# Showing papers in "Physics of Fluids in 1971"

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TL;DR: In this paper, a simple model for the nonlinear interaction of a low-density monoenergetic electron beam and a relatively cold infinite homogeneous one-dimensional plasma was proposed.

Abstract: Recently, a simple model was proposed for the nonlinear interaction of a low‐density monoenergetic electron beam and a relatively cold infinite homogeneous one‐dimensional plasma. The essential feature of this model is the observation that after several e‐foldings the bandwidth of the growing waves is so narrow that the electrons interact with a very nearly pure sinusoidal field. In terms of this single wave model, a properly scaled solution of the nonlinear beam‐plasma problem which depends analytically on all the basic parameters of the problem (i.e., plasma density, beam density, plasma thermal velocity, and beam drift velocity) is presented. This solution shows that the single wave grows exponentially at the linear growth rate until the beam electrons are trapped. At that time the wave amplitude stops growing and begins to oscillate about a mean value. During the trapping process the beam electrons are bunched in space and a power spectrum of the higher harmonics of the electric field is produced. Both the oscillation in wave amplitude and the power spectrum are given a simple physical interpretation.

384 citations

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TL;DR: In this paper, an algorithm for the rapid calculation of the convolution sums appearing in the Fourier-transformed Navier-Stokes equations in three dimensions is given. But the algorithm is only a factor of 4 more efficient than previously suggested algorithms.

Abstract: An algorithm is given for the rapid calculation of the convolution sums appearing in the Fourier‐transformed Navier‐Stokes equations. In three space dimensions, the new algorithm is a factor of 4 more efficient than previously suggested algorithms.

348 citations

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TL;DR: In this article, the fundamental derivative has been defined as the non-dimensional form Γ≡12ρ3c4(∂2Υ/∂P2)s.

Abstract: The quantity which is here called the fundamental derivative has been defined as the nondimensional form Γ≡12ρ3c4(∂2Υ/∂P2)s. The relation of Γ to other thermodynamic variables is discussed. It is already known that the existence of conventional compression shocks requires Γ>0. It is shown that other dynamic behavior of compressible fluids is fixed by the sign of Γ. Particular emphasis is given to phenomena corresponding to negative Γ. These phenomena include the area variation of a transonic passage, the form of a Prandtl‐Meyer wave, the behavior of adiabatic flow with friction, and nonlinear wave propagation. Formulas and numerical values are given for Γ in various substances.

315 citations

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TL;DR: In this article, the stability of a steady, axisymmetric, laminar, primary flow composed of two fluids flowing concentrically in a straight circular tube is investigated by the method of small perturbations.

Abstract: The stability of a steady, axisymmetric, laminar, primary flow composed of two fluids flowing concentrically in a straight circular tube is investigated by the method of small perturbations. Both asymmetric and axisymmetric disturbances to the primary flow are considered. It is demonstrated that, regardless of the size of the Reynolds number, no situations are encountered for which the primary flow is stable to both types of disturbances, simultaneously. The primary cause of instability is found to be the difference in viscosities of the two fluids.

259 citations

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TL;DR: In this paper, a pair of coupled, nonlinear, partial differential equations which describe the evolution of low-frequency, large-scale-length perturbations propagating parallel, or nearly parallel, to the equilibrium magnetic field in high-β plasma have been obtained.

Abstract: A pair of coupled, nonlinear, partial differential equations which describe the evolution of low‐frequency, large‐scale‐length perturbations propagating parallel, or nearly parallel, to the equilibrium magnetic field in high‐β plasma have been obtained. The equations account for irreversible resonant particle effects. In the regime of small but finite propagation angles, the pair of equations collapses into a single Korteweg‐de Vries equation (neglecting irreversible terms) which agrees with known results.

252 citations

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TL;DR: In this article, the velocity correlation function and the diffusion coefficient are calculated in detail using functional probabilities and compared with those from computer experiments and their significance for real plasma is discussed, and it is shown that in this model diffusion always exhibits the anomalous 1/B variation with magnetic field.

Abstract: Diffusion of plasma in two dimensions is studied in the guiding center model. It is shown that in this model diffusion always exhibits the anomalous 1/B variation with magnetic field. The velocity correlation function and the diffusion coefficient are calculated in detail using functional probabilities. In addition to the 1/B field dependence, the diffusion coefficient is unusual in that it depends weakly on the size of the system. These theoretical results are compared with those from computer experiments and their significance for real plasma is discussed.

239 citations

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TL;DR: In this paper, an experimental and theoretical investigation of the angular distribution of the electric field of a short radio frequency probe in a plasma in a magnetic field is described, where the field becomes very large along a resonance cone whose axis is parallel to the static magnetic field and whose opening angle is observed to vary with incident probe frequency, cyclotron frequency, and plasma frequency in agreement with cold plasma dielectric theory.

Abstract: An experimental and theoretical investigation of the angular distribution of the electric field of a short radio frequency probe in a plasma in a magnetic field is described. The field is observed to become very large along a resonance cone whose axis is parallel to the static magnetic field and whose opening angle is observed to vary with incident probe frequency, cyclotron frequency, and plasma frequency in agreement with simple cold plasma dielectric theory. The relationship of these cones to the limiting phase‐ and group‐velocity cones which appear in the theory of plane wave propagation is discussed. The addition of electron thermal velocities (warm plasma effects) is examined in the limit of a large static magnetic field. A fine structure appears inside the cones and is shown to result from an interference between a fast electromagnetic wave and a slow plasma wave. This interference structure is observed experimentally and measurements of the angular interference spacing are shown to agree with the warm plasma theory. The use of measurements of the resonance cones and structure as a diagnostic tool to determine the plasma density and electron temperature in a plasma in a magnetic field is discussed.

204 citations

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TL;DR: In this paper, the motion of a vapor gas in contact with interphase surfaces is studied on the basis of kinetic theory of gases, and the mass and energy fluxes are determined from the condition of the vapor far away from the interfaces.

Abstract: The motion of a vapor gas in contact with interphase surfaces is studied on the basis of kinetic theory of gases. The mass and energy fluxes are determined from the condition of the vapor far away from the interfaces. The result for a vapor gas between two interfaces shows that the temperature gradient in the vapor is opposite in sign to the externally maintained temperature difference.

179 citations

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TL;DR: In this article, the existence and structure of large amplitude, stationary, longitudinal plasma oscillations are studied for the case of a simple waterbag distribution of electrons and an immovable background of ions.

Abstract: The existence and structure of large amplitude, stationary, longitudinal plasma oscillations are studied for the case of a simple waterbag distribution of electrons and an immovable background of ions. The analysis employs the one‐dimensional Vlasov equation for a plasma of infinite spatial extent. An expression for the maximum amplitude of the oscillations is derived. This maximum amplitude decreases monotonically as the ratio of the electron thermal velocity to the wave phase velocity increases. The structure of the oscillations is expressed analytically in terms of hyperelliptic integrals.

164 citations

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TL;DR: The dynamical formation and structure of the plasma focus has been studied with a two-dimensional numerical fluid model in this paper, and the extremely high kinetic energy densities obtained in the numerical fluid experiment as the result of adiabatic compression and viscous heating agree well with experiment.

Abstract: The dynamical formation and structure of the plasma focus has been studied with a two‐dimensional numerical fluid model. The extremely high kinetic energy densities obtained in the numerical fluid experiment as the result of adiabatic compression and viscous heating agree well with experiment. Three features in the plasma focus are isolated: an anode cold source, a hot pinch region, and an axial shock. The anomalously long lifetime of the plasma focus is shown to be the result of axial flow, with stabilization of magnetohydrodynamic modes through the ion stress tensor in the intermediate collisionless, collision‐dominated regime. Estimates of the neutron yield based on the numerical fluid experiment concur with experimental yields, and are the result of thermally reacting deuterons in the hot pinch region. The plasma parameters of interest determined from the hot pinch region suggest that the ion distribution function will not have a simple Maxwellian form and this in particular may account for the discrepancy with experiment on the anisotropy in space of the neutron yield.

156 citations

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TL;DR: In this article, the threshold ac electric field required for the excitation of the purely growing "oscillating two-stream" and decay instabilities is found to increase when the density gradient scale length H becomes less than the mean free path.

Abstract: The threshold ac electric field required for the excitation of the purely growing “oscillating two‐stream” and decay instabilities is found to increase when the density‐gradient scale length H becomes less than the mean free path. For a plasma with no magnetic field, the threshold field is given by E2/4πnTe≈2(k‖H)−1, where k‖≲(1/4)D−1 is the wavenumber of the instability in the direction orthogonal to the density gradient. (D is the electron Debye length.) Qualitative arguments suggest that this result should hold in the presence of a magnetic field. Recent experimental measurements of the threshold field agree well with these theoretical calculations. The reason for the threshold field increasing is that, in an inhomogeneous plasma, the unstable region has a finite extent spatially and energy (in the form of electron plasma waves) can propagate out of this region, thus creating an energy loss not found in uniform plasmas.

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TL;DR: In this paper, a nonlinear analysis of collisional drift waves is presented in which a systematic expansion is made in powers of the wave amplitude, including the effects of resistivity, viscosity, and thermal transport.

Abstract: A nonlinear analysis of collisional drift waves is presented in which a systematic expansion is made in powers of the wave amplitude. The two‐fluid equations are used, including the effects of resistivity, viscosity, and thermal transport. The result, for the wave amplitude as a function of magnetic field in the linearly unstable region close to marginal stability, agrees reasonably well with experiment.

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TL;DR: The limits of very slow and very fast sheath growth in a low-pressure plasma can be described by Child's law and the ion matrix model, respectively as mentioned in this paper, and it is shown that the range between these two extremes can be treated analytically by transforming the equations to the sheath edge frame of reference.

Abstract: The limits of very slow and very fast sheath growth in a low‐pressure plasma can be described by Child's law and the ion matrix model, respectively. It is shown that if the sheath edge grows with constant speed, the range between these two extremes can be treated analytically by transforming the equations to the sheath edge frame of reference. This provides a continuous transition between the two extremes which reduces to Child's law for very slow growth and to the ion matrix model for very fast growth. The theoretical expression for sheath thickness calculated from the continuous transition model is found to be in good agreement with experimental results.

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TL;DR: In this paper, the linear stability problem for a fluid in a classical Benard geometry, when the temperature gradient has both a steady and a time-periodic component, is considered and the modulating effect of the oscillatory gradient on the stability characteristics of the basic configuration is examined.

Abstract: The linear stability problem for a fluid in a classical Benard geometry, when the temperature gradient has both a steady and a time‐periodic component, is considered. The modulating effect of the oscillatory gradient on the stability characteristics of the basic configuration is examined. It is found that, in general, there is enhancement of the critical value of a suitably defined Rayleigh number.

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TL;DR: In this paper, the structure of turbulent electric and magnetic field fluctuations is described, and the internal dynamics of the associated velocity fluctuations are compared with those of fluid turbulence, and it is suggested that the plasma turbulence is qualitatively similar to fluid turbulence but with the turbulent elements elongated along the mean magnetic field to form rolls.

Abstract: The structure of turbulent electric and magnetic field fluctuations is described, and the internal dynamics of the associated velocity fluctuations are compared with those of fluid turbulence. It is suggested that the plasma turbulence is qualitatively similar to fluid turbulence but with the turbulent elements elongated along the mean magnetic field to form rolls. This suggests that an appropriate comparison might be with the hypothetical two‐dimensional limit of fluid turbulence, in which energy is expected to be transferred toward long wavelengths rather than to short wavelengths as for isotropic turbulence. In the plasma case, the direction of energy flow is inferred to be toward short wavelengths but the measured form of the triple correlation is close to that expected for two‐dimensional turbulence; we cannot, therefore, make a clear choice between the two alternatives. Effects arising from the particle structure of the plasma do not appear to be important, except that at low pressures an increased damping occurs which may be ion Landau damping. The source of the turbulent energy is not primarily convection due to the pressure gradient but involves some mechanism of direct coupling with the plasma current associated with the “excess resistance” of the discharge.

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TL;DR: In this article, the energy stability limit for a Boussinesq ferrofluid in the presence of gravity and thermal and magnetic field gradients is obtained for a motionless fluid heated from below the limit.

Abstract: The energy stability limit is obtained for a Boussinesq ferrofluid in the presence of gravity and of thermal and magnetic field gradients. For a motionless fluid heated from below the limit is identical to the linear theory result.

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TL;DR: In this paper, a characteristic equation describing the stability of an electrified fluid cylinder is derived taking into account effects due to the electric field, surface tension, viscosity, and inertia.

Abstract: A characteristic equation describing the stability of an electrified fluid cylinder is derived taking into account effects due to the electric field, surface tension, viscosity, and inertia. The electric field causes the cylinder to be unstable subsequent to nonaxisymmetric deformations of the interface. For relatively inviscid fluids electrification can make the growth rates for nonaxisymmetric disturbances almost as large as the rates for axisymmetric forms. It is found that viscous effects tend to damp axisymmetric motions more than motions accompanying a sinuous deformation; thus, the sinuous mode may be the most unstable form of disturbance in many situations. The combined effects of viscosity and the electric field act so that the cylinder manifests the greatest instability at wavelengths about as large as the circumference of the cylinder, when viscous effects are paramount, in contrast to Rayleigh's result for an unelectrified fluid cylinder.

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TL;DR: In this paper, the effects of sensor length on measured one-dimensional temperature spectrum and temperature variance dissipation rate were evaluated numerically for isotropic turbulence with the Corrsin-Pao 3D temperature spectrum, providing response curves.

Abstract: Effects of sensor length on measured one‐dimensional temperature spectrum and temperature variance dissipation rate are studied. Expressions are evaluated numerically for isotropic turbulence with the Corrsin‐Pao three‐dimensional temperature spectrum, providing response curves.

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TL;DR: In this article, the structure of turbulent line vortices is examined and a general argument is constructed to show that the vortex must develop an overshoot of circulation if it entrains fluid at a rate greater than that due to molecular diffusion.

Abstract: The structure of turbulent line vortices is examined. A general argument is constructed to show that the vortex must develop an overshoot of circulation if it entrains fluid at a rate greater than that due to molecular diffusion. A weak hypothesis on the distribution of Reynolds stress leads to the logarithmic profile of Hoffman and Joubert and an estimate of the maximum Reynolds stress. The results of a turbulence model due to Saffman are presented and shown to be poor.

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TL;DR: In this paper, the authors used the numerical model of a spark discharge in air, described in the preceding paper, to compute the properties of the discharge channel of a lightning return stroke and its acoustic wave.

Abstract: The numerical model of a spark discharge in air, described in the preceding paper [Phys. Fluids 14, 2111 (1971)], is used to compute the properties of the discharge channel of a lightning return stroke and its acoustic wave. Initial conditions and electric current waveforms were varied over a wide range, to correspond with the observed variability of natural lightning events. Computed temperatures, pressures, and electron densities in the discharge channel agree well with available observational data. The model predicts temperatures near the upper limit of the range of spectroscopically determined temperatures. Discharge channel radii agree well with similarity solutions for the growth of a spark channel. The energy dissipated per unit length of the discharge channel and the strength of the pressure wave lie well below most literature estimates. The computed energy inputs fall between 10 and 60 J/cm, and depend primarily on the maximum values of the discharge current.

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TL;DR: In this article, the problem of evaporation from an interphase and heat conduction in a gas are studied on the basis of the linearized single-relaxation model equation.

Abstract: The problem of evaporation from an interphase and the problem of heat conduction in a gas are studied on the basis of the linearized single‐relaxation model equation. Both problems lead to the same set of integral equations. Exact values are obtained for the microscopic temperature and density jumps. It is found that the macroscopic temperature and density jumps of the evaporation problem can be expressed in terms of the macroscopic jumps of the heat conduction problem and vice versa. A proof for the existence and uniqueness of the solutions is also given.

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TL;DR: In this article, a numerical model is presented which describes the evolution with time of a short segment of a spark channel in air and its associated acoustic wave, assuming a straight, cylindrical conducting column in which local thermodynamic equilibrium exists at every point.

Abstract: A numerical model is presented which describes the evolution with time of a short segment of a spark channel in air and its associated acoustic wave. The model assumes a straight, cylindrical conducting column in which local thermodynamic equilibrium exists at every point. The electrical energy input to the column is determined by a prescribed electrical current waveform, coupled with a computation of the plasma conductivity. The evolution with time of the conducting column and its surrounding flow field is then found by numerical integration of the equations of gas dynamics. The model employs a realistic equation of state for air at high temperatures, and incorporates kinetic and radiative energy transport processes. It is shown that a satisfactory description of the properties of a spark channel cannot be achieved when radiative transport processes are neglected. The model agrees well with experimental measurements of spark channel radii, temperatures, pressures, and electron densities, and predicts the resultant shock wave strengths closely. The voltage gradients along the spark channel predicted by the model, and the total energy input to the channel, are not as uniformly in agreement with experiment. Possible reasons for these discrepancies are discussed.

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TL;DR: In this paper, the authors considered ion heating by a strong ion−ion two-stream instability perpendicular to a magnetic field in the presence of a relatively cold electron background (Te≪miVd2) is considered.

Abstract: Ion heating by a strong ion‐ion two‐stream instability perpendicular to a magnetic field in the presence of a relatively cold electron background (Te≪miVd2) is considered. The magnetic field strength is such that the ion trajectories are straight, whereas the electrons are bound to the field lines (krLe≪1≪krLi). Theory is presented for both quasilinear and nonlinear stages of the evolution of the system for the case that the instability is electrostatic [(B2/8π) (1+β) >nmiVd2/8] and is compared with a series of computer simulation experiments. It is found that the quasilinear theory gives a fairly accurate description of spatially averaged plasma properties until the ion beams have been sufficiently modulated for ions to be trapped by the waves. In the subsequent nonlinear stage, stabilization occurs when the ion trapping period is equal to the reciprocal growth rate associated with the instability. The directed ion beam energy is mainly converted into random ion energy. The possible role of this instabil...

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TL;DR: In this article, an approximate method for solving the slip problems in the kinetic theory of gases is proposed, and it is shown that extremely good results can be obtained by a simple modification of Maxwell's arguments.

Abstract: An approximate method for solving the slip problems in the kinetic theory of gases is proposed, and it is shown that extremely good results can be obtained by a simple modification of Maxwell's arguments.

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TL;DR: Based on experimentally observed scaling laws, the interaction between the motion in the sublayer with that of the wake-like outer flow was described in terms of a limit cycle as mentioned in this paper, where the inner flow was modeled as a linear combination of the two sublayer flows.

Abstract: Based on experimentally observed scaling laws, the interaction between the motion in the sublayer with that of the wakelike outer flow is described in terms of a limit cycle.

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TL;DR: In this paper, Doppler spectra in microwave scattering from wind waves have been measured at X band (3.2 cm) and K band (1.25 cm) in a short fetch wind wave tank and the scattering cross sections, mean frequency shifts, and bandwidths obtained are compared with results of measurements of the directional slope spectra, mean drift, and particle velocities of the wind waves.

Abstract: Doppler spectra in microwave scattering from wind waves have been measured at X band (3.2 cm) and K band (1.25 cm) in a short fetch wind wave tank. The scattering cross sections, mean frequency shifts, and bandwidths obtained are compared with results of measurements of the directional slope spectra, mean drift, and particle velocities of the wind waves. The scattering is attributed to low order Bragg scattering, the phase speeds of the wind waves are deduced and the dependence of the Doppler bandwidth on wind speed and Bragg wavenumber is given.

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TL;DR: In this paper, a numerical method of calculating the correct stable equilibrium shape of a conducting drop subject to electrical forces is described, and its application to the problem of the stability of an isolated water drop of undistorted radius R and surface tension T situated in an electric field is described.

Abstract: A numerical method of calculating the correct stable equilibrium shape of a conducting drop subject to electrical forces is described. Its application to the problem of the stability of an isolated water drop of undistorted radius R and surface tension T situated in an electric field E indicates that a drop is unstable if E(R/T)12 exceeds a value of 1.603 and its maximum stable deformation expressed as a/b, the ratio of the major to minor axes is 1.83. The numerical method is also applied to a study of the stability of drop pairs. The results show that the critical value of E(R/T)12 depends upon the initial separation X of the drops. For X = 0, E(R/T)12 is zero and its value increases as X increases, tending to a value of 1.603. Finally, the relevance of these results to the behavior of drops in the atmosphere is discussed.

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TL;DR: In this article, a one-dimensional continuum hydrodynamic theory is used to investigate the structure of the deflagration wave which occurs when a laser light beam impinges on a solid target.

Abstract: A one‐dimensional continuum hydrodynamic theory is used to investigate the structure of the deflagration wave which occurs when a laser light beam impinges on a solid target. It is shown that the nonlinear electron heat conduction is responsible for most of the structure; a region of density higher than the cutoff is strongly heated. Density and temperature profiles are calculated. An approximate solution for the thickness x of the overdense layer gives x = 45[(γ−1) / (5γ−1)](m/k)3/2A(Tc2/ρc) (m is the ion mass, k is the Boltzmann constant, A T5/2 is the nonlinear electron heat conduction coefficient, Tc is the temperature of the plasma at cutoff density ρc. The average density in the layer is about twice the cutoff density. The effects of viscosity and (since electrons are heated by the laser light) of ion‐electron relaxation are evaluated.

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TL;DR: In this article, a numerical analysis of the laminar boundary layer on a fixed circular disk of radius a whose axis is concentric with that of a vortex having circulation Γ is described.

Abstract: A numerical computation of the laminar boundary layer on a fixed circular disk of radius a whose axis is concentric with that of a vortex having circulation Γ is described. The computations were started at the edge of the disk and continued inward toward the axis until the properties of the terminal flow became evident. A two‐layer asymptotic expansion was formulated for the solution of the boundary‐layer equations near the axis, and the terminal‐flow properties revealed by the analysis are shown to be in excellent agreement with the numerical results. The structure of the terminal boundary layer consists of an inner layer next to the surface with thickness O[(ν/Γ)1/2r] in which the flow is primarily radial, and an outer layer with thickness O[(ν/Γ)1/2a] of predominantly inviscid nature in which the flow recovers to the external potential vortex. The mass flux in the outer layer does not vanish as r→0, indicating that the boundary layer must erupt from the surface at r=0in the manner envisioned by Moore.

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TL;DR: Transport cross sections and collision integrals are calculated over a wide range of energies and temperatures for the attractive and repulsive screened Coulomb potentials in this article, and results accurate to 1% over the whole range from classical to quantum-mechanical behavior are given in the form of numerical tables.

Abstract: Transport cross sections and collision integrals are calculated over a wide range of energies and temperatures for the attractive and repulsive screened Coulomb potentials. Results accurate to 1% over the whole range from classical to quantum‐mechanical behavior are given in the form of numerical tables. Results of lower accuracy are given as analytical expressions.