Equilibrium and stability of a cylindrical linear pinch with a homogeneous temperature distribution

The equilibrium and stability of a high-current discharge have been theoretically investigated in a dense optically gray plasma. The plasma is assumed to be completely opaque to longwave photons and completely transparent to short-wave photons. The threshold frequency is determined by setting the diameter of the plasma pinch equal to the mean free path of the photons. We solve the equations of magnetohydrodynamics together with the equation of radiative transfer. We show that in a gray plasma an equilibrium state can exist with a practically homogeneous temperature distribution over the discharge cross section. Temperature homogeneity is ensured by the large radiant thermal conductivity, which is related to the longwave radiation. The radiant thermal conductivity also causes the discharge to be stable with respect to superheating. We analyze the possibility of using such a discharge for the energy pumping of a laser. We show that for discharge currents of order 10⁶ A, the efficiency of a gray discharge exceeds the efficiency of an opaque discharge by a factor of three.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 33–39, January–February, 1971.     

Equilibrium states of an elastic conducting rod in a magnetic field

Our principal concern is an analysis of the equilibrium states of a nonlinearly elastic conducting rod in a magnetic field. We assume hyperelasticity so the equilibria formally appear as critical points of a potential energy functional on the strains. Fairly standard methods give existence of a minimum (not necessarily unique) with e.g., L₂-regularity. The assumptions imposed on the functional preclude the use of the usual techniques for justification of the formal necessary conditions for optimality. A new general technique is developed to justify these conditions; it then follows that minimizers satisfy the equilibrium conditions in the classical sense. (A feature of this technique is that the variations considered are homotopies so one can consider minimization within a homotopy class.) In the symmetric case, which admits trivial (straight and untwisted) solutions, we show that nontrivial solutions also exist if the field is strong enough.     

Hydrodynamic stability of dense plasma in a corrugated magnetic field

The hydrodynamic stability of plasma in a corrugated magnetic field is considered. A stability criterion is established for flute oscillations; it is valid for arbitrary values of β = 8πp/B². In a fairly long system unstable flute perturbations with a wavelength much greater that the period of corrugation always exist. The equations of motion are solved for these most dangerous perturbations, and the instability increments are derived for the case of an ideal plasma and also with due allowance for viscosity. The viscosity is considerable for large β and may lead to a reduction by a factor of ～ √β in the increments.     

Resonance frequencies and stability of a current-carrying suspended nanobeam in a longitudinal magnetic field

The resonance frequencies and stability of a nanobeam in a longitudinal magnetic field are investigated. To this aim, a three dimensional beam model is used in which the small-scale effect is taken into account based on the nonlocal elasticity theory. The Lorentz forces are obtained in terms of the local elastic rotations of the beam and the thermal stress due to current is modeled as an axial compressive force. Using the Galerkin method, the governing equations of motion are solved and the stability boundary of the nanobeam is determined.     

Convective stability of a rotating conducting fluid layer in an external magnetic field

The stability of the rest state of a conducting incompressible fluid forming a horizontal layer with rigid dielectric boundaries heated from below and rotating about a vertical axis, with a vertical magnetic field superimposed, is studied in the Boussinesq approximation. With increase in the Rayleigh number, depending on the relationship between the problem parameters (Taylor, Chandrasekhar and kinematic and magnetic Prandtl numbers), the eigenvalue of the critical mode of the linearization operator may be zero or imaginary, so that the instability of the rest state may be monotonic or oscillatory. The effect of the parameter values on the instability mode is investigated. In particular, the parameter ranges on which the critical eigenvalue is zero or imaginary are found.     

Equilibrium and stability of a whirling rod-mass system

A two-degree-of-freedom lumped-mass model is used to gain understanding of the equilibrium and stability of a circularly towed cable. Particular cases considered are those of no drag, viscous drag, and viscous drag with a crosswind.     

Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field

The nonlinear combined resonance problem of a ferromagnetic circular plate in a transverse alternating magnetic field is investigated. On the basis of the deformation potential energy, the strain potential energy, and the kinetic energy of the circular plate, the Hamilton principle is used to induce the magnetoelastic coupling transverse vibration dynamical equation of the ferromagnetic circular plate. Based on the basic electromagnetic theory, the expressions of the magnet force and the Lorenz force of the circular plate are presented. A displacement function satisfying clamped-edge combined with the Galerkin method is used to derive the Duffing vibration differential equation of the circular plate. The amplitude-frequency response equations of the system under various combined resonance forms are obtained by means of the multi-scale method, and the stability of the steady-state solutions is analyzed according to the Lyapunov theory.Through examples, the amplitude-frequency characteristic curves with different parameters, the amplitude of resonance varying with magnetic field intensity and excitation force,and the time-course response diagram, phase diagram, Poincar′e diagram of the system vibration are plotted, respectively. The effects of different parameters on the amplitude and stability of the system are discussed. The results show that the electromagnetic parameters have a significant effect on the multi-valued attribute and stability of the resonance solutions, and the system may exhibit complex nonlinear dynamical behavior including multi-period and quasi-periodic motion.     

Computer-aided study of the stability of parallel incompressible flows in a co-linear magnetic field

Summary The analysis is based on the differential equations governing infinitesimal disturbances of the form of three-dimensional waves [ e ^{im + ia(z–ct)}] superimposed on a parallel, plane or axisymmetric, laminar flow of viscous incompressible fluid in a uniform magnetic field aligned with the basic flow. A method which makes use of the fact that the governing vector eigenvalue equations for the perturbation quantities are solvable analytically in the case of constant basic-flow velocity, was employed to obtain approximate solutions of the problem. After testing this technique in the cases of the two-dimensional sech ² r profile and the Blasius profile, numerical results in form of curves of neutral stability were obtained for the three-dimensional (1 + r ²)^–2 profile with m = 1 (see Fig. 5).

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Übersicht Ausgangspunkt der Arbeit sind die Differentialgleichungen für infinitesimale Störgr;oßen von der Form dreidimensionaler Wellen [ e ^{im + ia(z–ct)}], die der parallelen, ebenen oder axisymmetrischen, Laminarströmung einer inkompressiblen Flüssigkeit nichtverschwindender Viskosität in einem zur Strömungsrichtung parallelen und im ganzen Raum konstanten Magnetfeld überlagert sind. Eine Methode, die davon Gebrauch macht, daß die Vektor-Eigenwertgleichungen für die Störgr;oßen im Falle räumlich konstanter Geschwindigkeit der Grundströmung exakt lösbar sind, wurde zur näherungsweisen Behandlung des Problems herangezogen. Nachdem diese Methode an den Testbeispielen des zweidimensionalen sech ² r-Profils und des Blasius-Profils erprobt worden war, wurden numerische Ergebnisse in Form von Stabilitätskurven für das dreidimensionale (1 + r ²)^–2-Profil mit m = 1 gewonnen (s. Abb. 5).

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This research was supported by the German Academic Exchange Service and by the National Science Foundation (Grant No. GK-72). The computations were performed at the Computation Center of the Massachusetts Institute of Technology, Cambridge, Mass., USA.     

Equilibrium and stability of tokamaks

Axisymmetric, ideal MHD configurations with steady flow are computed by the finite-element method. Rectangular elements with four to nine nodes are used. The equilibria are obtained in flux co-ordinates by mesh rearrangement. The properties of the linearized motions around an equilibrium state are studied by a normal-mode analysis. Dissipation is taken into account. Applying the Galerkin method leads to a large-scale complex eigenvalue problem ??x = λ ??x, which is solved by inverse iteration and by the Lanczos algorithm.     

Equilibrium configurations of the surface of a perfectly conducting fluid in the magnetic field of a current-carrying linear conductor

This paper considers the problem of the possible equilibrium configurations of the free surface of a perfectly conducting fluid deformed by a nonuniform magnetic field. A family of exact solutions of the problem is obtained using conformal mappings; equilibrium is achieved due to the balance of capillary and magnetic pressures. According to these solution, the surface strain amplitude increases with increasing current and the hole is transformed into a two-dimensional bubble covering the linear conductor.     

Influence of a magnetic field on the thermoacoustic stability of a high-temperature heat releasing gas

The conditions of thermoacoustic stability are found for a high-temperature electrically conducting gas with internal heat release in a constant magnetic field which transforms acoustic waves into fast and slow magnetoacoustic oscillations, and also introduces Joule dissipation. The investigation is by means of the energy balance method, and also by direct solution of the equations for small perturbations in the special case of wavelengths of the acoustic oscillations that are short compared with the inhomogeneity scales in the region of heat release. The limits of stability with respect to fast and slow magnetoacoustic oscillations are found.Translated from Izvestiya Akademii Nauk SSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 102–107, July–August, 1979.     

Yield stress of a magnetic fluid in a magnetic field

The yield stress ₀ of a magnetic fluid in a plane channel and the shape of the chains restraining the motion of the fluid are determined. The equilibrium problem for a magnetic fluid in a plane channel in the presence of an external magnetic field perpendicular to the plane of the channel and a pressure difference between the channel ends is solved within the framework of the structured fluid model. It is shown that equilibrium is possible only when the pressure difference does not exceed a certain limit p _cr at which the shear stress on the channel wall has a maximum. In weak fields p _cr and the corresponding ₀ depend quadratically on the field and in strong fields tend to saturation. The phenomenological parameters of the model are estimated for the case in which the microstructure is a system of chains of magnetic particles. The results obtained are found to be in qualitative agreement with the experimentally observed dependence of p _cr and ₀ on the field and the magnetic phase concentration.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 4–10, May–June, 1992.The authors are grateful to V. V. Gogosov for useful discussions and his interest in the work.     

Energy stability of modulation driven porous convection with magnetic field

Meccanica - Free convective onset in a bottom heated porous medium under the influence of transverse magnetic field is studied within the framework of nonlinear theory. Simple harmonic time...     

Equilibrium and stability of a nonlinear-elastic plate with a tapered disclination

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 3, pp. 157–163, May–June, 1992.     

The stability of the modified plane Poiseuille flow in the presence of a transverse magnetic field

The stability against small disturbances of the pressure-driven plane laminar motion of an electrically conducting fluid under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and not ferromagnetic, the appropriate boundary conditions on the magnetic field perturbations are presented. The Chebyshev collocation method is adopted to obtain the eigenvalue equation, which is then solved numerically. The critical Reynolds number R_c, the critical wave number α_c, and the critical wave speed c_c are obtained for wide ranges of the magnetic Prandtl number P_m and the Hartmann number M. It is found that except for the case when P_m is sufficiently small, the magnetic field has both stabilizing and destabilizing effects on the fluid flow, and that for a fixed value of M the fluid flow becomes more unstable as P_m increases.     

Flow stability of a conducting liquid flowing down an inclined plane in the presence of a magnetic field

The stability of a steady flow of incompressible, conducting liquid down an inclined plane in the presence of longitudinal and transverse magnetic fields is studied. Solutions of the linearized magnetohydrodynamic equations with corresponding boundary conditions are found on the assumption that the Reynolds number R_g and the wave number are small. It is shown that the longitudinal magnetic field plays a stabilizing role. It is known [1] that the flow of a viscous liquid over a vertical wall is always unstable. In this article it is shown that the instability effect at small wave numbers may be eliminated if the longitudinal magnetic field satisfies the conditions found. The case when the Alfvén number and the wave number are small and the Reynolds number is finite is also examined.