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Nov 1960

Volume 3, Issue 6, pp. 835-1036

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On Flow Duration in Low‐Pressure Shock Tubes

Anatol Roshko

Phys. Fluids 3, 835 (1960); http://dx.doi.org/10.1063/1.1706147 (8 pages) | Cited 73 times

Online Publication Date: 22 November 2004

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The severe decrease of flow duration in shock tubes operating at low pressures, previously reported by Duff, is confirmed by experiment and by an analysis of the effects of the laminar‐boundary layer behind the shock wave. The latter leads to a shock tube similarity length parameter X, which depends on the tube pressure, diameter and shock Mach number, and to a flow duration parameter T. The theoretical relation T = T(X) is determined and compared with experimental results. From the theoretical result Tmax = 1, the maximum possible flow duration τm in a shock tube is determined; it increases linearly with the initial pressure and the square of the tube diameter and decreases strongly with shock Mach number.

Spectroscopic Study of Helium Plasmas Produced by Magnetically Driven Shock Waves

E. A. McLean, C. E. Faneuff, A. C. Kolb, and H. R. Griem

Phys. Fluids 3, 843 (1960); http://dx.doi.org/10.1063/1.1706148 (14 pages) | Cited 61 times

Online Publication Date: 22 November 2004

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Measurements have been made of the temperature, density, and degree of ionization of plasmas produced by Mach 30 magnetically driven shock waves in helium (ambient pressure 1 mm Hg). Simultaneous photoelectric intensity measurements of the absolute spectral intensities of HeI λ3889, HeI λ5876, HeII λ4686, and HeII λ3203 indicate temperatures of 3.7 ev, electron and ion densities ∼1017 cm−3, degree of ionization ∼99.9%, and a density ratio of ∼4 across the shock front. The estimated error is ±2% for the temperature and ±12% for the electron and ion densities. The electron density was derived independently from the width of HeII λ4686 and agreed with the photoelectric density measurement to within the experimental error for the line width, thus providing a proof of the ionization and excitation equilibrium assumption used in analyzing the absolute intensity data. Continuum intensity measurements also provided a check on the consistency of the results. The temperature calculated from the measured shock velocity using the usual Rankine‐Hugoniot equations is lower by a factor of about 2, and the density ratio is higher by a factor of about 3. A plausible explanation of this discrepancy is that ultraviolet radiation emitted by the hot plasma in the arc region is absorbed in front of the shock wave.

Some Unexpected Results of Shock‐Heating Xenon

Per Gloersen

Phys. Fluids 3, 857 (1960); http://dx.doi.org/10.1063/1.1706149 (14 pages) | Cited 40 times

Online Publication Date: 22 November 2004

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Results from a detailed experimental study of the structure of pressure‐driven incident shock waves in very pure xenon contained in a thoroughly pumped Pyrex shock tube are presented and discussed. As a result of some optical studies of the luminous structure of the shocks, the following features of the luminosity delay time were discovered: independence of the pressure in the undisturbed xenon in the range 0.75 to 4.0 mm Hg, dependence on shock velocity in a way not explainable on the basis of reasonable volume processes alone, and dependence on shock tube diameter. The visible luminosity from the shock was found to terminate well in advance of the measured position of the xenon‐driver interface. This is indicative of severe radiation cooling. The visible luminosity was also found to be profoundly altered by the addition of impurities either in the xenon itself or in the driver. The spectrum of the delayed luminosity in the region from 3000 to 10 000 A was studied with an electronic‐recording time‐resolving spectrometer and found to consist of xenon atom lines superimposed on a strong continuum. The continuum may reasonably be attributed to dissociative transitions from bound excited states of the Xe2 molecule related to the xenon atom levels 7 pKJ and above to unbound Xe2 states related to the atomic levels 6s11 and 6s12. Positive electrical signals, observed during the passage of the shock through external metal rings, are attributed to ejection of electrons from the shock tube walls by photoelectric action and∕or metastable atoms. In addition, two different types of electrical precursors were observed. The first was observed under the usual conditions, namely that the shock was sufficiently strong to cause the delayed luminosity. The second was observed in some experiments in which the shock was too weak to cause the delayed luminosity. Both could be due to a photoelectric effect on the shock tube walls, but the precursor observed in the absence of the delayed luminosity may be also due to diffusion of electrons ahead of the shock front. The present experiments strongly indicate that shock tube experiments of others may need reinterpretation.

Some Variational Theorems for Non‐Newtonian Flow

M. W. Johnson

Phys. Fluids 3, 871 (1960); http://dx.doi.org/10.1063/1.1706150 (8 pages) | Cited 11 times

Online Publication Date: 22 November 2004

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A new variational theorem is formulated which has as its Euler equations and natural boundary conditions all of the differential equations and boundary conditions of the boundary value problem under consideration. Other variational theorems, including the classical theorems of Helmholtz, follow from this fundamental theorem. Conditions under which the theorems obtained are minimum or maximum principles are discussed and application to non‐Newtonian flow in a tube is made.

Fluid Flow through a Porous Channel

W. E. Wageman and F. A. Guevara

Phys. Fluids 3, 878 (1960); http://dx.doi.org/10.1063/1.1706151 (4 pages) | Cited 10 times

Online Publication Date: 22 November 2004

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Approximate solutions of the equations of motions governing laminar incompressible fluid flow through a cylindrical channel with a porous wall are derived. The invalidity of an approximation in the solution of these equations under certain circumstances is pointed out, and the results of a numerical integration in the region where the approximation is invalid are indicated. A brief description is given of an experiment to verify the calculations, and some interesting results are noted.

Theory of Gas Bubble Dynamics in Oscillating Pressure Fields

Milton S. Plesset and Din‐Yu Hsieh

Phys. Fluids 3, 882 (1960); http://dx.doi.org/10.1063/1.1706152 (11 pages) | Cited 21 times

Online Publication Date: 22 November 2004

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The behavior of a permanent gas bubble in a liquid with an oscillating pressure field is analyzed with a linearized theory. If the assumption is made that conditions within the bubble are uniform, the thermodynamic relations found are as expected; i.e., at low frequencies the bubble behaves isothermally and at high frequencies the behavior becomes adiabatic. However, a more detailed analysis, which allows the bubble interior to vary not only in time but also in space, leads to an average isothermal behavior for the bubble even in the high‐frequency limit.

On Complete Blast Scaling

Ulf Ericsson and Kjell Edin

Phys. Fluids 3, 893 (1960); http://dx.doi.org/10.1063/1.1706153 (3 pages) | Cited 6 times

Online Publication Date: 22 November 2004

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Experimental evidence is presented for the ability of Sachs' complete energy scaling to account for the influence of ambient pressure and temperature at not too small distances from the charge.

Sound Velocity, Phase Separation, and Lambda Transitions of He3☒He4 Mixtures

Thomas R. Roberts and Stephen G. Sydoriak

Phys. Fluids 3, 895 (1960); http://dx.doi.org/10.1063/1.1706154 (8 pages) | Cited 31 times

Online Publication Date: 22 November 2004

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The velocity of first sound has been measured in saturated liquid He3☒He4 mixtures ranging in composition from 20 to 90% He3 between 0.5° and 2.3°K. The velocities decrease with increasing He3 concentration due to the large increase in compressibility. The temperature Ts at which a liquid mixture first begins to separate into two layers of different composition is identified by a break in the temperature derivative of the sound velocity. The values of Ts are compared with and found to agree well with all other recent direct measurements. Breaks also occur in the temperature derivatives of first sound velocity and signal amplitude at temperatures which are in good agreement with most recent determinations of lambda transition temperatures for concentrations up to 30%. For higher concentrations, the values obtained are in good agreement with λ points determined from second sound measurements but differ markedly from results based on the disappearance of boiling. Possible reasons for the differences are discussed. A smoothed table of λ temperature as a function of concentration is given up to the intersection of the λ line and the phase separation region at 68% He3 and 0.86°K. The vapor pressure of pure He4 at its lambda point is 37.80 ± 0.01 mm Hg (at 0°C and standard gravity) based on the measured vapor pressure at the minimum sound signal and a calculated displacement of the sound minimum from the lambda point.

Kinetic Theory of Moderately Dense Gases: Rigid Sphere Limit

R. F. Snider and C. F. Curtiss

Phys. Fluids 3, 903 (1960); http://dx.doi.org/10.1063/1.1706155 (2 pages) | Cited 17 times

Online Publication Date: 22 November 2004

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The rigid sphere limiting forms of the expressions for the first‐order density corrections to the transport coefficients previously derived by the authors from the modified Boltzmann equation of Green and Bogoliubov are re‐examined. It is shown that in this limit these expressions approach the classic results of Enskog.

Solutions of Concentration‐Dependent Diffusion Equation

Ranjit Paul

Phys. Fluids 3, 905 (1960); http://dx.doi.org/10.1063/1.1706156 (3 pages) | Cited 5 times

Online Publication Date: 22 November 2004

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The concentration of a gas in a chamber of Ney and Armistead type, as a function of time, has been calculated for two simple cases of concentration dependence of the diffusion coefficient viz., (i) D = D0(1 + αc), and (ii) D = D0[1 + αc∕(1 + c)]. The correction due to the nonattainment of a quasi‐stationary state has also been calculated.

Kirkendall Effect in Gaseous Diffusion

K. P. McCarty and E. A. Mason

Phys. Fluids 3, 908 (1960); http://dx.doi.org/10.1063/1.1706157 (15 pages) | Cited 18 times

Online Publication Date: 22 November 2004

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An experiment is described which is the analog in gases of the well‐known Kirkendall effect in solids. The experiment can also be thought of as a measurement of the pressure differences arising in a diffusing gas mixture. Seven gas mixtures were measured at 30°C and over a maximum pressure range of about 0.2 to 1.7 atm as follows: H2☒D2, CO2☒C3H8, He☒Ar, He☒CO2, H2☒CO2, D2☒CO2, and H2☒(H2 + CO2 mixture). The results are discussed in terms of the usual phenomenological theory of the Kirkendall effect and the rigorous kinetic theory of gases. The phenomenological theory is shown to be incorrect, but the kinetic theory is capable of accounting for all the effects observed, usually quantitatively. The results can be used to determine gaseous diffusion coefficients from the observed marker motion, and the diffusion coefficients obtained are in reasonable agreement with those obtained by other methods.

Kinetic Equation with a Constant Magnetic Field

Norman Rostoker

Phys. Fluids 3, 922 (1960); http://dx.doi.org/10.1063/1.1706158 (6 pages) | Cited 53 times

Online Publication Date: 22 November 2004

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The collision operator is derived for the case of a spatially homogeneous plasma subject to a constant external magnetic field. A generalization of Lenard's method is employed.

Electric Field Distribution in a Dense Plasma

J. L. Jackson

Phys. Fluids 3, 927 (1960); http://dx.doi.org/10.1063/1.1706159 (5 pages) | Cited 4 times

Online Publication Date: 22 November 2004

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A calculation is presented of the probability distribution function of the electric field at the center of an ion or atom in a plasma in the high ion density limit. In this limit, it is possible to take into account rigorously the effect of the Coulomb interactions on the distribution function. The distribution function in the high‐density limit is Gaussian. The Coulomb interaction decreases the mean square electric field by the multiplicative factor [1 + κa0 + ⅓(κa0)2]−1, where κ is the reciprocal Debye length and a0 the radius of the ion or atom at whose center the field is evaluated.

Long‐Wavelength Beam Instability

Marshall N. Rosenbluth

Phys. Fluids 3, 932 (1960); http://dx.doi.org/10.1063/1.1706160 (5 pages) | Cited 28 times

Online Publication Date: 22 November 2004

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The kinematics of a self‐pinched neutralized relativistic stream is developed in the limit of oscillations whose frequency in the rest frame of the beam is low compared to the betatron frequency of oscillation of the particles of the beam. It is shown that in this limit the beam is able to displace itself more or less rigidly giving rise to the possibility of kink‐type instabilities depending on its coupling with the external medium. In particular, the case where the beam passes through a low conductivity plasma is considered here. In this case there is a viscous drag when the beam displaces and pulls its magnetic field through the resistive plasma. This leads to an instability of the beam.

Ion Wave Instabilities

Ira B. Bernstein and Russell M. Kulsrud

Phys. Fluids 3, 937 (1960); http://dx.doi.org/10.1063/1.1706161 (9 pages) | Cited 59 times

Online Publication Date: 22 November 2004

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The dispersion relation for electrostatic oscillations in a magnetic field is derived on the basis of the Boltzmann equation for arbitrary velocity distributions and for propagation in an arbitrary direction under the following restrictions: (1) The thermal velocities of the particles and the phase velocity of the wave are small compared to those of light; (2) the component of the wave vector k perpendicular to the magnetic field is small compared to the reciprocal of the gyration radius of an ion at the larger of the mean ion, and electron thermal, energies; (3) the magnetic field B is uniform. The dispersion relation is formally identical with that for electrostatic oscillations in the absence of a magnetic field. The dispersion relation is examined for stability under the further restrictions that: (4) The mean thermal energy of the ions is small compared to that of the electrons; (5) the electron distribution function for the component of the velocity v along B has a single maximum. It is found that a very small shift along B of this maximum relative to the ions leads to an unstable ion oscillation. The growth rates and frequencies of these oscillations are determined; their possible applications are discussed. Some further results on the ``two stream'' instability are given.

Dynamic Stabilization of a Plasma Column

Erich S. Weibel

Phys. Fluids 3, 946 (1960); http://dx.doi.org/10.1063/1.1706162 (15 pages) | Cited 34 times

Online Publication Date: 22 November 2004

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A cylindrical plasma column is confined by a magnetic field whose z component is constant while the φ component oscillates sinusoidally. The skin depth is assumed to be negligible so that the boundary can be considered as sharp and the interior of the plasma as field free. The plasma itself is treated as an assembly of noncolliding particles which are specularly reflected at the plasma surface. The oscillating pressure of the applied field causes the plasma surface to execute a small periodic motion independent of φ and z. All other components of the displacement decay in time. Thus the plasma is positively stable against all deformations. For magnetostatic confinement this treatment yields the usual instabilities, but, in contrast to earlier work, the growth rates are bounded as the deformation wavelength approaches zero. This appears to be in better agreement with experiment.
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Plasma Motion Across Magnetic Fields

George Schmidt

Phys. Fluids 3, 961 (1960); http://dx.doi.org/10.1063/1.1706163 (5 pages) | Cited 75 times

Online Publication Date: 22 November 2004

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The analysis of plasma flow problems in magnetic fields is usually based on a hydromagnetic fluid model. In low‐density collisionless plasmas, however, the limitations of the applicability of this method are not clearly understood. In this paper a simplified self‐consistent field method is used, with particle motion considered in the guiding center approximation. In this case the high ``dielectric constant'' of the magneto‐plasma plays the role of the infinite conductivity of the fluid model. Several experiments are analyzed on the basis of this model, and the limitations and shortcomings of the hydromagnetic treatment discussed.

Charge Excitation of Plasma Motion in a Magnetic Field

R. A. Pappert

Phys. Fluids 3, 966 (1960); http://dx.doi.org/10.1063/1.1706164 (7 pages) | Cited 1 time

Online Publication Date: 22 November 2004

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The wake of a massive point charge traversing a low‐density plasma of vanishing ion temperature, in a constant external magnetic field, is investigated by means of linearized Boltzmann theory. The speed of the exciting charge is taken to be less than the root‐mean‐square thermal speed of the electrons and the electron thermal energy is taken to be much less than the ion energy associated with the ambient flow velocity, that is, the ion flow velocity as seen in a reference frame moving with the external charge. The magnetic field is taken along the direction of motion of the exciting charge. One of the effects of the magnetic field is the production of oscillations of both the charge density and the potential. Over some regions of space the frequency of these oscillations is approximately equal to the ion cyclotron frequency.

Plasma Acceleration in a Radio‐Frequency Field Gradient

G. A. Swartz, T. T. Reboul, G. D. Gordon, and H. W. Lorber

Phys. Fluids 3, 973 (1960); http://dx.doi.org/10.1063/1.1706165 (4 pages) | Cited 3 times

Online Publication Date: 22 November 2004

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Experiments have been performed at 140 Mc to determine the acceleration of a mercury plasma in an rf field gradient. With a maximum rf field amplitude of 170 v∕cm, a tenuous plasma (of density less than a critical density) was accelerated from 5 × 105 to 25 × 105 cm∕sec. A denser plasma (of density greater than the critical density) was decelerated in the same field. These experimental results are consistent with a theory based on energy considerations.

Sufficient Conditions for Hydromagnetic Stability of a Diffuse Linear Pinch

Harold P. Furth

Phys. Fluids 3, 977 (1960); http://dx.doi.org/10.1063/1.1706166 (5 pages) | Cited 3 times

Online Publication Date: 22 November 2004

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From the energy principle a sufficient stability condition can be derived that involves a disposable function. The existence of some choice of the disposable function such as to satisfy the sufficient condition is a necessary condition for stability. The disposable function may also be chosen so as to yield sufficient conditions that are not necessary, but that may be useful from the points of view of simplicity and generality. This technique is illustrated for ``maximal‐force configurations,'' where the current density is everywhere orthogonal to the magnetic field.

Stabilization of Pinch Discharges

Stirling A. Colgate and Harold P. Furth

Phys. Fluids 3, 982 (1960); http://dx.doi.org/10.1063/1.1706167 (19 pages) | Cited 22 times

Online Publication Date: 22 November 2004

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The hydromagnetic stability properties of ``hard‐core'' pinches are shown to be more favorable than those of conventional ``stabilized pinches.'' Linear ``hard‐core'' pinch experiments with a wide range of configurations show a basic consistency with hydromagnetic theory, but all configurations studied so far become unstable at sufficiently high current densities and low particle densities, even if possessed of theoretical hydromagnetic stability. Pinches with nulls in Bθ tend to be much less stable than those with nulls in Bz. ``Inverse stabilized pinches'' yield perfectly reproducible magnetic probe traces at power levels several times those at which ``stabilized pinches'' are unstable. A technique for the study of plasma density distributions is given, based on the propagation of radial compressional waves. The theory of the steady‐state pinch is enlarged to include convective effects and is documented by experiment. The role of electrodes is considered, and an anode‐cathode asymmetry in instability behavior is demonstrated for the linear ``stabilized pinch.'' The prospects of stable, high‐temperature operation with the toroidal hard‐core pinch, or ``levitron,'' are discussed.

Superfast Pinch Experiment

Vernal Josephson

Phys. Fluids 3, 1001 (1960); http://dx.doi.org/10.1063/1.1706135 (7 pages) | Cited 2 times

Online Publication Date: 22 November 2004

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An assembly designed to accelerate deuterons to about 10 kev by means of a radially converging magnetic piston has been constructed and tested. The assembly operates with 300 kv on the energy storage condenser, the ringing frequency is 18 Mc, and present operation uses hydrogen gas at 50 μ. Both slow and fast pinches have been observed, the former being possibly thermal shock waves, and the latter possibly being associated with inward motion of the plasma sheath.

Ionization Growth in a Gas with a Constant Electric Field

E. Gerjuoy and G. W. Stuart

Phys. Fluids 3, 1008 (1960); http://dx.doi.org/10.1063/1.1706136 (8 pages) | Cited 14 times

Online Publication Date: 22 November 2004

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A model for the growth of ionization in a gas with a uniform electric field is studied by Laplace transform techniques. The total electron‐molecule cross section is taken inversely proportional to the velocity; the ionization cross section may have arbitrary velocity dependence; all secondary electrons are created with zero energy; all collisions scatter electrons isotropically but produce no energy loss. With this model, in the steady‐state spatially dependent case that primary electrons are continuously furnished by a cathode at z = 0, an exact solution is found for Townsend's α and for the asymptotic (valid at large z) electron distribution function integrated over velocity orientation. Similarly, in the spatially homogeneous time‐dependent case that primary electrons are furnished at t = 0 by a uniformly distributed source, an exact solution is found for the time rate of ionization growth and for the asymptotic (valid at long times) electron distribution function integrated over velocity orientation. Numerical results, including effective electron temperatures are presented for several assumed ionization cross sections. Because of the neglect of energy loss, applicability of the model is limited to the high E∕p (electric field∕pressure) range.

Nonlinear Behavior of the Electrical Conductivity of a Slightly Ionized Gas

Melvin Epstein

Phys. Fluids 3, 1016 (1960); http://dx.doi.org/10.1063/1.1706137 (3 pages) | Cited 10 times

Online Publication Date: 22 November 2004

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Attention is directed to the nonlinear relationship between the current density and field strength of an applied oscillating electromagnetic field. It is pointed out that the general solution of this problem obtained by Margenau is too complex for use in further analyses except for the limiting case of vanishing field strength. A further approximation is introduced which yields a relatively simple expression for the current‐field strength relationship, valid for large frequencies and moderate field strengths. It is found that significant nonlinear effects may occur even at small field strengths if the gas density is sufficiently low. The effect of the nonlinearity is to increase that part of the current in phase with the applied field.

Kinematics of Ohmic Heated Plasmas in the B‐1 Stellarator

W. Bernstein, A. Z. Kranz, and F. Tenney

Phys. Fluids 3, 1019 (1960); http://dx.doi.org/10.1063/1.1706138 (7 pages) | Cited 16 times

Online Publication Date: 22 November 2004

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A kinematic description of the B‐1 stellarator plasma during various stages of Ohmic heating discharges is presented. It is shown that for specific ranges of plasma current, for a given confining field, the plasma assumes a shape which has an azimuthal distribution of the form exp im θ. It is further shown that the plasma column rotates at high velocity while the distortion is present. The possible relationship between these current ranges and the predictions of hydromagnetic instability theory is considered.
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