisionless Boltzmann equation and Maxwell's equations to obtain the eigenfrequencies of the oscillations. Previous work on this problem was based on a longitudinal approximation, i.e., the oscillation was assumed to have electric vector parallel to its direction of propagation. The present work does not make this approximation and finds: (i) a mode which corresponds to the mode found in longitudinal approximation, with the same eigenvalue as given by that approximation; (ii) modes with polarization similar to the ``longitudinal'' mode which are purely oscillatory; (iii) a purely transverse mode which is unstable. These results are compared with magnetohydrodynamic and other approximations.

ISSN:0031-9171    

DOI:10.1063/1.1706723

出版商:AIP    

年代:1963

数据来源: AIP

摘要:

Nonlinear effects from Vlasov's equation are studied for a one‐dimensional electron gas with fixed neutralizing background. The electron distribution function is expanded in Hermite polynomials and a closure condition is introduced to cut off the resulting infinite system of differential equations for the expansion coefficients. Then a nonlinear partial differential equation can be derived for the electric field. The consequences of some different closure conditions are studied. It turns out that their use implies severe restrictions on the physical situations described. In the discussed closed polynomial approximations no nonlinear instabilities are found. In particular, it results that with vanishing heat flux one has only stable linear or nonlinear oscillations showing that unstable situations are necessarily nonadiabatic. In an analogous manner, the condition that the plasma follows a state law with pressure proportional to density, like a perfect gas, eliminates unstable modes.

ISSN:0031-9171    

DOI:10.1063/1.1706724

出版商:AIP    

年代:1963

数据来源: AIP

摘要:

The angular dependence of the absorption coefficient of cyclotron radiation is obtained from the Vlasov equation and evaluated numerically, and the total radiation from both slab and cylindrical plasmas is calculated. The critical harmonicm* (below which the radiation is effectively contained) is presented graphically as a function ofTe, and the combination &bgr;eBL/(1−R). The results for the total radiation are in agreement with the earlier results of Trubnikov and Kudryavtsev. The energy production in a thermonuclear reactor is computed for several different models and compared with the cyclotron radiation loss to find the critical size. The critical size depends on (1/&bgr;e)3and for &bgr; = &bgr;e+ &bgr;i= 0.40, the critical size is moderate (less than a meter) for all but the most unfavorable assumptions. For smaller &bgr; reflectors must be used to obtain reasonable critical sizes.

ISSN:0031-9171    

DOI:10.1063/1.1706725

出版商:AIP    

年代:1963

数据来源: AIP