Abstract: The stability of a liquid layer flowing down an inclined plane is investigated. A new perturbation method is used to furnish information regarding stability of surface waves for three cases: the case of small wavenumbers, of small Reynolds numbers, and of large wavenumbers. The results for small wavenumbers agree with Benjamin's result obtained by the use of power series expansion, and the results for the two other cases are new. The results for large wavenumbers, zero surface tension, and vertical plate contradict the tentative assertion of Benjamin. The three cases are then re‐examined for shear‐wave stability, and the results compared with those for confined plane Poiseuille flow. The comparison serves to indicate the vestiges of shear waves in the free‐surface flow, and to give a sense of unity in the understanding of the stability of both flows. The case of large wavenumbers also serves as a new example of the dual role of viscosity in stability phenomena.The topological features of the ci curves for...

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

Abstract: The rate of molecular dissociation behind strong shock waves is calculated with the assumption that dissociation can occur preferentially from the higher vibrational levels. An exponential probability of dissociation from the various vibrational levels is employed using an anharmonic oscillator model. Results for the dissociation of oxygen in an argon diluent are presented. Vibrational non‐equilibrium introduces a T−3 temperature dependence into the oxygen dissociation rate constant in the range 4000°–8000°K. A dissociation lag‐time of the order of the extrapolated vibrational relax ation time is predicted immediately behind the shock front. The computed results are shown to be in agreement with available experimental results.

Abstract: A weakly ionized plasma in a uniform magnetic field is considered. Application of a potential across the magnetic field results in a steady current flow, owing to the finite conductivity. It is shown that this steady state is unstable if the plasma density is nonuniform in the direction of the applied electric field and if the applied potential is large enough. It is necessary that the sign of the product of the electric field and the density gradient be positive.

Abstract: The reduction of test time in low pressure shock tubes, due to a laminar wall boundary layer, has been analytically investigated. In previous studies by Roshko and Hooker the flow was considered in a contact surface fixed coordinate system. In the present study it was assumed that the shock moves with uniform velocity, and the flow was investigated in a shock fixed coordinate system. Unlike the previous studies, the variation of free stream conditions between the shock and contact surface was taken into account. It was found that β, a parameter defined by Roshko, is considerably larger than the estimates made by Roshko and Hooker except for very strong shocks. Since test time is proportional to β−2, previous estimates of test time are too large, particularly for weak shocks. The present estimates for β appear to agree with existing experimental data to within about 10 percent for shock Mach numbers greater than 5. In other respects, the basic theory is in general agreement with the previous results of Roshko.

Abstract: A continuum theory for spherical electrostatic probes in a slightly ionized plasma is developed. The density of the plasma is taken to be sufficiently high such that both ions and electrons suffer numerous collisions with the neutrals before being collected by an absorbing probe. A general discussion of probes at an arbitrary potential is given. It is found that for very negative probe potentials the sheath thickness can be comparable to the probe radius. Two explicit forms of current‐voltage characteristics are given; one for very negative probes, the other for probes at nearly plasma potential. Both of these are based on the assumption that the probe radius is large compared with the Debye length. Numerical computation is also given for negative probes of a wider range of probe sizes.

Abstract: In a plasma the ionization energy is decreased due to the presence of the microfield. In the past several attempts have been made to calculate this effect. These calculations, which use statistical and thermodynamical procedures, give different results. They produce either a ``polarization term'' or a ``lattice term'' or both of them. Moreover there are quantitative discrepancies. The lowering of the ionization energy is here derived by a statistical method, which is physically conceivable. The results cover a wide density range below and above the so‐called critical density. The new results are compared with the results of all earlier calculations and reveal the cause of their discrepancies.

Abstract: The breakup of drops exposed to blasts of air is studied in a shock tube. Results are obtained for water, methyl alcohol, and three viscous oils. An acoustical drop holder has been developed in which radiation pressure is used to support the drops at rest in the shock tube. Photographs showing new details in the breakup process are presented.

Abstract: The problem of an electrostatic probe in a dense, slightly ionized gas is treated by techniques of asymptotic analysis. In particular, the asymptotic limits ρp = rp/λd → ∞ and e = T+/T− → 0 are treated in considerable detail. (rp, λd, T−, T+ are probe radius, Debye length, electron, and ion temperature, respectively.) Sample integral curves for electrostatic potential, ion and electron density are given. Probe characteristic curves for three values of e (finite and small) and ρp (large) are also given. It is noted that these characteristics do not saturate for large probe potential because the influence of the probe is felt to very great distances from the probe; the shielding due to the space‐charge sheath is incomplete.

Abstract: The motion of an ideal collisionless plasma subject to periodic boundary conditions, with velocity distribution given initially, is studied. The minimum value which the kinetic energy can subsequently attain is determined, on the assumption that the system is subject only to the constraint that phase‐space volume is conserved.

Abstract: The stability of a free laminar layer between parallel streams is examined. The neutral curve and the curves of constant amplification are obtained by a numerical method for Reynolds numbers ranging from 0 to 40 as well as for the inviscid case. No minimum Reynolds number is found. The eigenfunctions are discussed. The case of a layer of increasing thickness is considered, and it is concluded that turbulent transition will occur when the Reynolds number reaches 150.

Abstract: An exact kinetic equation for plasma and the electromagnetic field is derived. This equation describes the fluctuations of the fields and particle distributions. The solution is obtained by expanding in a parameter which characterizes the amplitude of these fluctuations. A systematic procedure is given for generating the solution to arbitrary order in the expansion. Some typical applications of the theory are presented. These include calculations of a collision integral, incoherent scattering, and bremsstrahlung emission and absorption.

Abstract: The optical reflectivity method was used to investigate the structure of shock fronts in argon from Mach 1.70 to Mach 4.85 and in nitrogen from Mach 2.01 to Mach 3.72. Experimental data were obtained at two wavelengths and over a wide range of initial pressures. The reflectivities, corrected empirically for shock curvature, were fitted to a bimodal profile to yield a maximum‐slope density thickness. The reciprocal of the thickness in argon (expressed in terms of the Maxwellian mean free path in the undisturbed gas) rises rapidly to a maximum of approximately 0.31 at about Mach 3.5 and decreases gradually thereafter. Above Mach 3 the thickness is about 50% greater than calculated from the Navier‐Stokes equations, using a realistic viscosity‐temperature relationship. There is excellent agreement, especially at the higher shock strengths, with recent bimodal calculations carried out by Muckenfuss, using realistic intermolecular potentials. In nitrogen, the shocks are thinner than in argon and appear to attain a minimum value of 2.5 initial mean free paths at about Mach 3.7. Rotational relaxation appears to be as rapid in the strong shocks as previously observed in weak shocks; it appears to be completed within the shock front. The experimental density thicknesses are approximately 50% greater than those calculated from the Navier‐Stokes equations, using the experimental shear viscosity μ and a bulk viscosity of 2μ/3. The agreement with these Navier‐Stokes solutions is about as good as those in argon.

Abstract: The exact motion of a one‐dimensional system interacting through the electrostatic field is followed using a fast computer. The properties of the model are discussed and related to the three‐dimensional plasma. New results are presented for the drag and diffusion of test particles and the space and time correlations of the charge density. These ``experiments'' agree with the conventional theory. In an appendix is derived the kinetic equation for this model to first order in the small parameter, the reciprocal of the number of particles in the Debye length. To first order a velocity distribution is constant in time.

Abstract: The drag coefficients for cylinders normal to the flow have been determined experimentally at Mach ∼2, Mach ∼4, and Mach ∼6 with Knudsen numbers extending from continuum conditions to free‐molecule‐flow conditions. The results indicate a smooth transition from inviscid values at low Knudsen numbers to free‐molecule‐flow predictions for diffuse reflection at high Knudsen number. Small departure theories which are applicable to the ``near free molecule flow'' regime are compared to the experimental data.

Abstract: A shock tube has been developed capable of producing a gas sample of known conditions by shocks with velocities as high as 43 000 ft/sec. The driver of this shock tube employs a capacitor bank which discharges electrical energy into helium, heating the helium to temperatures of 10 000–20 000 °K, and raising the pressure to 10 000–20 000 psi. The high‐pressure bursts the scribed diaphragm and the resulting shock wave propagates into the test gas. Diagnostic techniques have been employed to investigate the resulting hot gas sample. The growth of this sample has been observed optically and correlations have been achieved with theoretical calculations. The observed radiation has been compared with and can be used to extend the known radiative properties of high temperature air. Time‐resolved luminous pictures and spectra have also been taken to show the purity of the test gas. The speed and attenuation of the shock front have been measured. The observed operation of this shock tube has been compared with theoretical predictions, and although no precise correlation can be made, the driver gas energy transfer and losses in the shock tube boundary layer can be accounted for.

Abstract: The one‐dimensional initial‐value problem of a monatomic single component gas is considered. Using the linearized Boltzmann equation the dispersion relation is studied. In addition to the usual gas‐dynamic sound waves, one finds an infinity of decaying propagating waves. The phenomenon exhibits itself as a sequence of epochs, the last state of which is hydrodynamic. With reference to the same problem, macroscopic equations such as Euler, Navier‐Stokes, Burnett, moments equations, etc., are considered. In addition, the recently considered ``kinetic models'' of Gross et al. are applied to the problem. These various formulations are critically analyzed and compared with each other and with the Boltzmann analysis. Lastly, several modifications are offered which remedy some of the shortcomings which appear in the approximate theories.

Abstract: A series of experiments with electromagnetically driven shock waves in various gases have shown that the luminous emission observed along the tube should often be interpreted as being due to direct ionization produced by the driver discharge and not to thermal ionization produced by the shock wave. The experiments also show that for high Mach numbers obtained at low initial pressures (of the order of 1 mm Hg), the shock front and the front of the discharge plasma jet (contact front) practically coincide. Thus, in contradiction with the conclusions reached by Kolb, Griem, and others, the state of the luminous gas following ths shock front cannot be calculated by the Rankine‐Hugoniot equations from the velocity of the front.The results presented have been obtained by spectrographic investigation of the light emitted along a special H‐shaped glass tube in which the gas between one of the electrode pairs is initially different from that filling the rest of the tube.

Abstract: The two‐dimensional motion of a plasma in an inhomogeneous magnetic field with circular symmetry and a homogeneous external field under the influence of an electric field was studied previously [E. T. Karlson, Phys. Fluids 5, 476 (1962)]. The influence of space charges was taken into account. However, the treatment was generally valid only when no forbidden regions for the particles exist. When the inhomogeneity is strong enough, such forbidden regions will arise. The treatment is now generalized, so that it is valid for this case also. A discussion is given of the possibility to explain geomagnetic storms and auroral phenomena within the framework of the present theory.

Abstract: The dimensionless ratio f = λM/ηCv relating the thermal conductivity, molecular weight, viscosity, and constant volume molar heat capacity has been determined for several nonpolar polyatomic gases in the neighborhood of room temperature (270°–295° K). The experimental method, due to Eckert and Irvine, provides a direct determination of f by measurement of the subsonic temperature recovery factor. A recent theory of Mason and Monchick has been used to calculate collision numbers for rotational relaxation from the experimental data as follows: CH4, 9.4; CF4, 3.0; SF6, 2.5; C2H4, 2.4; C2H6, 4.0; O2, 12; N2, 7.3; CO2, 2.4; and C2H2, 1.8. Collision numbers for the near‐spherical molecules were in close accord with a classical theory for rough sphere molecules with attractive forces; ethylene, which deviates appreciably from spherical symmetry, exhibited a smaller collision number. The data on linear molecules were in qualitative agreement with a quantum treatment. In general, collision numbers for rotational relaxation are determined by the following factors: (1) The molecular mass distribution, (2) the strength of the intermolecular attractive forces, and (3) the molecular asymmetry.

Abstract: A conducting limiter brings about equilibrium of a toroidal plasma by anchoring magnetic field lines. Charge separation arising from the curvature effect can be short circuited by the limiter. This way of achieving equilibrium is not possible if the charge separation current exceeds the maximum ion saturation current to the limiter. Experiments with the Model C stellarator are consistent with this model. An observed instability is related to the loss of equilibrium due to the excess charge separation current.

Abstract: The general theory for high‐temperature plasmas, in the presence of an external uniform magnetic field, is developed by means of a Laplace‐transform technique. The dispersion relations for the longitudinal and the transverse oscillations, for a relativistic Maxwellian distribution with mass motion as well as for a non‐Maxwellian distribution in the absence of magnetic field are derived. The stability of two streams is investigated in detail both in the nonrelativistic and in the extreme relativistic limits. The longitudinal and the transverse modes are uncoupled only when the stream velocity and the wave velocity are parallel.

Abstract: Two types of hot plasmas were produced and confined between magnetic mirrors by plasma injection followed by slow magnetic compression in the Table Top III device: (1) a ``hot‐electron plasma,'' in which the plasma was highly compressed and the electron temperature was 10 to 25 keV, and (2) a ``hot‐ion plasma'' in which the compression was less but the ions were hotter than the electrons. In both cases, regimes could be found in which gross instability occurred. The plasma column was observed to leave the axis and spiral, as a whole, outward to the chamber walls with a radial velocity of about 5 × 105 cm/sec. On the basis of the observations and a comparison with theory this behavior is interpreted as a simple azimuthal mode number M = 1 interchange instability accompanied by a rotation arising from the fact that the plasma has a net charge. No multiple flutes (M > 1) have been observed during this motion, although occasionally simultaneous existence of two plasma columns has been detected. The lower critical density for the onset of the instability was approximately that density for which the Debye distance was equal to the plasma diameter. Stabilized regimes were also obtained by operation with high injected densities or at higher base pressures and the plasma remained confined for many milliseconds. There was also an intermediate regime in which instability could be deliberately ``triggered'' by a finite electrostatic perturbation in a plasma that had been quiescent for milliseconds. Some aspects of the observed stabilization, particularly the nonobservance of multiple flutes, occurring under conditions in which simple hydromagnetic theory predicts instability for all values of M > 0, seem to be explainable in terms of a ``finite‐orbit'' theory, although other effects seem also to be present.

Abstract: As the curvature of shock waves increases, the shock structure becomes two dimensional, and the usual Hugoniot jump conditions no longer hold. An equation has been derived for the structure of such a two‐dimensional non‐Hugoniot shock in the case of weak shocks with Mach numbers close to one. The development of this equation from the Navier‐Stokes equations is based on the assumptions that the vertical velocity is of order (M1* − 1)3/2 and that the flow within the shock is irrotational. From the derivation it appears that the non‐Hugoniot region behaves as an acoustic wave driven by higher‐order viscous effects. The properties of the above equation, which has been called the viscous‐transonic or V‐T equation have been investigated. The V‐T equation appears to be a combination of Burgers' equation for weak normal shock structure and the transonic equation. It is shown that the structure of oblique shocks is a similarity solution of the V‐T equation. Proper formulation of boundary conditions is considered a...

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

Abstract: A pipe closed at both ends contains a pressure‐sensitive plane heat source at its midsection. The heat release rate is assumed to be proportional to the pressure fluctuation at the heater. When the feedback coefficient e is positive, the system is unstable and a self‐excited oscillation grows exponentially in time until shock waves are formed in the system. The ensuing self‐sustained nonlinear periodic vibration is calculated for the case e « 1. It is shown that the pressure‐time curve at a fixed point in the pipe has the form of a sawtooth function, while the velocity‐time curve has the form of a square wavefunction. To the order of e, the frequency of the self‐sustained vibration is equal to the resonance frequency of the system. The shock waves propagate with constant strength and the shock loci in the xt space are calculated. The effects produced by a time lag in heat source's response are discussed.

Abstract: A factorization theorem is demonstrated for special types of contributions to the time‐dependent collision operator G00(τ) which plays a central role in the classical master equation derived by I. Prigogine and P. Resibois; this theorem expresses the mutual independence of two subgroups of particles when these groups are noninteracting with each other during a given time interval (τi, τj) of the collision process. The theorem is applied to rederive in a very direct way the kinetic equation of an homogeneous stable plasma, obtained first by R. Balescu. Other possible applications of the method are also discussed.

Abstract: A presentation of transport coefficients in a fully ionized plasma in terms of effective collision, gyration, and oscillation frequencies is given. It has the virtue of reducing the task of presenting the data to the tabulation of certain dimensionless functions of restricted variation. Moreover the quantities involved are directly related to the reciprocals of the coefficients usually considered, for instance to the resistivity rather than the conductivity.