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Dec 1967

Volume 10, Issue 12, pp. 2509-2717

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Rotational Temperature Measurements in Electron‐Beam Excited Nitrogen

Harry Ashkenas

Phys. Fluids 10, 2509 (1967); http://dx.doi.org/10.1063/1.1762070 (12 pages) | Cited 22 times

Online Publication Date: 9 December 2004

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Measurements of the rotational temperature of electron‐beam excited nitrogen have been made along the axis of a free jet under various flow conditions. Temperatures calculated from the isentropic flow equations are compared with those derived from the rotational spectra. Discrepancies between the two temperature determinations led to an investigation of the rotational spectrum of nitrogen in thermodynamic equilibrium at 300 and at 78°K. Results of the ``static'' gas measurements are presented which indicate that existing models of the electron‐beam excitation—de‐excitation process may be incorrect.

Transport Properties of a Nonequilibrium Partially Ionized Gas in a Magnetic Field

Robert M. Chmieleski and Joel H. Ferziger

Phys. Fluids 10, 2520 (1967); http://dx.doi.org/10.1063/1.1762071 (11 pages) | Cited 8 times

Online Publication Date: 9 December 2004

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A modified Chapman—Enskog technique developed for a partially ionized gas in an electric field is generalized to include the effects of a magnetic field. The method allows the electron and heavy particle temperatures to differ while requiring all species to have the same macroscopic velocity to zero order. Expressions are obtained for the transport properties of such a gas and a complete set of self‐contained magnetohydrodynamic equations is given. As an example, the transport properties of a fully ionized gas containing a single ion species are evaluated using the cutoff Coulomb potential. In the limit of equal electron and ion temperatures, the results are in agreement with previous results.

Second Virial Coefficient for the Exponential Repulsive Potential

Ludwig W. Bruch

Phys. Fluids 10, 2531 (1967); http://dx.doi.org/10.1063/1.1762072 (9 pages) | Cited 4 times

Online Publication Date: 9 December 2004

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The classical second virial coefficient and its first two quantum corrections are evaluated for a repulsive exponential potential by the use of Laplace transforms. The results are used to estimate the effect of the intermolecular attraction in the high‐temperature second virial coefficient of 4He and in its first quantum correction. An approximation is found for this first quantum correction at temperatures above 100°K for 4He. A simplified expression for the S‐wave phase shift of the exponential repulsion is derived and is used to calculate a quantum‐mechanical turning point for wave packets or effective hard core radius. Similar methods are applied to the Morse potential to evaluate the classical second virial coefficient and its first quantum correction and the S‐wave phase shift.

Stability of Laminar Boundary Layers at Separation

Ahmed R. Wazzan, T. T. Okamura, and A. M. O. Smith

Phys. Fluids 10, 2540 (1967); http://dx.doi.org/10.1063/1.1762073 (6 pages) | Cited 28 times

Online Publication Date: 9 December 2004

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The stability of laminar boundary layers at separation is considered. The velocity distribution is represented by (1) a Pohlhausen fourth‐degree polynomial P4, and (2) by a Falkner—Skan similarity profile at separation, Hartree β = − 0. 1988. The Orr—Sommerfeld equation is integrated using Runge—Kutta with Gram—Schmidt orthonormalization. Using single precision arithmetic, the method leads to satisfactory answers at Reynolds numbers R1 = 100 000 and larger. In either case the minimum critical Reynolds number is found to be of the order of 100. It is found that the neutral curves for the P4 profile obtained by solving the Orr—Sommerfeld equation, whether exactly (numerically) or within the framework of the asymptotic approximations, approach along the upper branch the same asymptotic value, namely α1 = 0.8028 (in the case of the β = − 0. 1988 profile, however, the corresponding values is α1 = 1.240). It is also found that the asymptotes to the lower branch in both cases vary with α1 according to R11∕3∼α1 −7∕9 (in the case of the β = − 0. 1988 profile however, the corresponding variation is R11∕3 ∼ α1 ‐ 0. 477).

Flow of Micropolar Fluids between Two Concentric Cylinders

T. Ariman, A. S. Cakmak, and L. R. Hill

Phys. Fluids 10, 2545 (1967); http://dx.doi.org/10.1063/1.1762074 (6 pages) | Cited 14 times

Online Publication Date: 9 December 2004

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The flow of micropolar fluids between two concentric cylinders, is analyzed first for the use of Couette flow with ∂p∕∂θ, then for the case of Poiseuille flow. The results are presented graphically and compared with the classical ones, and the differences are discussed.

Theory of Simple Deformable Directed Fluids

S. J. Allen, C. N. DeSilva, and K. A. Kline

Phys. Fluids 10, 2551 (1967); http://dx.doi.org/10.1063/1.1762075 (5 pages) | Cited 25 times

Online Publication Date: 9 December 2004

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The governing equations are given for a fluid with deforming microstructure by postulating an energy balance, an entropy production inequality, and a principle of material frame indifference. A canonical form of the energy equation is derived which serves as a guide to define a simple deformable directed fluid. A theory for such fluids is developed.

Class of Exact Magnetohydrodynamic Solutions

Vivian O'Brien and Sheldon Weinbaum

Phys. Fluids 10, 2556 (1967); http://dx.doi.org/10.1063/1.1762076 (10 pages) | Cited 1 time

Online Publication Date: 9 December 2004

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Exact magnetohydrodynamic solutions for incompressible fluids, valid for all values of conductivity and viscosity, are found. These collinear magnetohydrodynamic fields have magnetic surfaces corresponding to stream surfaces.

Flows Induced by Nonuniform Magnetohydrodynamic Pinch

Chuen‐Yen Chow and Randolph D. Brunell

Phys. Fluids 10, 2566 (1967); http://dx.doi.org/10.1063/1.1762077 (5 pages)

Online Publication Date: 9 December 2004

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Ring vortices are induced after the passage of an axial current through a stagnant conducting liquid contained between two concentric, corrugated glass tubes with a strong current flowing along the common axis of the tubes. Experiments have been conducted using transparent aqueous solution of copper sulfate suspended with fine aluminum powder for flow visualization. Flow patterns are photographed for the cases that the sinusoidal deformations on the tube walls are either in phase or completely out of phase, and one of the tubes is straight. The result agrees considerably well with that obtained previously from a linearized theory.

Application of a Microwave Technique to the Measurement of Electron Density and Ionization Time

Samuel Lederman and Edward F. Dawson

Phys. Fluids 10, 2570 (1967); http://dx.doi.org/10.1063/1.1762078 (9 pages) | Cited 2 times

Online Publication Date: 9 December 2004

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The measurement of ionization rates, electron density, and ionization time in thermally ionized gases, by means of a microwave technique, is discussed. The technique used in this work utilizes a so‐called ``end‐wall'' microwave resonant cavity. It is essentially a microwave cavity operated in the TE011 mode with a plasma forming one end wall. Since the resonant frequency of such a cavity is a function of the axial length of the cavity, a variable density plasma with its corresponding variable conductivity is utilized to tune the cavity to different resonant frequencies. It is shown that by applying two frequencies to the cavity separated from each other by a known Δf, which is small compared to the resonant frequency of the closed cavity, a simple method is arrived at to measure electron density, ionization rate, and ionization time.

Plasma Wind Tunnel Studies of Collision‐Free Flows and Shocks

E. Pugh and R. Patrick

Phys. Fluids 10, 2579 (1967); http://dx.doi.org/10.1063/1.1762079 (7 pages) | Cited 13 times

Online Publication Date: 9 December 2004

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A plasma wind tunnel has been constructed and the super‐Alfvénic flow over a magnetic obstacle studied. The wind tunnel produces a steady (10−3 sec) flow of high enthalpy (∼100 eV), high density (1014) plasma. The flow of this high conductivity plasma has been studied using magnetic field probes. The interface and shock wave produced in front of a magnetic obstacle in the flow have been studied. The shock wave is shown to satisfy the usual magnetohydrodynamic Rankine—Hugoniot relations.

Study of Shock Waves in a Nonequilibrium Plasma

Walter H. Christiansen

Phys. Fluids 10, 2586 (1967); http://dx.doi.org/10.1063/1.1762080 (10 pages) | Cited 5 times

Online Publication Date: 9 December 2004

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Probe and recombination‐radiation spectrum measurements were used to measure the electron number density and temperature profiles in front of a flat disc in a supersonic seeded‐gas plasma. The region of interest extended several body diameters upstream of the disc and through the shock wave. The experiment covered a free stream electron density range of 4 × 1011 to 1.6 × 1012 cm−3 and an electron temperature range of 0.06 to 0.1 eV. The region of reduced luminosity upstream of the shock wave is shown to be associated with electron heating at constant electron density. The electron thermal conductivity is derived from measurements in the upstream region and gives reasonable agreement with the Spitzer formula. Measurements are made of the electron temperature change across the heavy‐particle shock wave and of the over‐all electron temperature ratio across the entire process. It is shown that the over‐all electron‐temperature ratio could not be predicted on the basis of an adiabatic one‐demensional flow process in this experiment.

Unified Shock Profile in a Plasma

Harold Grad and Pung Nien Hu

Phys. Fluids 10, 2596 (1967); http://dx.doi.org/10.1063/1.1762081 (7 pages) | Cited 31 times

Online Publication Date: 9 December 2004

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Previously [P.N.Hu, Phys. Fluids 9, 89 (1966)], a universal monotonic profile was found for a weak plasma shock propagating perpendicular to a magnetic field using an elaborate set of moment equations which included electron and ion viscosity, coupled electron and ion heat conduction, thermal diffusion, and resistivity, together with distinct Hall terms for the resistivity, stress, and heat flow. Inclusion of an additional term related to electron inertia now gives a one‐parameter family of profiles which spans the entire range from collision dominated monotonic profiles to damped oscillatory profiles and the limiting collisionless case of purely oscillatory and solitary wave profiles.

Dynamical Properties of Truncated Wiener‐Hermite Expansions

Steven A. Orszag and L. R. Bissonnette

Phys. Fluids 10, 2603 (1967); http://dx.doi.org/10.1063/1.1762082 (11 pages) | Cited 40 times

Online Publication Date: 9 December 2004

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The use of truncated Wiener‐Hermite functional expansions as a basis for the theory of turbulence is critically examined. An account is given of the application of such expansions to Burgers' model equation. The nature of Wiener‐Hermite expansions is such that certain important consistency requirements, such as realizability, follow immediately. In order to gain insight into other aspects of the predicted turbulence dynamics, a simpler model problem in which there are only three interacting modes is first studied. It is shown that Wiener‐Hermite closures do not faithfully represent the dynamics for this latter model. Then, the analysis is extended to Burgers' equation. It is shown that Wiener‐Hermite closures do not preserve fundamental properties of the exact dynamics in equilibrium. Inviscid equipartition solutions do not survive in the closures. In addition, it is shown that these closures do not treat the effects of large scales on small scales properly. Numerical calculations be extended to the corresponding theory of Navier‐Stokes turbulence. We conclude that truncated Wiener‐Hermite expansions are unsuitable for the theory of high Reynolds‐number turbulence.

Functional Integration Theory for Incompressible Fluid Turbulence

Gerald Rosen

Phys. Fluids 10, 2614 (1967); http://dx.doi.org/10.1063/1.1762083 (6 pages) | Cited 6 times

Online Publication Date: 9 December 2004

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A purely deductive approximation theory for incompressible fluid turbulence is presented, based exclusively on the Hopf characteristic functional space‐time formulation and without any additive statistical postulate. Certain real symmetric solenoidal tensors called ``stationary functionalities'' are defined in terms of the characteristic functional by functional integration, quantities which vanish in the neighborhood of an exact physical characteristic functional, hence, for an approximate physical characteristic functional. By explicit functional integration, the first and second tensorial rank stationary functionalities are evaluated with a zero‐mean velocity field Gaussian approximation for the characteristic functional. Equating the resulting expressions to zero produces a subsidiary equation and a dynamical equation for the two‐point velocity correlation tensor, equations which are identical to the Navier‐Stokes expectation value equation for a zero‐mean velocity field and to a specific two‐point Navier‐Stokes expectation value equation with a zero‐mean velocity field probability distribution such that the fourth‐order velocity field product expectation values are related to lower‐order product expectation values in the same way as for a zero‐mean Gaussian probability distribution. Specialized forms of the nonlinear integrodifferential dynamical equation are worked out for the cases of homogeneous and isotropic homogeneous turbulence.

Diffusion, Scattering, and Acceleration of Particles by Stochastic Electromagnetic Fields

D. E. Hall and P. A. Sturrock

Phys. Fluids 10, 2620 (1967); http://dx.doi.org/10.1063/1.1762084 (9 pages) | Cited 71 times

Online Publication Date: 9 December 2004

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The behavior of charged particles in an electromagnetic field composed of two parts is treated. The first is an unperturbed part, for which the solution of the Vlasov equation is assumed to be known; the second part is a perturbation which is assumed to be random and of small amplitude. Such a ``stochastic'' or ``turbulent'' electromagnetic field leads to diffusion, pitch‐angle scattering, and acceleration. Equations governing these processes may be derived by test‐particle calculations. The problem is first posed in a general way, and solved by a method similar to that used in quasi‐linear theory. The connection of this method with the Fokker‐Planck formulation is discussed. The case which is analyzed in detail is that of relativistic particles moving in a uniform magnetic field under the influence of a steady homogeneous spectrum of electromagnetic fluctuations. Results are given which are valid for any (weak) fluctuation spectrum and for finite test‐particle gyroradii. The physical meaning of the results is discussed, and it is shown that in certain limits there is agreement with earlier, less general analyses.

Macroscopic Quasi‐Linear Theory of the Two‐Stream Instability

Heinrich J. Völk

Phys. Fluids 10, 2629 (1967); http://dx.doi.org/10.1063/1.1762085 (5 pages) | Cited 6 times

Online Publication Date: 9 December 2004

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The quasi‐linear development of the instability of two counter‐streaming electron beams is discussed from a hydrodynamic point of view. While it turns out that the closure assumptions for the system of moment equations have an appreciable influence on the time asymptotic result, the nonresonant diffusion mechanism allows a number of interesting interpretations in the fluid picture.

Plasma Heating by the Fast Hydromagnetic Wave

K. Chung and M. A. Rothman

Phys. Fluids 10, 2634 (1967); http://dx.doi.org/10.1063/1.1762086 (8 pages) | Cited 14 times

Online Publication Date: 9 December 2004

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The fast (right‐handed) hydromagnetic wave is generated by a Stix coil installed on the Model C stellarator. Maximum wave generation and heating occurs, as expected, under conditions such that the wavelength of the propagating wave equals the wavelength of the coil structure. Measurements of diamagnetic pressure determine the efficiency for conversion of rf power into thermal energy. Theoretical calculations of heating rates shows that plasma resistivity and Landau damping are insufficient to account for the observed heating rates if only the body wave is considered. Introduction of surface waves associated with finite electron mass results in increased Ez fields which enhance the heating rate.

Effect of a High‐Frequency Spatially Uniform External Electric Field on a Plasma

R. Prasad

Phys. Fluids 10, 2642 (1967); http://dx.doi.org/10.1063/1.1762087 (8 pages) | Cited 10 times

Online Publication Date: 9 December 2004

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Dispersion relation for transverse waves in a fully ionized plasma in the presence of a spatially uniform field E0 cos vt has been obtained and the possibility of instabilities has been discussed. It has been shown that the analysis also gives the Dawson‐Oberman formula for the force of an ion on the electrons for the uncoupled longitudinal field.

Observation of a Ballooning‐Type Instability in a Plasma Confined by a Quadrupole Magnetic Field

D. M. Meade and S. Yoshikawa

Phys. Fluids 10, 2649 (1967); http://dx.doi.org/10.1063/1.1762088 (4 pages) | Cited 14 times

Online Publication Date: 9 December 2004

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Instabilities of a ballooning type were observed in the ∮ dl∕B stable region of a linear quadrupole device. The amplitude distribution of the fluctuation along a field line behaved like a standing wave that had a node where ▽B2 ⋅ ▽p < 0 and was peaked where ▽B2 ⋅ ▽p > 0. The quantity k∥λmfpe was typically 5–10. The observed fluctuation is in qualitative agreement with the theory of collisionless drift waves modified to take into account the change of field curvature and field strength along the field line. The threshold for the instability appears to be strongly dependent on the ratio of the plasma radius to connection length. The plasma confinement was not sensitive to the presence of the ballooning instability.

Single Particle Motion in Toroidal Stellarator Fields

A. Gibson and J. B. Taylor

Phys. Fluids 10, 2653 (1967); http://dx.doi.org/10.1063/1.1762089 (7 pages) | Cited 28 times

Online Publication Date: 9 December 2004

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The discussion of the motion of single particles in stellarators is simplified by identifying three distinct groups of particles. The drift equations are used to treat these groups by analytical and numerical methods. It is shown that, unlike an axisymmetric system the region of a stellarator with surfaces closed about a magnetic axis need not form a perfect trap. In fact, unless the toroidal curvature is very small, particles in one of the groups, those mirrored between maxima in a single period of the helical field, will drift out beyond the separatrix no matter how large the magnetic field strength.

Stability of a Dense Plasma Confined by a Rotating Magnetic Field

Francis Troyon

Phys. Fluids 10, 2660 (1967); http://dx.doi.org/10.1063/1.1762090 (15 pages) | Cited 13 times

Online Publication Date: 9 December 2004

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The stability of a dense plasma (negligible skin depth) confined by a rotating magnetic field which is parallel to the surface is analyzed for plane and cylindrical geometries. This analysis culminates in an integrodifferential equation for each mode of deformation. Specific properties of the plasma enter only through the normal acoustic impedance. The concept of average stability is introduced. Very general restrictions on the acoustic impedance of the plasma are introduced which allow the derivation of various stability criteria. These criteria are applied to different cases of interest. The plasma is described successively by a fluid model and a free particle model. Numerical estimates of the frequency needed to achive stability for each mode are given.

Direct Measurement of Mass Pickup in a Low‐Energy Theta Pinch

Robert W. Harris

Phys. Fluids 10, 2675 (1967); http://dx.doi.org/10.1063/1.1762091 (5 pages)

Online Publication Date: 9 December 2004

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See Also: Erratum

Show Abstract
A balanced magnetic probe technique has been used to determine the drive force acting on the plasma sheath during the first pinch. The sheath motion was studied with the probe technique, and with image converter framing pictures. The sheath mass was determined from the velocity and the time integral of the drive force. The results indicate an average pickup coefficient of 0.82 ± 0.06 for the first half of the pinch. Electron density and temperature were determined during the first half of the pinch by measurement of the Hβ line profile. Combination of the spectroscopic and mass data indicates an average sheath ionization lower than predicted by theory, but the experimental error in the ionization determination precludes a definite conclusion to this effect.

Radiation Losses from a Test Particle in a Plasma

J. F. McKenzie

Phys. Fluids 10, 2680 (1967); http://dx.doi.org/10.1063/1.1762092 (15 pages) | Cited 23 times

Online Publication Date: 9 December 2004

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A formula is derived for the average rate of energy loss suffered by a charged particle moving in a helical orbit in a homogeneous medium. In a cold, magnetized plasma it is shown that the Čerenkov excitation of the whistler and ion cyclotron waves depends critically on certain characteristic wave speeds relative to the component of the particle's velocity parallel to the magnetic field. The power spectrum of Čerenkov radiation exhibits sharp peaks not only at the frequencies of the lower and upper hybrid resonances but also at those frequencies at which the ray phase velocity equals the particle's velocity, v∥, parallel to the magnetic field. An intriguing feature of this radiation is that the more slowly the particle moves along the magnetic field, the greater is the emitted radiation; the power emitted in the whistler mode also becomes large when v∥ equals the group speed of those waves that are beamed along the magnetic field. By using a simple graphical construction one can readily locate those directions in which the angular distribution of synchrotron radiation is sharply peaked.

Stimulated Brillouin Effect in Magnetoplasmas

Y. Pomeau

Phys. Fluids 10, 2695 (1967); http://dx.doi.org/10.1063/1.1762093 (7 pages) | Cited 5 times

Online Publication Date: 9 December 2004

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The manner in which a pumping wave of high amplitude can produce parametric instabilities in a plasma is investigated. A very simple derivation of the nonlinear polarizabilities is given for the case when the pumping wave is homogeneous. From this nonlinear polarizability, a derivation of the accurate nonlinear dispersion relation is given. One finds unstable waves by solving this nonlinear dispersion relation. The critical level of the pumping wave is very much reduced if the proper modes allow carrying out the stimulated Brillouin effect. A pumping wave with a cyclotron resonance gives a particularly low critical level.
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Correction to Stokes Law for a Charged Sphere Moving in a Polarizable, Compressible Liquid

G. T. Schappert

Phys. Fluids 10, 2702 (1967); http://dx.doi.org/10.1063/1.1762095 (3 pages)

Online Publication Date: 9 December 2004

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It is shown that Stokes law for a charged sphere of radius R moving slowly in a compressible polarizable liquid is
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