Ordinary differential equations of the first order linear differential equations complex numbers and linear algebra simultaneous linear differential equations numerical methods the descriptive theory of nonlinear differential equations mechanical systems and electric circuits Fourier series Fourier integrals and Fourier transforms the Laplace transformation partial differential equations Bessel functions and Legendre polynomials applications and further properties of matrices vector analysis the calculus of variations analytic functions of a complex variable infinite series in the complex plane the theory of residues conformal mapping.

Advanced Engineering Mathematics. By C. R. Wylie. Pp. xiv, 813. 1966. (McGraw-Hill.)

A review is given on arc cathode spots, mainly based on the investigation of arcs in a vacuum with cold cathodes. For the latter and after a short description of general features and theoretical concepts, experiments are presented that study the temporal and spatial behaviour of the spots with high time and space resolution of less than 10 ns and less than 5 µm, respectively. With the help of these observations the various spot types described in the literature are ordered into three levels: level A corresponding to the proper spot with typical diameters of 50-100 µm, level B associated with spot fragments having a size of 10-20 µm and level C comprising a substructure of the fragments. The structures undergo periodic fluctuations of brightness and position with characteristic times that can be arranged in a hierarchy from a few nanoseconds through about 100 µs. The analysis of these fluctuations shows that the spot operates in cycles that include both extremely non-stationary periods with time constants of less than 10 ns and more stationary periods in the microsecond range. In the presence of an external magnetic field, the latter periods lead to unstable plasma configurations that give rise to retrograde motion. Finally, for vacuum arc spots the basic parameters are summarized. After that, the peculiarities of spots in gases with cold electrodes are discussed, followed by a presentation of spots with hot cathodes at high pressures.

https://iopscience.iop.org/article/10.1088/0022-3727/34/17/202/pdf

Topical Review: Cathode Spots of Electric Arcs

Theoretical predictions are given of the development of the current and the distributions of charge and electric field in negative corona, or Trichel current pulses [G W Trichel, Phys Rev 54, 1078 (1938)], in oxygen at a pressure of 667 kPa (50 Torr) For a 10-mm-diam negative sphere located 20 mm from a positive plane, the calculated current pulse has a rise time of 11 ns, a pulse width of 50 ns, and a peak amplitude of 13 mA These results agree satisfactorily with experimental values The predicted velocity of the cathode-directed light pulse also agrees well with observations The theory is based on the accurate numerical solution of Poisson's equation in conjunction with the continuity equations for electrons, positive ions, and negative ions The effects of ionization, attachment, recombination, electron diffusion, and photoemission and ion secondary-electron emission from the cathode are all included The initial steep rise of the current pulse is largely due to rapid ionization and electron motion in the high Laplacian field near the cathode As the discharge develops, a dense plasma forms near the cathode, leading to strong space-charge distortion of the field A prominent cathode fall region is formed immediately adjacent to the cathode, an almost zero field is formed within the plasma and the field is enhanced over the region to the anode The current pulse is quenched because the low electric field in the plasma immobilizes the majority of the electrons which then undergo three-body attachment; furthermore, the cathode fall region becomes reduced to such a short distance that insignificant current is produced from this region Because of the low mobility of the negative ions, the current remains low and the structure of the space-charge fields changes only slowly with time between pulses

Theory of negative corona in oxygen.

Biological decontamination using a nonthermal gas discharge at atmospheric pressure in air is the subject of significant research effort at this time. The mechanism for bacterial deactivation undergoes a lot of speculation, particularly with regard to the role of ions and reactive gas species. Two mechanisms have been proposed: electrostatic disruption of cell membranes and lethal oxidation of membrane or cytoplasmic components. Results show that death is accompanied by cell lysis and fragmentation in Gram-negative bacteria but not Gram-positive species, although cytoplasmic leakage is generally observed. Gas discharges can be a source of charged particles, ions, reactive gas species, radicals, and radiation (ultraviolet, infrared, and visible), many of which have documented biocidal properties. The individual roles played by these in decontamination are not well understood or quantified. However, the reactions of some species with biomolecules are documented otherwise in the literature. Oxidative stress is relatively well studied, and it is likely that exposure to gas discharges in air causes extreme oxidative challenge. In this paper, a review is presented of the major reactive species generated by nonthermal plasma at atmospheric pressure and the known reactions of these with biological molecules. Understanding these mechanisms becomes increasingly important as plasma-based decontamination and sterilization devices come closer to a wide-scale application in medical, healthcare, food processing, and air purification applications. Approaches are proposed to elucidate the relative importance of reactive species

Bactericidal Action of the Reactive Species Produced by Gas-Discharge Nonthermal Plasma at Atmospheric Pressure: A Review

Significant departures from standard transport coefficients [e.g., Sov. Phys. JETP 6, 338 (1958)] have been found for the electron current and heat flux in a fully ionized plasma. These have been discovered by carrying out a direct and accurate numerical solution of the linearized Fokker–Planck equation using a Cartesian tensor expansion of the distribution function. The results, which were carried out for plasmas of various atomic numbers, show the presence of major inaccuracies (errors of up to 65%) in Braginskii coefficients β■, κ⊥, and κ■ (as conventionally defined) for Hall parameters ωτ in the range 0.3≲ωτ≲30. Surprisingly, α■ and β⊥ are found to depend on τ/(ωτ)2/3 and 1/(ωτ)5/3, and not on τ/(ωτ) and 1/(ωτ)2, respectively, as ωτ→∞. An analytic expansion for large ωτ verifies this result, showing that the relatively cold unmagnetized electrons in the distribution function play a dominant role in the cross‐field transport. The numerical results are fitted to within 15% to a polynomial in ωτ for vari...

Plasma transport coefficients in a magnetic field by direct numerical solution of the Fokker–Planck equation

The formula derived by Nobuyasu Sato (1980 J. Phys. D: Appl. Phys. 13 L3-L6), for the current flowing in an external circuit due to the motion of charged particles in a gap is extended to include a time-dependent applied voltage. The effects of negative ions, and electron, positive ion and negative ion diffusion are also included. It is found that Sato's original equation still describes the contribution of charged particle motion to the external circuit current. The total circuit current is determined by adding the displacement current that effectively flows across the gap to the current determined by Sato's equation. Some comments are also made about the interpretation of Sato's equation in three dimensions.

https://iopscience.iop.org/article/10.1088/0022-3727/32/5/005/pdf

The discharge current induced by the motion of charged particles in time-dependent electric fields; Sato's equation extended

A non-equilibrium one-dimensional model is proposed for the plasma sheaths near the electrodes of electric arcs where the cathode is a thermionic emitter. Three coupled equations are solved: (a) the charge continuity equation, which accounts for ambipolar diffusion, recombination and ionization, (b) a form of Ohm's law relating the effective electrical conductivity with a local electric field and (c) the energy balance equation, accounting for thermal conduction, Ohmic heating, plasma radiation and the effects which occur at the electrode surface, i.e. thermionic cooling, ion heating, radiative heating from the plasma and radiative cooling by thermal emission. The three equations give distributions of temperature, electric field and charge density within the sheath. Results are given for the sheath at the thoriated-tungsten cathode and anode for a 200 A arc in argon. Calculations have also been made of sheath properties for various electrode materials to determine the critical current density sustainable by a cathode without cathode melting. Critical current densities for anode melting are less than for cathode melting because of the cooling effect at the cathode of thermionic emission. The authors find that the effect of ion heating tends to keep the cathode temperature near to the temperature for which the current density from thermionic emission equals the imposed current density.

http://iopscience.iop.org/article/10.1088/0022-3727/26/4/016/pdf

A one-dimensional theory for the electrode sheaths of electric arcs

An improved finite-element flux-corrected transport (FE-FCT) scheme, which was demonstrated in one dimension by the authors, is now extended to two dimensions and applied to gas discharge problems. The low-order positive ripple-free scheme, required to produce a FCT algorithm, is obtained by introducing diffusion to the high-order scheme (two-step Taylor-Galerkin). A self-adjusting variable diffusion coefficient is introduced, which reduces the high-order scheme to the equivalent of the upwind difference scheme, but without the complexities of an upwind scheme in a finite-element setting. Results are presented which show that the high-order scheme reduces to the equivalent of upwinding when the new diffusion coefficient is used. The proposed FCT scheme is shown to give similar results in comparison to a finite-difference time-split FCT code developed by Boris and Book. Finally, the new method is applied for the first time to a streamer propagation problem in its two-dimensional form.

R Morrow

Papers

The discharge current induced by the motion of charged particles in time-dependent electric fields; Sato's equation extended

A one-dimensional theory for the electrode sheaths of electric arcs

A two-dimensional, finite-element, flux-corrected transport algorithm for the solution of gas discharge problems

Numerical solution of hyperbolic equations for electron drift in strongly non-uniform electric fields

Flux-corrected transport and diffusion on a non-uniform mesh