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Dec 1966

Volume 9, Issue 12, pp. 2309-2544

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Linearized Initial Value Problem for a Gas

John K. Buckner and Joel H. Ferziger

Phys. Fluids 9, 2309 (1966); http://dx.doi.org/10.1063/1.1761619 (6 pages) | Cited 11 times

Online Publication Date: 9 December 2004

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A complete set of eigenfunctions of the linearized modeled kinetic theory Boltzmann equation appropriate to the initial value problem is constructed. These functions may be used to solve the initial value problem with arbitrary initial conditions and may find application in the construction of the correlation functions required for the analysis of slow‐neutron scattering experiments.

Linearized Boundary Value Problem for a Gas and Sound Propagation

John K. Buckner and Joel H. Ferziger

Phys. Fluids 9, 2315 (1966); http://dx.doi.org/10.1063/1.1761620 (8 pages) | Cited 33 times

Online Publication Date: 9 December 2004

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A complete set of eigenfunctions of the linearized modeled kinetic theory Boltzmann equation appropriate to the boundary value problem is constructed. These functions are then used to solve a boundary value problem which contains the essential features of sound propagation experiments. The calculated pressure field correctly gives the hydrodynamic solution as its high‐density limit and modified free flow behavior as its low‐density limit. Even for very simple models the results are in remarkably good agreement with experiments.

Mixtures of Maxwell Molecules

Lawrence Sirovich

Phys. Fluids 9, 2323 (1966); http://dx.doi.org/10.1063/1.1761621 (4 pages) | Cited 17 times

Online Publication Date: 9 December 2004

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The linearized collision integrals for mixtures of Maxwell molecules are considered. A simple proof of the eigentheory is given. Relations for and amongst eigenvalues are obtained. A brief discussion of the corresponding relaxation theory is also included.

Rarefied Gas Channel Flows for Three Molecular Models

A. B. Huang and R. L. Stoy

Phys. Fluids 9, 2327 (1966); http://dx.doi.org/10.1063/1.1761622 (10 pages) | Cited 13 times

Online Publication Date: 9 December 2004

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An analytic solution of the Boltzmann equation for the problem of a rarefied gas flowing between two parallel, infinite plates is developed for three different molecular models. They are the (1) Bhatnagar‐Gross‐Krook model, (2) hard‐sphere model, and (3) Maxwellian model. The half‐range moment method, which is used in the analysis for the three models, yields satisfactory results (with the expected minimum in the volume flow rate) for the transition, slip, and continuum regions. For the Bhatnagar‐Gross‐Krook model, the Willis iteration method provides good results for the near‐free‐molecular region even when, as a first guess, the first approximation by the half‐range moment method is used. The comparisons between the analytic results and the experimental data indicate that the Bhatnagar‐Gross‐Krook model is a satisfactory approximation to the collision integral for the purpose of this study, and that the Maxwellian model provides an upper limit for the volume flow rate, and the hard‐sphere model, a lower limit.

Effects of Turbulence on Laminar Skin Friction and Heat Transfer

Mahlon C. Smith and Arnold M. Kuethe

Phys. Fluids 9, 2337 (1966); http://dx.doi.org/10.1063/1.1761623 (8 pages) | Cited 65 times

Online Publication Date: 9 December 2004

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Experiments were performed in two low‐turbulence wind tunnels (u∞′ < 0.1%) to determine the effects of turbulence on heat transfer from plates and circular cylinders in incompressible flow. Grid turbulence up to 6% was imposed. Heat transfer was increased about 30% in the laminar region of a flat plate and up to 70% on a circular cylinder; smaller though still significant increases in shearing stress at the wall were measured by hot wires near the surface. A phenomenological theory is given which shows good agreement with the experiment.

Test Time and Particle Paths in Low‐Pressure Shock Tubes

J. N. Fox, T. I. McLaren, and R. M. Hobson

Phys. Fluids 9, 2345 (1966); http://dx.doi.org/10.1063/1.1761624 (6 pages) | Cited 14 times

Online Publication Date: 9 December 2004

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Measurements are reported on shocked‐gas‐flow duration in small‐diameter shock tubes at low initial pressures. They are correlated with expressions derived from Mirels' theory for such flows, and substantial agreement is obtained, with closer correlation in the smaller‐diameter shock tube. An expression is developed from Mirels' theory to relate particle and laboratory time coordinates, and a direct experimental measurement of particle paths in the flow is described. The time transformation introduces a significant correction to the ρ2∕ρ1 linear transformation more generally used, and this is seen to be important in investigations of relaxation processes and rate kinetic measurements carried out in shock tubes. Some of the values of rate coefficients previously reported in the literature may well require revision in the light of this correction.

Shock‐Wave Structure in a Gas Containing Ablating Particles

Ali Hasan Nayfeh

Phys. Fluids 9, 2351 (1966); http://dx.doi.org/10.1063/1.1761625 (6 pages) | Cited 8 times

Online Publication Date: 9 December 2004

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The one‐dimensional equations governing the structure of a shock wave in a mixture of a gas and an ablating dust using the transfer properties for free molecular flow are derived. The analysis takes into account the simultaneous effects of the solid particle drag, convective heat transfer, and ablation. These equations are programmed for numerical solution on an IBM 7094. The output of the program is flow properties as functions of the distance behind the shock. The results show that the relaxation distance decreases as the dust mass fraction increases. The pressure, velocity, and temperature in the equilibrium region (i.e., where the dust is completely vaporized, but not chemically reacted) are also obtained by globally conserving mass, momentum, and energy. These results are in reasonable agreement with those obtained from the numerical integration of the full equations. The equilibrium values are applied to the flow in a shock tube. The results show that the presence of the dust increases the driving pressure ratio needed in a shock tube to attain a given shock speed.

Pressure Wave Generation in a Fissioning Gas. IV. Analysis by the Method of Characteristics

Walter N. Podney, Harold P. Smith, and A. K. Oppenheim

Phys. Fluids 9, 2357 (1966); http://dx.doi.org/10.1063/1.1761626 (8 pages) | Cited 2 times

Online Publication Date: 9 December 2004

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Nonsteady, one‐dimensional, fissionable gas motion induced in a semi‐infinite, cylindrical tube of constant cross section by a spatially uniform neutron pulse, irradiating an initial segment of the tube, is formulated in terms of the method of characteristics, and a numerical solution describing the motion, prior to development of strong shock discontinuities, is obtained by finite differences with the use of an IBM 7094 computer. Resulting pressure pulses are compared with those obtained from a ``bulk expansion'' model, which assumes spatial uniformity within the irradiated gas. Although physical validity of this model has been ascertained by a satisfactory agreement between the two, an improved version, in the form of a ``constant pressure gradient,'' model is proposed. Pressure pulses predicted by these models, which neglect wave interactions, are accurate for neutron pulse widths of the order of twice the irradiation length divided by the initial sound speed in the undisturbed gas.

Weakly Radiative Acoustic Flow Induced by Radiation from a Stationary Wall

Alexander Solan and Ira M. Cohen

Phys. Fluids 9, 2365 (1966); http://dx.doi.org/10.1063/1.1761627 (6 pages) | Cited 6 times

Online Publication Date: 9 December 2004

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The radiative wave flow induced by radiation from a stationary solid wall is treated for large ratio of photon mean free path to molecular mean free path, lph∕λmfp ≫ 1, and large Boltzmann number, 1 ≪ Bo ≤ (lph∕λmfp)☒ (weak radiation). For times within the range (λmpf∕lph)☒ ≪ ta∞∕lph ≪ Bo, where a∞ is the isentropic speed of sound, the problem is split into a radiationless heat‐conducting boundary layer and a radiation‐induced acoustic outer region. The representations of the two regions are reduced to a heat‐conduction equation and an inhomogeneous linear wave equation, respectively, and these are solved in closed form. The flow in the acoustic region differs from ordinary acoustic flow in that it is induced throughout the flow field instantaneously, rather than progressively. The wavelike behavior manifests itself in a weak discontinuity (in the second derivatives of the flow variables) which propagates outwards from the plate at the isentropic speed of sound.

Asymptotic Law of Decay of Homogeneous Magnetoturbulence

Jacques C. J. Nihoul

Phys. Fluids 9, 2370 (1966); http://dx.doi.org/10.1063/1.1761628 (7 pages) | Cited 2 times

Online Publication Date: 9 December 2004

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Solutions of the magnetohydrodynamic equations for the spectral energy tensors are obtained by assuming that the turbulent field is homogeneous and weak enough for triple correlations to be negligible. The total turbulent magnetic and kinetic energies are derived by integrating over all wave number space. The asymptotic law of decay (for times larger than the ``inhibition time'') is obtained in an analytical form by the method of steepest descent, and it is shown that there are two important contributions to the total energy, one which decays as t−5∕2, as in ordinary turbulence and corresponds to equipartition between magnetic and kinetic modes, and one which decays as t−3 and leads to the partition of energy between the two modes in the inverse ratio of their respective diffusivities. In the course of the decay, the latter becomes negligible, but the time at which the t−5∕2 component dominates depends upon the ratio n = ν∕λ of the two diffusivities and is found to fall beyond any reasonable time of observation if n is much less or much larger than one; in which cases, equipartition is not approached before the turbulence is damped out or has attained such a large scale that, in experimental situations, the limit of validity of the homogeneity hypothesis is reached.
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Velocity Space Diffusion from Weak Plasma Turbulence in a Magnetic Field

C. F. Kennel and F. Engelmann

Phys. Fluids 9, 2377 (1966); http://dx.doi.org/10.1063/1.1761629 (12 pages) | Cited 389 times

Online Publication Date: 9 December 2004

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The quasi‐linear velocity space diffusion is considered for waves of any oscillation branch propagating at an arbitrary angle to a uniform magnetic field in a spatially uniform plasma. The space‐averaged distribution function is assumed to change slowly compared to a gyroperiod and characteristic times of the wave motion. Nonlinear mode coupling is neglected. An H‐like theorem shows that both resonant and nonresonant quasi‐linear diffusion force the particle distributions towards marginal stablity. Creation of the marginally stable state in the presence of a sufficiently broad wave spectrum in general involves diffusing particles to infinite energies, and so the marginally stable plateau is not accessible physically, except in special cases. Resonant particles with velocities much larger than typical phase velocities in the excited spectrum are scattered primarily in pitch angle about the magnetic field. Only particles with velocities the order of the wave phase velocities or less are scattered in energy at a rate comparable with their pitch angle scattering rate.

Ionization Phenomena in a Gas‐Particle Plasma

Edward G. Gibson

Phys. Fluids 9, 2389 (1966); http://dx.doi.org/10.1063/1.1761630 (11 pages) | Cited 12 times

Online Publication Date: 9 December 2004

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Particles in a plasma can appreciably change the electron density from the value it would assume if the particles were not present. The case of pure particle ionization, in which there is only thermionic emission from the particles and no gas ionization, is first considered. It is established that the potential and the charge distributions can be divided into a strong shielding regime, in which most of the free electrons are packed close to the particle surfaces in regions of high potential, and its direct opposite, a weak shielding regime. In both regimes, the free‐electron content of the plasma is most readily altered by variations in the particle size, rather than in the work function or particle temperature. The suppression of one form of ionization by the other when both particle and gas contribution to the electron density are comparable is next investigated. In the case of gaseous ionization enhancement it is shown that, if the thermionically emitting particles are hotter than the gas, the electron temperature will also be higher than that of the gas and the gaseous ionization thereby enhanced. Lastly, it is shown that in some transient situations, the particles are able to control the time rate of change of the electron density.

Wave‐Excited Anomalous Diffusion in a Fully Ionized Magnetoplasma

B. J. Eastlund, K. Josephy, R. F. Leheny, and T. C. Marshall

Phys. Fluids 9, 2400 (1966); http://dx.doi.org/10.1063/1.1761631 (8 pages) | Cited 12 times

Online Publication Date: 9 December 2004

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The transverse diffusion coefficient D⊥ of a fully ionized plasma in a Q machine was observed to depend on the frequency and amplitude of low‐frequency (kilocycle) waves. These waves, observed with a floating Langmuir probe and spectrum analyzer, could be impressed on the plasma (1) by a weak electron beam injected along the plasma axis, (2) by applying an ac signal of appropriate frequency to a plate at one end of the plasma, or (3) were found to occur naturally. D⊥ was determined from the radial profile of potassium plasma pulses. In all cases, D⊥ was strongly influenced by the amplitude of the low‐frequency components of the spectrum and was observed to be proportional to the rms power of the fluctuations. The impedance of the plasma as a function of the impressed ac signal showed minima at frequencies close to that of the ``drift instability,'' where the value of D⊥ exhibited strong maxima.

Extension of the Fokker‐Planck Equation

John C. Price

Phys. Fluids 9, 2408 (1966); http://dx.doi.org/10.1063/1.1761632 (4 pages) | Cited 3 times

Online Publication Date: 9 December 2004

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A simple collision term valid to order 1∕ln Λ is derived for small‐amplitude waves in a uniform plasma. This result extends the validity of Fokker‐Planck type equations from the domain 0 ≤ ω ≪ ωp, 0 ≤ k ≪ kd to the domain 0 ≤ ω ≪ Λωp, 0 ≤ k ⪝ kd.

Negative Mass Instability

Ronald W. Landau and V. Kelvin Neil

Phys. Fluids 9, 2412 (1966); http://dx.doi.org/10.1063/1.1761634 (16 pages) | Cited 27 times

Online Publication Date: 9 December 2004

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A canonical formalism has been developed for the description of the negative mass instability. The formalism has been applied to the formulation of a kinetic equation which is solved by the method of characteristics (or integration over orbits). It is found that a spread in betatron oscillation amplitudes contributes to stability, in addition to the spread in the canonical angular momentum. The negative mass instability equation for two streams is also obtained, and the result is the same as though each stream were present separately. It is also shown that the linear two‐stream equation does not apply for a circular beam geometry; instead, the negative mass instability two‐stream equation applies.

Kinetic Equation for an Unstable Plasma in Parallel Electric and Magnetic Fields

Burton D. Fried and Sidney L. Ossakow

Phys. Fluids 9, 2428 (1966); http://dx.doi.org/10.1063/1.1761635 (8 pages) | Cited 8 times

Online Publication Date: 9 December 2004

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The response of a homogeneous plasma, in a strong uniform magnetic field, B0, to the sudden application of a strong, uniform electric field is studied in a self‐consistent formulation which treats the coupled equations for the one‐particle and two‐particle distribution functions resulting from neglect of three‐particle correlations. As in earlier work on the case B0 = 0, the electric field is assumed to be large compared to that which produces a ``runaway current,'' so that particle‐wave interactions should dominate. In the limit of strong magnetic field, use of the Bernstein‐Kulsrud expansion in k⊥Ric makes the problem effectively one‐dimensional, eliminating the need for factorization assumptions of the form f(v) = f∥(v∥)f⊥ (v⊥) used in the B0 = 0 case. Only the dominant contributions to the diffusion in velocity space, arising from the fastest growing waves, are retained. Other simplifications result from the assumption of unequal electron and ion temperatures, but it proves necessary to specify the ratio Te∕Ti, rather than simply assuming it to be large. In sharp contrast to the case B0 = 0, it is found that the diffusion affects only those electrons which have, in the ion rest frame, velocities of order (m∕M)☒ of the electron thermal speed. No appreciable ``quasi‐linear'' effects, such as the current limitation found in the B0 = 0 case, are therefore to be expected; the results of numerical calculations confirming this are summarized.

Streaming Instabilities in Plasmas with Zero‐Order Density Gradients

Kai Fong Lee

Phys. Fluids 9, 2435 (1966); http://dx.doi.org/10.1063/1.1761636 (9 pages) | Cited 5 times

Online Publication Date: 9 December 2004

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Current‐carrying plasmas with zero‐order density gradients are analyzed for instability using the Vlasov and Maxwell's equations. The plasmas are confined by a strong axial magnetic field, and the densities are nonuniform in the transverse directions. By means of a variational method, it has been found that the density inhomogeneity (1) decreases the real frequency of oscillation, (2) reduces the maximum unstable wave number, and (3) increases the minimum current required for instability and imposes an upper limit on the streaming velocity beyond which the plasma is stable against perturbations of all wavelengths. Explicit results are given for Gaussian and exponential density profiles with resonant, Maxwellian, and delta function velocity distributions. The physical mechanisms that are responsible for these changes are discussed.

Plasma Oscillations Perpendicular to a Weak Magnetic Field

D. E. Baldwin and G. Rowlands

Phys. Fluids 9, 2444 (1966); http://dx.doi.org/10.1063/1.1761637 (10 pages) | Cited 46 times

Online Publication Date: 9 December 2004

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The propagation of electrostatic waves perpendicular to a uniform magnetic field, the so‐called Bernstein modes, are discussed. It is shown, in the limit as the strength of the magnetic field approaches zero, that these modes, which themselves are undamped, collectively act so as to form a single damped mode identical to the usual Landau damped mode appropriate to zero magnetic field. The behavior of the plasma for small but finite magnetic field is also considered. It is shown that damping due to phase mixing exists for a time short compared to ωc−1, the reciprocal of the cyclotron frequency. In times long compared to this, the plasma exhibits a quasi‐periodic behavior with period 2π∕ωc.

Wave Propagation near Electron Cyclotron Harmonics in Nonuniform Plasmas

Gary A. Pearson

Phys. Fluids 9, 2454 (1966); http://dx.doi.org/10.1063/1.1761638 (10 pages) | Cited 41 times

Online Publication Date: 9 December 2004

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Equations are derived that describe the excitation and propagation of longitudinal waves (Bernstein modes) in nonuniform plasmas with a uniform applied magnetic field perpendicular to the direction of wave propagation and to the gradients of the unperturbed plasma. Certain features of the microwave absorption by plasma columns have recently been explained on the basis of such modes. Comparison shows that the equations derived by most other authors are correct only when the scale of the plasma inhomogeniety is long compared to the wavelength considered; this is primarily because they ignore the ambipolar electric field and the anisotropy of the unperturbed electron velocity distribution. The inclusion of these effects does not complicate the equations.

Longitudinal Modes in Magnetized Nonuniform Plasma Columns

H. L. Frisch and Gary A. Pearson

Phys. Fluids 9, 2464 (1966); http://dx.doi.org/10.1063/1.1761639 (8 pages) | Cited 12 times

Online Publication Date: 9 December 2004

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Certain resonant frequencies of magnetized cylindrical plasma columns were calculated using the theory of Buchsbaum and Hasegawa as well as an improved theory that takes into account the ambipolar electric field and the anisotropy of the electron velocity distribution. Attention is restricted to longitudinal modes with no propagation along the column axis and with frequencies near the second harmonic of the electron cyclotron frequency. For azimuthally symmetric modes (and for analogous planar models), the resonant frequencies calculated numerically using these two theories differ little; both theories agree with experiment to within the experimental uncertainties. Modes that are not azimuthally symmetric are described by a fourth‐order differential equation that has not been solved exactly. An approximate perturbation theory treatment we have carried out suggests that the true eigenfrequencies are close to those obtained from a much simpler second‐order equation.

Stability of the Theta‐Pinch

F. A. Haas and J. A. Wesson

Phys. Fluids 9, 2472 (1966); http://dx.doi.org/10.1063/1.1761640 (6 pages) | Cited 15 times

Online Publication Date: 9 December 2004

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A model of a β = 1 axisymmetric plasma in which the equilibrium quantities are assumed to vary slowly in the axial direction is set up. The energy principle is used to investigate stability, and it is shown that the theta‐pinch configuration is stable to the m = 1 mode.

Particle Loss in a Three‐Dimensional Cusp

William Grossmann

Phys. Fluids 9, 2478 (1966); http://dx.doi.org/10.1063/1.1761641 (8 pages) | Cited 18 times

Online Publication Date: 9 December 2004

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The loss of particles through an axisymmetric cusped magnetic field containment geometry is considered. An appropiate adiabatic invariant is derived for the plasma considered to be a high β, collisionless collection of particles contained within a field‐free region bounded by a mathematically sharp interface. The particles are allowed to undergo soft collisions with the interface thus giving rise to a boundary layer. A velocity space loss cone criterion analogous to the loss cone criterion in magnetic mirror machines is derived. Particle loss rates are calculated for a special form of the particle distribution function, and the results are compared with relevant experimental data. A previous treatment of the present problem considering two‐dimensional geometry is re‐examined in view of the present work concerning adiabatic invariance.

Static Theory of a Discharge Column at Intermediate Pressures

S. A. Self and H. N. Ewald

Phys. Fluids 9, 2486 (1966); http://dx.doi.org/10.1063/1.1761642 (7 pages) | Cited 159 times

Online Publication Date: 9 December 2004

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A steady‐state theory of the discharge column is derived which is applicable in the pressure range where the ion mean free path is neither much greater than nor much less than the column radius, and which goes over in the low‐ and high‐pressure limits to the free‐fall and ambipolar diffusion theories, respectively. Solutions for planar and cylindrical geometry are given for the density and potential profiles. The plasma‐sheath boundary is discussed and the sheath potential drop is estimated. The theory is shown to agree well with measurements in mercury discharges over the pressure range 1–20 μ Hg.

Experimental Results on the Ion‐Beam Excitation of Drift Waves in an Alkali Plasma

L. Enriques

Phys. Fluids 9, 2493 (1966); http://dx.doi.org/10.1063/1.1761643 (7 pages) | Cited 4 times

Online Publication Date: 9 December 2004

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An experiment is described in which azimuthal drift waves were excited in the quiescent plasma of a Q machine by means of a modulated low‐density ion beam. The characteristics of propagation of a sinusoidal signal and their variations with the velocity of the beam and with the frequency of modulation are described. The signal is found to propagate azimuthally in the direction of the electron drift and axially with phase velocity close to the ion‐beam velocity. This is consistent with the predictions of a simplified theoretical model of the ion‐beam interaction with a nonuniform plasma.

Scattering of Energetic Particles by Laboratory Plasmas

David Montgomery, Celso Roqúe, and Igor Alexeff

Phys. Fluids 9, 2500 (1966); http://dx.doi.org/10.1063/1.1761644 (4 pages) | Cited 4 times

Online Publication Date: 9 December 2004

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Possible experiments on the scattering of energetic particles by laboratory plasmas are discussed. Expressions for scattering coefficients in terms of the electric field autocorrelation function are derived. Sample situations are considered. Whereas incoherent scattering of electromagnetic waves is governed by the charge density autocorrelation function, energetic particle scattering is governed by the electric field autocorrelation function.
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