Gasdynamics of a plasma in a multiple-mirror magnetic trap with “point” mirrors

Equations are derived for the gasdynamics of a dense plasma confined by a multiple-mirror magnetic field. The limiting cases of large and small mean free paths have been analyzed earlier: ₀ and k, where is the length of an individual mirror machine, ₀ is the size of the mirror, and k is the mirror ratio. The present work is devoted to a study of the intermediate range of mean free paths ₀ k. It is shown that in this region of the parameters the process of expansion of the plasma has a diffusional nature, and the coefficients of transfer of the plasma along the magnetic field are calculated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 14–19, November–December, 1974.The authors thank D. D. Ryutov for the statement of the problem and interest in the work.     

Collision integrals of ionized air components for a screened coulomb potential

Results of a numerical evaluation of the integrals ^,s ( = 1, 2, 3, 4 and s = ... 8 -), in terms of which transport coefficients are expressed [1], are given in this paper. The results obtained are of practical use for calculating kinetic properties of ionized gases within the Chapman-Enskog theory, including third-order terms in the Sonine polynomial expansion. The integrals ^,s were calculated for singly-ionized colliding air particles at temperatures T = 10,000–40,000 °K and pressures p = 0.1, 1, and 100 atm. The screened Coulomb potential for attraction and repulsion was the model for electron-ion and ionic interactions, and the Debye screening length was chosen by taking into account screening both by electrons and by multiply-charged ions. Quantum effects are not important in the temperature and pressure ranges considered for air, and can, therefore, be neglected in calculating kinetic properties.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 168–171, September–October, 1971.     

Effect of inelastic electron — Atom collisions on nonequilibrium ionization rate

We study possible formulations of the processes taking place near the cathode of the low-voltage arc as a function of the relationship between the electron Coulomb free path (_ee) and the free path for elastic scattering of electrons by atoms (₀) on the one hand, and for inelastic scattering (₁), on the other hand. Expressions are obtained for the correction to the Maxwellian distribution function, the local and overall nonequilibrium ionization rate with atoms. Results of computer numerical calculations are presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 42–46, May–June, 1970.     

Theory of nonlinear waves in a plasma

Stationary nonlinear waves propagating in a cold rarefied plasma composed of electrons and two types of ions are considered. The structure of isolated waves and shock waves is found. In recent years an intensive study has been made of finite-amplitude waves and collisionless shock waves in a rarefied plasma, in connection with laboratory experiments [1] and astrophysical applications (the problem of the interaction of the solar wind with the Earth's magnetosphere [2]). When allowance is made for dispersion effects associated with the departure of the dispersion law =(k) from the linear, and for the compensating nonlinear twisting of the wave profile, we are able to obtain the profile of stationary nonlinear waves of finite amplitude, and when allowance is made for damping we can also obtain the structure of a collisionless shock wave [3]. Such waves have been studied fairly fully for the case of a two-component plasma. The present paper examines stationary nonlinear waves propagating across a magnetic field in a cold rarefied quasi-neutral plasma composed of electrons and two types of ions.     

Electric field excited in air by a pulse of gamma rays

The characteristics of the electric field produced by air polarization during the passage of nonstationary Compton currents excited by a -ray pulse in low-density air are discussed. The influence of the field on the motion of the Compton electrons is taken into account. The amplitude and relaxation time of the field are evaluated. A polarization electric field is created through the action of a directed current of -rays in air because of the movement of the Compton electrons. This paper discusses the basic characteristics of the resultant field in low-density air. A similar problem was raised in [1], where the electromagnetic field excited by a nonstationary source of -radiation in the upper atmosphere was considered. In that case, the Compton-electron currents were specified and their magnitude was assumed to be proportional to the ratio between the gas kinetic ranges of Compton electron and -ray (this ratio is of the order of 0.01 and is indepenent of height). With an increase in electron range, however, the decelerating action of the resultant electric field on the motion of the Compton electron becomes important (eE/ is a criterion for the effect; E is the field intensity, and and are the range and energy of the Compton electron).Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 3–8, July–August, 1970.In conclusion, the authors thank G. M. Gandel'man for several discussion.     

Motion of a conductive piston in a channel with variable inductance

The motion of a conductive piston in the channel of a magnetohydrodynamic (MHD) generator of the conduction type with compound electrodes is considered. Formulas are obtained for calculation of the energy characteristics of the pulse MHD generator for various operational regimes. It is shown that in an MHD generator at magnetic Reynolds number values Re_m = ₀u₀ 1 (where ₀ is the permeability of a vacuum, is the electrical conductivity of the piston, u₀ is the initial velocity, and is the characteristic dimension), the energy transferred to an ohmic load may significantly exceed the values obtained in [1, 2]. Conditions for high-efficiency transformation of piston kinetic energy to electrical energy are considered for limiting values of the ratio of the latter to initial magnetic field energy in the generator channel.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 41–46, November–December, 1973.The authors thank V. I. Yakovlev for his helpful evaluation.     

On the Temperature of a Rarefied Gas in Non‐Equilibrium

This paper addresses the problem of the proper definition of temperature of a gas in nonequilibrium. It shows that the mean kinetic energy of the atoms of a rarefied gas is not a good measure for thethermodynamic temperature, because in general it jumps at a wall, and because it is nonmonotone in a onedimensional process of stationary heat conduction. The jump of the kinetic temperature is calculated and found to be about 5K in a rarefied gas. The basis for the calculations is provided by the arguments of extended thermodynamics of 14 moments. An essential tool is the minimax principle of entropy production recently postulated by Struchtrup Weiss [1], because it furnishes one important boundary condition.Sommario. Il lavoro riguarda la corretta definizione della temperatura di un gas in condizioni di nonequilibrio. Si mostra come lenergia cinetica media degli atomi di un gas rarefatto non sia una buona misura della temperatura termodinamica poiché in generale, essa risulta discontinua su una parete e nonmonotona in un processo unidimensionale di conduzione stazionaria del calore. Viene calcolato il salto della temperatura cinetica che risulta pari a circa 5K in un gas rarefatto. La base per il calcolo è fornita dal contesto della termodinamica estesa di 14 momenti. Uno strumento essenziale è rappresentato dal principio di minimax di produzione di entropia recentemente postulato da Struchtrup and Weiss [1], che fornisce unimportante condizione alcontorno.     

Waves of finite amplitude in a hot plasma

A study is made of a plane shock wave of arbitrary strength propagating in a hot rarefied plasma across the magnetic field. The question of the propagation of nonstationary waves of finite but small amplitude under these conditions is examined.Fairly detailed studies have been made of waves of finite amplitude in a cold rarefied plasma. The profile of such waves is formed as the result of nonlinear and dispersion effects, the dispersion effects being caused by electron inertia and plasma anisotropy. If the gas-kinetic pressure of the plasma is taken into account, then dispersion effects appear which are associated with the fact that the Larmor radius of the ions is finite. Stationary waves of small but finite amplitude propagating across the magnetic field in a hot plasma (when the gas-kinetic pressure p is comparable with the magnetic pressure H²/87) have been treated in [1, 2]. In [1] an isolated rarefaction wave was found in a hot plasma, instead of the compression wave characteristic of a cold plasma, and a qualitative picture of the shock wave structure was given. In [2] a study was made of a small-amplitude shock wave with the finite size of the ion Larmor radius taken into account. The present paper investigates the structure of shock waves of arbitrary strength which propagate across the magnetic field in a fairly hot rarefied plasma, and also examines nonstationary waves of finite but small amplitude excited in a plasma by a magnetic piston acting over a limited time interval.Notation p gas-kinetic pressure - H magnetic field - u, v macroscopic velocities along the x and y axes - density - m_e(m_i) mass of electron (ion) - plasma conductivity - _H ion-cyclotron frequency - V_A Alfvèn velocity - c velocity of light - adiabatic exponent - V specific volume - _0e(_0i) electron (ion) plasma frequency - S₀ velocity of sound. In conclusion the author thanks R. Z. Sagdeev and N. N. Yanenko for discussing the paper, and also R. N, Makarov for helping with the numerical computations.     

Invariant solutions of the equations of a multicomponent charged-particle beam

The concept of the invariant-group solution (H-solution) was introduced and a general method for obtaining it was developed in [1–3]. The group properties of the equations of a monoenergetic charged-particle beam with the same value and sign of the specific charge, assuming univalency of the velocity vector V, were studied in [4–6], where all essentially different H-solutions were also constructed. Below, the results of [4–6] are extended to the case of a beam in the presence of a fixed background of density ₀ (§1), and also to the case of multivelocity (V is an s-valued function) and multicomponent beams (i.e., beams formed by particles of several kinds) (§2). A number of analytic solutions that describe some nonstationary processes in devices with plane, cylindrical, and spherical geometry —among them a continuous periodic solution for a plane diode with a period determined by the background density -are obtained in §1. A transformation that contains arbitrary functions of time and preserves Vlasov's equations is given (§2). The equations studied can be treated as the equations of a rarefied plasma in the magnetohydrodynamic approximation, when the pressure gradients are negligible as compared with forces of electromagnetic origin.     

Existence and Conditional Energetic Stability of Capillary-Gravity Solitary Water Waves by Minimisation

Firstly, the two-dimensional stationary water-wave problem is considered. Existence of capillary-gravity solitary waves is proved by minimising a functional related to Smales amended potential. We first establish the existence of periodic solutions of arbitrarily large periods, leading to a minimising sequence in L²() that stays away from the boundary of the neighbourhood of 0 W^2,2() in which the analysis is carried out. With the help of the concentration-compactness principle, we then show that every minimising sequence has a subsequence that, after possible shifts in the propagation direction, converges in L²() to a minimiser. Secondly, for the evolutionary problem, we prove that the set of minimal solitary waves as a whole is energetically conditionally stable. Energetically means that the distance to the set of all minimisers is defined in terms of the total energy, and conditionally means that we consider solutions to the evolutionary problem that do not explode instantaneously but could perhaps explode in finite time (e.g., via the explosion of another norm). We work in some bounded set in W^2,2() that contains the quiescent state and we are not interested in the fate of solutions that leave this set.     

Effect of surface potential and intrinsic magnetic field on resistance of a body in a supersonic flow of rarefied partially ionized gas

The character of flow over a body, structure of the perturbed zone, and flow resistance in a supersonic flow of rarefied partially ionized gas are determined by the intrinsic magnetic field and surface potential of the body. The effects of intrinsic magnetic field and surface potential were studied in [1–4]. There have been practically no experimental studies of the effect of intrinsic magnetic field on flow of a rarefied plasma. Studies of the effect of surface potential have been limited to the case R/_d<50 [1, 3]; this is due to the difficulty of realization of flowover regimes at R/_d>10² (where R is the characteristic dimension of the body and X is the Debye radius). At the same time R/_d>10², the regime of flow over a large body, is of the greatest practical interest. The present study will consider the effect of potential and intrinsic magnetic field on resistance of a large (R/_d>10²) axisymmetric body (disk, sphere) in a supersonic flow of rarefied partially ionized gas.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 41–47, May–June, 1986.     

Waves in a plasma with Hall dispersion

In a magnetohydrodynamic approximation, an investigation is made of the propagation of waves in a plasma, whose characteristic frequency is much less than the collision frequency of the electrons _e ^–1. It is assumed that the magnetic field is sufficiently strong so that the equality _ee1 will be satisfied, where _e is the cyclotron frequency of the rotation of the electrons. With large magnetic Reynolds numbers (R_m1), which are characteristic for many astrophysical problems, this latter condition leads to a need to take account of dispersion effects connected with Hall currents, in the absence of Joule dissipation. The dispersion equation for the propagation of small perturbations is analyzed in the limiting cases of weak dispersion and of a wave propagating along the magnetic field. In the case of weak dispersion, an equation is derived for nonlinear waves. The solutions are found in the form of stationary solitons. The region of such solutions is analyzed. A typical example of a medium with Hall dispersion is an interplanetary plasma, in which the parameter _ee is generally great.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 108–113, May–June, 1974.     

Nonlinear theory of the propagation of pulses of electromagnetic waves through a plasma boundary

A solution is obtained for the problem of the propagation of electromagnetic waves of arbitrary form through a plasma boundary on condition that the length of the wave train is much greater than the wave length. A solution is found both for the case of a wide spectrum of width much greater than the plasma frequency ₀, as well as for a narrow spectrum. The results obtained enable us to draw conclusions about the time and space variation of the shape of electromagnetic pulses in a plasma.The passage of high frequency electromagnetic waves through a plasma is similar to that of a beam of charged particles [1, 2]. This is associated with the fact that decay processes are similar to Cerenkov radiation effects. The dynamics of the development of transverse wave instabilities in a uniform Isotropic plasma were studied in [2] assuming that the wave phase behaves stochastically. It was calculated here that instabilities develop quite differently in the case of a wide frequency spectrum than in the case of a narrow monochromatic spectrum. If we can speak of transverse quanta diffusion effects in the field of the generated longitudinal quanta in the first case, and if the resulting effects are closely similar to the nonlinear effects arising when beam instability develops [3, 4], then the development of instabilities in the case of a narrow spectrum leads to the appearance of red satellites in the transverse wave spectrum differing from the basic frequency by a quantity ₀ (=1, 2, 3,...). In this case the development of the instability corresponds to a tendency for a plateau over the satellites to appear.Attention should however be drawn to the fact that the dynamics of instability development in a semibounded plasma may be quite different. This is associated first with the different values of group velocities of transverse and longitudinal waves, and what is also important, with the effect of longitudinal wave accumulation in the boundary region if the length of the wave train is sufficiently large. The treatment of a similar problem for beam instabilities in paper [5] showed that a narrow transition layer may arise with a transverse wave energy density greatly in excess of the energy density of the injected beam. In what follows we examine the part played by boundary effects in the passage of pulses of electromagnetic waves through the boundary of the plasma. The cases of both narrow and wide spectra are considered. We note that in the case of narrow spectra the wave train must necessarily be greatly in excess of ^–1, and the effects of the accumulation of oscillations will be appreciable.The phases of both transverse waves, and also generated longitudinal waves are assumed to be stochastic quantities. The boundary effects which have been treated may be applied both in the generation of longitudinal waves necessary for the effective acceleration of particles in a plasma as well as in the modulation and alteration of the initial transverse wave spectrum. It should also be stressed that these effects which have been considered could be applied for turbulent plasma diagnostics, as has already been pointed out in [2].The authors are grateful to Ya. B. Fainberg, M. S. Rabinovich, I. S. Danilkin, and M. D. Raizer for their interest in the paper and for valuable criticisms.     

Travelling, non-periodic patterns in nonlinear stability problems

In this short note we present results on the existence of several classes of travelling, non-periodic solutions of the complex Ginzburg-Landau equation. First we give a very short introduction to the G-L equation and show its importance in nonlinear stability theory. We then study the G-L equation with complex coefficients and establish the existence of a 2-parameter family of quasi-periodic solutions and two different types of one-parameter families of heteroclinic orbits; all members of these families travel with a well-defined wave-speed. The heteroclinic solutions correspond to (travelling) soliton-like localized structures which connect different (stable) periodic patterns. Mathematically, these families of travelling solutions (quasi-periodic and heteroclinic) are continuations into the complex case of the stationary solutions of the real G-L equation.     

A general model for the effective viscosity of pseudoplastic and dilatant fluids

Summary A new and very general expression is proposed for correlation of data for the effective viscosity of pseudoplastic and dilatant fluids as a function of the shear stress. Most of the models which have been proposed previously are shown to be special cases of this expression. A straightforward procedure is outlined for evaluation of the arbitrary constants.

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Zusammenfassung Eine neue und sehr allgemeine Formel wird für die Korrelation der Werte der effektiven Viskosität von strukturviskosen und dilatanten Flüssigkeiten in Abhängigkeit von der Schubspannung vorgeschlagen. Die meisten schon früher vorgeschlagenen Methoden werden hier als Spezialfälle dieser Gleichung gezeigt. Ein einfaches Verfahren für die Auswertung der willkürlichen Konstanten wird beschrieben.

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Nomenclature b arbitrary constant inSisko model (eq. [5]) - n arbitrary exponent in eq. [1] - x independent variable - y(x) dependent variable - y ₀(x) limiting behavior of dependent variable asx 0 - y(x) limiting behavior of dependent variable asx - z original dependent variable - arbitrary constant inSisko model (eq. [5]) andBird-Sisko model (eq. [6]) - arbitrary exponent in eqs. [2] and [8] - effective viscosity = shear stress/rate of shear - _A effective viscosity at = _A - _B empirical constant in eqs. [2] and [8] - ₀ limiting value of effective viscosity as 0 - ₀() limiting behavior of effective viscosity as 0 - limiting value of effective viscosity as - () limiting behavior of effective viscosity as - rate of shear - arbitrary constant inBird-Sisko model (eq.[6]) - shear stress - _A arbitrary constant in eqs. [2] and [8] - ₀ shear stress at inBingham model - _1/2 shear stress at = ( ₀ + )/2 With 8 figures     

Transformation and some solutions of the equations of motion of an ideal gas

The equations of one-dimensional and plane steady adiabatic motion of an ideal gas are transformed to a new form in which the role of the independent variables are played by the stream function and the function introduced by Martin [1, 2], It is shown that the function retains a constant value on a strong shock wave (and on a strong shock for plane flows). For one-dimensional isentropic motions the resulting transformation permits new exact solutions to be obtained from the exact solutions of the equations of motion. It is shown also that the one-dimensional motions of an ideal gas with the equation of state p=f(t) and the one-dimensional adiabatic motions of a gas for which p=f() are equivalent (t is time, is the stream function). It is shown that if k=s=–1, m and n are arbitrary (m+n0) and =1, the general solution of the system of equations which is fundamental in the theory of one-dimensional adiabatic self-similar motions [3] is found in parametric form with the aid of quadratures. Plane adiabatic motions of an ideal gas having the property that the pressure depends only on a single geometric coordinate are studied.     

Flow over a body with development of a steady separation zone as re → ∞

This paper discusses formulation of the total problem of flow of an incompressible liquid over a body, with formation of a closed stationary separation zone as Re . The scheme used is based on the method of matched asymptotic expansions [1]. Following [1], it is postulated that the separated zone is developed (i.e., it is not infinitely fragmented and does not vanish as Re ), and the flow inside it has a definite degree of regularity with respect to Re. With these hypotheses we can use the Prandtl-Batchelor theorem [2], which states that, in the limit as Re , a region of circulating flow becomes vortex flow of an inviscid liquid with constant vorticity . Therefore, a basis for constructing matched asymptotic expansions is the vortex-potential problem (the problem of determining a stream function , satisfying the equation = 0 in the region of translational motion and the equation = in a certain region, unknowna priori, of circulating motion). In the general case the solution of the vortex-potential problem depends on two parameters: the total pressure p_o and the vorticity in the separated zone. These parameters appear in the condition for matching the solutions of the first and second boundary-layer approximations (at the boundary of the separated zone for the end Re values) with the corresponding solutions for the inviscid flow. It is shown in the present paper that the conditions for matching the cyclic boundary layer with the external translational flow are the same additional relations which allow us to close the total problem. Thus, in using the method of matched asymptotic expansions to solve the problem of flow over a body with closed stationary separation zones one must simultaneously consider no less than two approximations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 28–37, March–April, 1978.The authors thank G. Yu. Stepanov for discussion of the paper and valuable comments.     

Asymptotic solution of the thin viscous shock layer equations at low Reynolds numbers for a cold surface

Hypersonic three-dimensional viscous rarefied gas flow past blunt bodies in the neighborhood of the stagnation line is considered. The question of the applicability of the gasdynamic thin viscous shock layer model [1] is investigated for the transition flow regime from continuum to free-molecular flow. It is shown that for a power-law temperature dependence of the viscosity coefficient T the quantity (Re)^1/(1+), where = ( – 1)/2 and is the specific heat ratio, is an important determining parameter of the hypersonic flow at low Reynolds numbers. In the case of a cold surface approximate asymptotic solutions of the thin viscous shock layer equations are obtained for noslip conditions on the surface and generalized Rankine-Hugoniot relations on the shock wave at low Reynolds numbers. These solutions give simple analytic expressions for the thermal conductivity and friction coefficients as functions of the determining flow parameters. As the Reynolds number tends to zero, the values of the thermal conductivity and friction coefficients determined by this solution tend to their values in free-molecular flow for an accommodation coefficient equal to unity. This tending of the thermal conductivity and friction coefficients to the free-molecular limit takes place for both two-and three-dimensional flows. The asymptotic solutions are compared with numerical calculations and experimental data.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 159–170. Original Russian Text Copyright © 2004 by Brykina.     

Stationary travelling waves on a film flowing down an inclined plane

At small flow rates, the study of long-wavelength perturbations reduces to the solution of an approximate nonlinear equation that describes the change in the film thickness [1–3]. Steady waves can be obtained analytically only for values of the wave numbers close to the wave number _n that is neutral in accordance with the linear theory [1, 2]. Periodic solutions were constructed numerically for the finite interval of wave numbers 0.5_{n n} in [4]. In the present paper, these solutions are found in almost the complete range of wave numbers 0 _n that are unstable in the linear theory. In particular, soliton solutions of this equation are obtained. The results were partly published in [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 142–146, July–August, 1980.     

Construction of exact numerical solutions of the stationary traveling wave type for viscous thin films

An effective numerical procedure, based on the Galerkin method, for finding solutions of the stationary traveling wave type in the complete formulation is proposed for the case of viscous liquid films. Examples of a viscous film flowing freely down a vertical surface have been calculated. The calculations have been made for various values of the dimensionless surface tension , including =0. The method makes it possible to predict a number of bifurcations that occur as decreases. The existence of numerous families of stationary traveling waves when 1 was demonstrated in [6]. The present study shows that as 1 all but one of these families of wave solutions disappear. The shape of the periodic and solitary waves and the pressure distribution in the film are found for various . When =0 and the wave number is fairly small, the periodic solution has a singularity, as predicted in [14]: at the crest of the wave a corner point appears; the angle between the tangents at this point =140–150. The method proposed can be used to calculate other wavy film flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–100, May–June, 1990.