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Dec 1970

Volume 13, Issue 12, pp. 2881-3068

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Contributions to Nonequilibrium Thermodynamics. II. Fluctuation Theory for the Boltzmann Equation

Ronald Forrest Fox and George E. Uhlenbeck

Phys. Fluids 13, 2881 (1970); http://dx.doi.org/10.1063/1.1692878 (10 pages) | Cited 49 times

Online Publication Date: 8 August 2003

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The theory of the nonequilibrium phenomena in a dilute gas as described by the Boltzmann equation is extended in order to also include the fluctuations of the distribution function of the molecules. It is shown that in a first approximation this extension leads to the Landau‐Lifshitz fluctuation terms in the hydrodynamical equations.

Navier‐Stokes Initial Value Problem for Boundary‐Free Incompressible Fluid Flow

Gerald Rosen

Phys. Fluids 13, 2891 (1970); http://dx.doi.org/10.1063/1.1692879 (13 pages) | Cited 10 times

Online Publication Date: 8 August 2003

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Convergence proofs are reported for a general local iteration solution to the Navier‐Stokes initial value problem and estimates of the accuracy of the nth iterative approximation are derived. Without appeal to methods of functional analysis, it is shown that a Kiselev‐Ladyzhenskaya weak solution is, in fact, a classical solution. Any one of three alternative conditions on the initial velocity field is found to be sufficient to guarantee the existence of a global solution. Breakdown phenomenon which may prevent a local solution from being continued for all t  ≥  0 to a global solution is analyzed. The mathematical theory suggests that breakdown is precluded for a suitably smooth initial velocity field, irrespective of the over‐all initial velocity field magnitude.

Stability of Steady Flow in a Diverging Channel

C. Nakaya and E. Hasegawa

Phys. Fluids 13, 2904 (1970); http://dx.doi.org/10.1063/1.1692880 (3 pages) | Cited 1 time

Online Publication Date: 8 August 2003

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An angle formed by two intersecting planes and maximum velocity are chosen as two characteristic parameters for steady flow. It is shown that for a given maximum velocity there exists a certain critical angle at which steady perturbation appears.

Some Results on the Nonoscillation of Salt Fingers

Chia‐Shun Yih

Phys. Fluids 13, 2907 (1970); http://dx.doi.org/10.1063/1.1692881 (5 pages) | Cited 2 times

Online Publication Date: 8 August 2003

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Salt‐finger convection in water discovered by Stommel, Arons, and Blanchard is discussed, and conditions under which the principle of exchange of stabilities holds are given.

Decay of Weak Turbulence

S. C. Ling and T. T. Huang

Phys. Fluids 13, 2912 (1970); http://dx.doi.org/10.1063/1.1692882 (13 pages) | Cited 34 times

Online Publication Date: 8 August 2003

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A detailed experimental investigation of the structure of weak turbulence generated by various single and multiple‐stage grids has been made. The weak turbulent fields were limited to those having no interaction with the linear momentum wake of the grid. The present study covers a range of Reynolds numbers of turbulence between 70 and 3. Components of turbulence were found to have equipartition of energy within this range. When the Reynolds number of turbulence is <30, the kinetic energy of turbulence decreases inversely as the square of decay time. All the length scales of turbulence increase as the square root of time, while the Reynolds number of turbulence decreases as the square root of time. Both the measured spectral and correlation functions show self‐preserving forms. The longitudinal correlation function is closer to a Cauchy than to a Gaussian distribution function. The resultant three‐dimension energy spectrum shows a weak but nonvanishing transfer of energy from low to high wavenumbers.

Influence of High‐Pass Filtering on Third‐Order Correlation Measurements

K. N. Helland and G. R. Stegen

Phys. Fluids 13, 2925 (1970); http://dx.doi.org/10.1063/1.1692883 (7 pages) | Cited 6 times

Online Publication Date: 8 August 2003

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An experimental investigation has been made of the influence of high‐pass filtering on third‐order correlations in grid turbulence. It has been found that for sufficiently small cutoff frequencies, the third‐order correlations remain unchanged by high‐pass filtering.

Finite‐Amplitude Instability of the Compressible Laminar Wake. Strongly Amplified Disturbances

J. T. C. Liu and Lester Lees

Phys. Fluids 13, 2932 (1970); http://dx.doi.org/10.1063/1.1692884 (7 pages) | Cited 8 times

Online Publication Date: 8 August 2003

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The interaction between mean flow and finite‐amplitude disturbances in certain experimentally observed unstable, compressible laminar wakes is considered theoretically without explicitly assuming small amplification rates. Boundary‐layer form of the two‐dimensional mean‐flow momentum, kinetic energy and thermal energy equations and the time‐averaged kinetic energy equation of spatially growing disturbances are recast into their respective von Kármán integral form which show the over‐all physical coupling. The Reynolds shear stresses couple the mean flow and disturbance kinetic energies through the conversion mechanism familiar in low‐speed flows. Both the mean flow and disturbance kinetic energies are coupled to the mean‐flow thermal energy through their respective viscous dissipation. The work done by the disturbance pressure gradients gives rise to an additional coupling between the disturbance kinetic energy and the mean‐flow thermal energy. The compressibility transformation suggested by work on turbulent shear flows is not applicable to this problem because of the accompanying ad hoc assumptions about the disturbance behavior. The disturbances of a discrete frequency which corresponds to the most unstable fundamental component, are first evaluated locally. Subsequent mean‐flow and disturbance profile‐shape assumptions are made in terms of a mean‐flow‐density Howarth variable. The compressibility transformation, which cannot convert this problem into a form identical to the low‐speed problem of Ko, Kubota, and Lees because of the compressible disturbance quantities, nevertheless, yields a much simplified description of the mean flow.

First‐Order Treatment of Higher‐Order Boundary‐Layer Effects

Clark H. Lewis

Phys. Fluids 13, 2939 (1970); http://dx.doi.org/10.1063/1.1692885 (11 pages)

Online Publication Date: 8 August 2003

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The effects of transverse curvature, shock‐generated external vorticity, boundary‐layer displacement, and wall slip and temperature jump are considered as first‐order boundary‐layer effects. The classical boundary‐layer equations were modified to include the higher‐order effects, and flows over a 9‐deg half‐angle blunt cone were considered at M∞  =  9 and 18. Comparisons are made with second‐order theory and experimental data. Primary interest is given to predicting the higher‐order effects on zero‐lift drag and comparsion with experimental data. Range of applicability of higher‐order boundary‐layer theory is indicated based upon the ability to predict zero‐lift drag. Vorticity was the dominant higher‐order effect, and the theory is most applicable to relatively short slender bodies. At very low Reynolds numbers, strong coupling of the higher‐order effects was found to exist.

Traveling Waves of Arbitrary Amplitude in Compressible Hydrodynamics under Gravity: An Exact Solution

Y. T. Chiu

Phys. Fluids 13, 2950 (1970); http://dx.doi.org/10.1063/1.1692886 (8 pages) | Cited 11 times

Online Publication Date: 8 August 2003

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Exact traveling‐wave solutions to the hydrodynamic equations for an isothermal atmosphere stratified by a uniform gravitational field are obtained in two Cartesian spatial coordinates and one temporal coordinate and represent simple Riemannian waves of arbitrary amplitude and wave form which reduce directly into acoustic waves in the absence of gravity. While a number of characteristics are similar to the theory of small amplitude acoustic‐gravity waves, these solutions exhibit features such as shock formation and phase dispersion which cannot be treated in small‐amplitude theory. Solutions are given in terms of relations between physical wave characteristics and hydrodynamic variables. Comparison with traveling atmospheric disturbances are proposed.

Viscous Flow through Porous Media. III. Upper Bounds on the Permeability for a Simple Random Geometry

Harold L. Weissberg and Stephen Prager

Phys. Fluids 13, 2958 (1970); http://dx.doi.org/10.1063/1.1692887 (8 pages) | Cited 35 times

Online Publication Date: 8 August 2003

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The variational treatment developed previously is applied to a porous medium constructed of randomly overlapping spheres. Upper bounds on the permeability are obtained for flow both in the absence and in the presence of slip. In the limit of zero solid fraction, these bounds approach the exact permeability in each case.

Electrostatic Probe in a Reacting Gas

G. F. Carrier and F. E. Fendell

Phys. Fluids 13, 2966 (1970); http://dx.doi.org/10.1063/1.1692888 (17 pages) | Cited 6 times

Online Publication Date: 8 August 2003

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A spherical perfectly catalytic conductor with negative potential bias lies in a quiescent slightly ionized gas whose density is large enough to warrant the use of continuum theory. The Debye number ϵ, the ratio of the Debye length to the conductor radius, is small. Principal attention is given to the case, amenable to asymptotic analysis, of probe potential ψp > 0(ln ϵ). The relation between the current collected by the probe and the electrode potential is sought; the current virtually saturates (increases very slowly relative to increases in applied potential difference) for ψp ≫ 0(ln ϵ). A reconstruction of the chemically frozen case with three novel features is furnished: a transformation of variables that renders the transition‐zone equations almost uniformly valid; a very accurate, easily obtained closed‐form approximation to the solution of the resulting boundary‐value problem; and a current‐potential relationship that is derived on the basis of the gross behavior of this approximate closed‐form solution. Then the increase in current is determined for the case of finite‐rate gas‐phase ionization and recombination in which the negative‐charge carrier is an electron and in which an electron does not serve as the third body for recombination. It is found that ionization can play a role in a thick sheath in addition to the more conventional effect of both ionization and recombination in the quasineutral region. When an electron serves as the third body for recombination, the sheath is frozen.

Properties of High Current Free‐Burning Arcs in Air

Leland M. Nicolai

Phys. Fluids 13, 2983 (1970); http://dx.doi.org/10.1063/1.1692889 (4 pages) | Cited 6 times

Online Publication Date: 8 August 2003

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The measured characteristic curves for free‐burning arcs in air at pressures from 0.019 to 1.0 atm and high currents from 190 to 400 A are presented. The experimental results show that the characteristic curves are flat (E  =  const) at these high currents. Characteristic arc dimensions were determined from arc photographs. The results show the arc diameter to be proportional to I½. Heated solid cylinder heat transfer ideas are applied to the free‐burning arc data and the functional form for the Nusselt number in terms of the Grashof and Prandtl number is determined.

Electrohydrodynamic Stability: Fluid Cylinders in Longitudinal Electric Fields

D. A. Saville

Phys. Fluids 13, 2987 (1970); http://dx.doi.org/10.1063/1.1692890 (8 pages) | Cited 58 times

Online Publication Date: 8 August 2003

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The stability of a fluid cylinder, stressed by an axial electric field, has heretofore been studied assuming that the cylinder and its surroundings behave as perfect dielectrics and viscous effects are ignored. With many fluids this is unrealistic since the existence of even a small conductivity implies that the deformed interface carries an electric charge. Upon deformation of the interface the charge interacts with the field to produce an electrical shearing stress. Thus, to avoid singular behavior viscous shear must be considered from the outset. The analysis is applicable to situations where the relaxation time for free charges is short compared with the time scale for fluid motion. It is found that electrical shearing forces can, under some conditions, completely stabilize the cylindrical interface to axisymmetric deformations. On the other hand, the conditions under which these same forces produce instability are delineated. In instances where both fluids have low viscosities a boundary layer effect produces additional damping and, in the presence of an axial electric field, electrical shearing stresses in this boundary layer may render an otherwise stable oscillation unstable.

General Solution for the Linearized Ekman‐Hartmann Layer on a Spherical Boundary

David E. Loper

Phys. Fluids 13, 2995 (1970); http://dx.doi.org/10.1063/1.1692891 (4 pages) | Cited 11 times

Online Publication Date: 8 August 2003

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A general solution for the linearized, hydromagnetic Ekman‐Hartmann boundary layer on a spherical boundary is presented. The fluid is assumed to be incompressible, viscous, electrically conducting, and in a state of motion close to rigid body rotation. The conductivity of the boundary is unspecified and an arbitrary magnetic field permeates the fluid and boundary. The solution obtained is found to be independent of both the conductivity of the boundary and the magnetic field components parallel to the boundary. If the magnetic field is assumed to be frozen into the fluid outside the boundary layer, the general solution may be integrated for large time to obtain a stationary solution. It is found that hydromagnetic effects preclude the possibility of boundary‐layer breakdown whenever a local normal magnetic field exists.

Steady Hydromagnetic Boundary Layer near a Rotating, Electrically Conducting Plate

David E. Loper

Phys. Fluids 13, 2999 (1970); http://dx.doi.org/10.1063/1.1692892 (4 pages) | Cited 17 times

Online Publication Date: 8 August 2003

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Gilman and Benton's study of the steady fluid motions and magnetic fields induced by a differentially rotating, electrically insulating infinite flat plate in the presence of a uniform magnetic field is generalized by allowing the electrical conductivity of the plate to be an arbitrary function of distance from the fluid‐plate interface. It is found that the important measure of the conductivity of the plate is the ratio of plate conductance to conductance of a layer of fluid whose thickness is one Ekman depth. The form of the steady Ekman‐Hartmann layer is found to be completely independent of the conductivity of the plate. The steady perturbation magnetic field within the plate is found to be entirely azimuthal and its magnitude is directly proportional to the conductance ratio. For values of parameters approximating conditions within the earth, the perturbation field may not be small. This azimuthal field causes an axial electric current to be drawn into the plate from the fluid, effectively strenghtening the coupling between fluid and plate. If the conductance ratio is large, as it is within the earth, spin‐up is accomplished primarily by this current and the Ekman‐Hartmann layer no longer plays a dominant role in coupling the fluid with its boundary.

Ion Cyclotron Instability

Erich S. Weibel

Phys. Fluids 13, 3003 (1970); http://dx.doi.org/10.1063/1.1692893 (4 pages) | Cited 26 times

Online Publication Date: 8 August 2003

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Electrostatic ion cyclotron waves are investigated for a plasma consisting of two groups of counter‐streaming ions in a background of stationary electrons. Under certain conditions such waves are weakly amplified. Analytic and numerical conditions for instability are presented.

Ordinary‐Mode Electromagnetic Instability in Counterstreaming Plasmas with Anisotropic Temperatures

M. Bornatici and Kai Fong Lee

Phys. Fluids 13, 3007 (1970); http://dx.doi.org/10.1063/1.1692894 (5 pages) | Cited 47 times

Online Publication Date: 8 August 2003

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The instability of the electromagnetic linearly polarized mode propagating perpendicular to a uniform magnetic field is studied by using the Vlasov equation for a counterstreaming electron plasma with anisotropic temperatures. An instability occurs if the streaming velocity exceeds a certain threshold value which can be below that required to excite the electrostatic two‐stream instability. It is found that temperature perpendicular to the field has a stabilizing effect, while parallel temperature enhances the electromagnetic instability. Typical growth rates are of the order of the electron cyclotron frequency.

Lyapunov Stability of an Inhomogeneous Two‐Stream Plasma

Robert M. Chervin and Amiya K. Sen

Phys. Fluids 13, 3012 (1970); http://dx.doi.org/10.1063/1.1692895 (8 pages)

Online Publication Date: 8 August 2003

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Lyapunov's direct method has been adapted to the study of an electrostatic instability of a plasma by using Zubov's extension of the method to distributed parameter systems. Specifically, the linear stability of two inhomogeneous interpenetrating electron streams in a finite one‐dimensional system is considered. Proper Lyapunov functionals are constructed for both stability and instability. It is seen that, as in the homogeneous case, the stability or instability depends on whether or not the thermal speed is greater or less than the streaming speed. In contrast to the methods of energy principle, the complete generality of Lyapunov's direct method is discussed.

Thermal Equilibrium and Stability of Tokamak Discharges

H. P. Furth, M. N. Rosenbluth, P. H. Rutherford, and W. Stodiek

Phys. Fluids 13, 3020 (1970); http://dx.doi.org/10.1063/1.1692896 (11 pages) | Cited 35 times

Online Publication Date: 8 August 2003

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Steady‐state temperature and magnetic field profiles are derived in cylindrical geometry, including classical electron‐ion equilibration, Ohmic heating, and the classical ion thermal transport appropriate to tokamak geometry. To fit the experimental data, a direct electron heat loss must accompany any appreciable resistivity anomaly. The classical equilibrium can be thermally stable only if Ti/Te ≳ 2/3. Direct electron heat loss by radiation is further destabilizing; but for typical parameters, stability can be achieved by a moderate amount of anomalous electron thermal conduction. The growth time of thermal instabilities is generally limited by the resistive skin time. The presence of a runaway component of the current allows shorter growth times, and this may be related to the present high‐density limit in tokamaks.

Explosive Plasma Instabilities

J. Fukai, S. Krishan, and E. G. Harris

Phys. Fluids 13, 3031 (1970); http://dx.doi.org/10.1063/1.1692897 (7 pages) | Cited 24 times

Online Publication Date: 8 August 2003

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The interaction of three monochromatic waves (two positive energy longitudinal waves and one negative energy wave) is investigated. When only the first‐order nonlinear interactions are considered, an explosive instability occurs. However, this instability is stabilized by the higher order nonlinear interactions. The matrix elements of the three‐wave and the four‐wave interactions in a magnetized plasma are calculated.

High‐Frequency Flutelike Instabilities in Multicomponent Plasmas

G. E. Guest, W. M. Farr, and R. A. Dory

Phys. Fluids 13, 3038 (1970); http://dx.doi.org/10.1063/1.1692898 (3 pages) | Cited 6 times

Online Publication Date: 8 August 2003

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Plasma in a magnetic field B is subject to unstable electrostatic gyroharmonic waves propagating across B, if the distribution of f0 of particle velocities v⊥(⊥ to B) has a region where ∂f0/∂v⊥ > 0. Stability limits are found for f0 representative of mirror‐confined plasmas having a “warm” Maxwellian species, an otherwise identical “hot” species with a region of positive ∂f0/∂v⊥, and a neutralizing background. The waves are stable if the ratio of mean energies of the warm to hot species exceeds 0.1, or if the ratio of densities does not fall in an unstable interval roughly characterized by 0.01 ≲ NW/NH ≲ 1, with more precise criteria given by Eqs. (5) and (6). It is important for fusion experiments that the conditions necessary for stability are less restrictive than the sufficient condition: ∂f0/∂v⊥ < 0 for all the species.

Ion Distribution Functions in Collisionless Surface Ionized Plasmas

J. M. Buzzi, H. J. Doucet, and D. Gresillon

Phys. Fluids 13, 3041 (1970); http://dx.doi.org/10.1063/1.1692899 (9 pages) | Cited 42 times

Online Publication Date: 8 August 2003

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For a low‐density surface‐ionized plasma (e.g., in a Q machine) where Coulomb or binary collisions are negligible, a simple steady‐state model of the plasma is obtained through the Vlasov and Poisson equations. Among the main predictions of this model, the electron‐rich regime involves a truncated Maxwellian distribution function for the ion velocities parallel to the magnetic field. Experiments have been carried out in a short single ended Q machine for a cesium plasma. Curves of density versus neutral flux are plotted. With the help of a heated probe and an electrostatic analyzer, whose work functions are carefully taken into account, the ion distribution functions are deduced. In our observations they correspond to beams of 1 km∕sec to 1.5 km∕sec mean velocity and velocity spread which can be characterized by a parallel temperature of 150‐300° K. These experiments and the theoretical model are in good agreement.
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Circulation‐Preserving Flow of an Inviscid Prim Gas

S. L. Passman

Phys. Fluids 13, 3050 (1970); http://dx.doi.org/10.1063/1.1692900 (2 pages)

Online Publication Date: 8 August 2003

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It is shown that, in the circulation‐preserving flow of an inviscid Prim gas, the entropy depends on the pressure and time only. In the special case where the entropy is steady and there is no heat flux, the pressure is constant along each particle path.

Density Field near a Wedge in Rarefied Flow

R. S. Hickman

Phys. Fluids 13, 3051 (1970); http://dx.doi.org/10.1063/1.1692901 (2 pages)

Online Publication Date: 8 August 2003

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The density field near a 10° wedge in a Mach 9.4 rarefied flow of nitrogen and a flow field picture obtained with the electron beam fluorescence technique are presented. The geometrical resolution is less than the hard sphere mean free path.

Propagation of Ion Waves in a Radio Frequency Electric Field

S. Takamura, S. Aihara, and K. Takayama

Phys. Fluids 13, 3052 (1970); http://dx.doi.org/10.1063/1.1692902 (3 pages) | Cited 6 times

Online Publication Date: 8 August 2003

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See Also: Erratum

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The wavelength of an ion wave decreases in an rf electric field. This phenomenon is explained by the additional pressure which is produced by the spatial modulation of the rf field by the ion wave.
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