Nonstationary shock waves in a low-density plasma

Numerical methods are used to study nonstationary collisionless shocks in a plasma propagating at some arbitrary angle to the unperturbed magnetic field in cases where the electrical conductivity and electron thermal conductivity have finite values.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 10–15, May–June, 1973.The author thanks Yu. A. Berezin for valuable discussions.     

Vertically propagating hydromagnetic waves in an isothermal atmosphere with a horizontal magnetic field

The problem considered is that of vertically propagating hydromagnetic waves of small amplitude and frequency 2π/ω in a horizontally stratified, perfectly conducing, isothermal atmosphere with a horizontal magnetic field. In an ideal fluid the boundary value problem for such waves is not well determined. However, the presence of small viscosity is sufficient to determine a unique solution. The resulting differential equation can be solved in terms of hypergeometric functions. The solution shows that the acoustic-gravity waves are modified by the effects of the viscosity and of the magnetic field in such a way as to be partly reflected downward.     

Nonstationary conducting free flow in a transverse magnetic field

In magnetohydrodynamic flow the viscous friction at the walls can be substantial. The role of viscous friction can be considerably reduced by using a free or a semirestricted flow of the conducting fluid. Nonstationary phenomena in one-dimensional motion of a free plane incompressible fluid flow in a transverse magnetic field are examined. The narrow sides of the flow come into contact with the sectional electrodes connected through external circuits with an active-inductive load. The magnetic Reynolds number and the magnetody-dynamic interaction parameter are assumed to be large. When the electric field due to electromagnetic induction in the channel is much smaller than the field due to the external circuits, the problem can be reduced to the characteristic Cauchy problem for a quasilinear hyperbolic system of first-order equations which can be solved by the method of characteristics using a computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 34–39, July–August, 1970.     

Theoretical and experimental study of waves propagating along the edge of a wedge

The properties of a surface acoustic wave moving along the edge of an elastic wedge are studied. The equations defining the character of the wave motion under consideration are obtained. A method is proposed to determine field parameters that adequately describe the experimentally confirmed acoustic phenomena taking into account the characteristics of wave propagation in the wedge-shaped region. The results of the theoretical study are compared with experimental data obtained on an original device used to determine displacements in small surface areas.     

Nonlinear waves propagating over a conducting ideal fluid surface in an electric field

For wave perturbations of a heavy conducting fluid in an electric field orthogonal to the undisturbed surface evolutionary equations quadratically nonlinear in amplitude are obtained. Equations for the long-wave approximation are derived. A method of deriving the nonlinear and simple-wave equations is proposed. Solutions for solitary waves are considered. It is shown that even a weak electric field significantly affects the form of the soliton solution, which is related with fundamental changes in the spectrum of the linear waves.     

Propagation of weak shock waves in a magnetic field

This paper gives a solution of the problem of the propagation of weak shock waves in an inhomogeneous conducting medium in the presence of a magnetic field. The width of the perturbed region is taken to be small compared with the characteristic dimensions of the problem. The magnetic Reynolds number is also assumed small, which allows one to neglect the induced magnetic field. The method of solution employed is similar to that used in [1–3],The author is grateful to B. I. Zaslavskii for useful advice and for discussing the paper.     

Velocity field generated by wing vibrations propagating along the surface

The velocity field generated by wing vibrations propagating along an elastic wing surface with finite velocity is studied.The gasdynamic problem is reduced to a mixed boundary-value problem with a moving boundary for the three-dimensional wave equation. The solution is obtained in closed form when the wing travels at supersonic velocity following an arbitrary law, the vibration propagation front is an arbitrary curve displacing along the wing surface, and the wing edges are supersonic.     

Nonstationary waves in a film of viscous liquid

A numerical investigation is made of the development of initial perturbations in a thin film of viscous liquid. It is shown that the resulting wave structure passes through complicated intermediate shapes such as solitons in which there are elevations and depressions of the surface, At large times, a wave regime is formed that is close to the optimal regime with respect to the wave number. The picture of the intermediate wave forms depends on the initial data, while the final result of the development depends weakly on the initial data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–154, May–June, 1981.We thanks A. A. Bunov for assistance in the calculations.     

Propagation of elastic waves in a rod in a longitudinal magnetic field

The propagation of harmonic waves in discussed for an ideally conducting continuous elastic cylindrical rod within an ideally conducting cylindrical rube. The annulus contains a steady homogeneous longitudinal magnetic field. The dispersion equation is derived. The case of bending vibrations is considered.     

Dispersion of a plasma cloud in a uniform magnetic field

The numerical solution of the two-dimensional gasdynamical problem of the dispersion of a plasma cloud in a magnetic field which is uniform to infinity is described. The disturbance of the field and the deformation of the cloud are taken into account self-consistently.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 146–148, July–August, 1974.In conclusion the authors thank A. I. Barchenkov, G. V0 2harova, S. A. Kuchai, V. G. Rogachev, and V. P. Statsenko for valuable advice and assistance in the work.     

Kelvin-Helmholtz instability of a compressible plasma in a magnetic field

The Kelvin-Helmholtz instability of a compressible plasma in a uniform magnetic field with respect to disturbances propagating along the flow is considered. First, the case with the magnetic field parallel to the direction of streaming is considered. The result given by Sen [4] that the compressibility effects destabilize an otherwise neutrally stable system even in the hydrodynamic limit is apparently erroneous. Re-examination of the dispersion relation in the limit of small compressibility effects shows that the latter reduce the growth rate of an otherwise unstable disturbance. Attention is also drawn to errors in the calculations of Fejer [2] in the limit of small compressibility effects. Next, the case with the magnetic field transverse to the direction of streaming is considered. It is found that the transverse magnetic field does influence the stability of the system when the compressibility effects are present, contrary to the result given by Chandrasekhar [1] for the case of an incompressible plasma. However, interestingly enough, the compressibility effects are effectively reduced if a transverse magnetic field is present! It is further shown that the transverse magnetic field reduces the stability of the system.     

Time-of-flight probing of a plasma jet in a magnetic field

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 8–11, January–February, 1991.     

The viscosity of a plasma in a strong magnetic field

The viscosity of a plasma is studied under conditions in which a magnetic field influences particle collisions. The expressions obtained for the viscosity coefficients differ significantly from those obtained in the normal theory. It is shown that in sufficiently strong magnetic fields a temperature difference arises between the electron and ion plasma components which is proportional to the drift velocity and depends logarithmically on the magnetic field strength.     

Plane stationary flows with ionizing shock waves in a magnetic field

The formulation of stationary, plane, and self-similar problems is considered when the flow parameters depend only on the polar angle, and the magnetic field lies in the flow plane. The case in which the magnetic field is perpendicular to the flow plane has been examined in [1]. The conditions are found under which the solution depends on an arbitrary parameter and the reasons for this nonuniqueness are explained. Self-similar solutions are constructed to describe the flow around an insulating wedge and a wall.     

Stabilization of plasma flow in a transverse magnetic field

Results are given of a theoretical and experimental investigation of the intensive interaction between a plasma flow and a transverse magnetic field. The calculation is made for problems formulated so as to approximate the conditions realized experimentally. The experiment is carried out in a magneto-hydrodynamic (MHD) channel with segmented electrodes (altogether, a total of 10 pairs of electrodes). The electrode length in the direction of the flow is 1 cm, and the interelectrode gap is 0.5 cm. The leading edge of the first electrode pair is at x = 0. The region of interaction (the region of flow) for 10 pairs of electrodes is of length 14.5 cm. An intense shock wave S propagates through argon with an initial temperature T_o = 293 °K and pressure p_o = 10 mm Hg. The front S moves with constant velocity in the region x < 0 and at time t = 0 is at x = 0. The flow parameters behind the incident shock wave are determined from conservation laws at its front in terms of the gas parameters preceding the wave and the wave velocity W_S. The parameters of the flow entering the interaction region are as follows: temperature T ₀ ¹ = 10,000 °K, pressure P ₀ ¹ = 1.5 atm, conduction ₀ ¹ = 3000 ^–1·m^–1, velocity of flow u ₀ ¹ = 3000 m·sec^–1, velocity of sounda ₀ ¹ = 1600 m·sec^–1, degree of ionization = 2%, 0.4. The induction of the transverse magnetic field B = [0, B_y(x), 0] is determined only by the external source. Induced magnetic fields are neglected, since the magnetic Reynolds number Re_m 0.1. It is assumed that the current j = (0, 0, jz) induced in the plasma is removed using the segmented-electrode system of resistance R_e. The internal plasma resistance is R_i = h(A)^–1 (h = 7.2 cm is the channel height; A = 7 cm² is the electrode surface area). From the investigation of the intensive interaction between the plasma flow and the transverse magnetic field in [1–6] it is possible to establish the place x* and time t* of formation of the shock discontinuity formed by the action of ponderomotive forces (the retardation wave R_T), its velocity W_T, and also the changes in its shape in the course of its formation. Two methods are used for the calculation. The characteristic method is used when there are no discontinuities in the flow. When a shock wave R_T is formed, a system of nonsteady one-dimensional equations of magnetohydrodynamics describing the interaction between the ionized gas and the magnetic field is solved numerically using an implicit homogeneous conservative difference scheme for the continuous calculation of shock waves with artificial viscosity [2].Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 112–118, September–October, 1977.