Wilton ripples between two uniform streaming magnetic fluids

An analysis is made of the small-amplitude capillary-gravity waves which occur on the interface of two incompressible inviscid magnetic fluids of different densities. The waves arise as a result of second harmonic resonance. The fluids moving with uniform velocities parallel to their interface are stressed by an oblique magnetic field. The linear relations between the oblique magnetic field and the instability criteria of the linear waves are analyzed. At the stability region (away from the neutral curve) of the linear theory, a pair of coupled non-linear partial differential equations are presented. On the neutral curve, a pair of coupled non-linear partial differential equations are introduced. The last pair of equations may be regarded as the counterparts of the single Klein-Gordon equation which occurs in the non-resonant case. In all cases, the wave profile and its stability conditions are obtained. These conditions are discussed analytically and graphically.     

Steady magnetic waves

Steady simple waves are investigated in an incompressible conducting ideal inhomogeneously and isotropically magnetizable fluid moving along the lines of force of a magnetic field. The integration of the system of equations describing such waves is reduced to the calculation of quadrature expressions in the case of an arbitrary magnetization law. It is shown that, depending on the magnetic properties of the medium, different types of steady waves are possible: magnetizing waves in a diamagnetic fluid and demagnetizing waves in a paramagnetic fluid. The results are given of calculations of demagnetizing waves in a conducting ferromagnetic fluid. An analysis is made of the various possible flow regimes of a conducting magnetizable fluid at the point of a perfectly conducting corner.     

Korteweg-de vries-type equations in the theory of surface waves in electrically conducting liquids

Equations are obtained which describe the propagation of long waves of small, but finite amplitude in an ideal weakly conducting liquid and on the basis of these equations the influence of MHD interaction effects on the characteristics of the solitary waves is investigated. The wave equations are derived under less rigorous constraints on the external magnetic field and the MHD interaction parameter than in [1–3]. It is shown that the evolution of the free surface is described by the KdV-Burgers or KdV equations with a dissipative perturbation, and that the propagation velocity of the solitary waves depends on the strength of the external magnetic field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1989.     

Influence of magnetic field on wave propagation at liquid-microstretch solid interface

The reflection and refraction of a longitudinal wave at an interface between a perfectly conducting nonviscous liquid half-space and a perfectly conducting microstretch elastic solid half-space are studied. The appropriate solutions to the governing equations are obtained in both the half-spaces satisfying the required boundary conditions at the interface to obtain a system of five non-homogeneous equations in the amplitude ratios of various reflected and transmitted waves. The system is solved by a Fortran program of the Gauss elimination method for a particular example of an interface between water and aluminum-epoxy composite. Numerical values of the amplitude ratios are computed for a certain range of the incidence angle both in the presence and absence of an impressed transverse magnetic field. The effects of the presence of the transverse magnetic field on the amplitude ratios of various reflected and transmitted waves are shown graphically.     

Simple waves and small perturbations in magnetizable or polarizable media

The system of equations describing the behavior of the electrically polarizable media in the case of a sufficiently weak magnetic field and the system of equations for magnetizable media in a weak electric field coincide except for the notations. These equations are used for the investigation of Riemann waves and small perturbations for different given dependences of ε and Μ on ρ and T. The case when the velocity of propagation of simple waves takes complex values is considered. Similar investigation has been carried out in [1, 2] in the case where the dependence of Μ on ρ and T is in the form Μ?1/Μ=cρT (Mossoti formula) and μ=1+4πρk(θ-T)/H. The same problems are investigated here for an infinitely conducting magnetizable gas, whose behavior is described by another system of equations reminiscent of the equations of magnetohydrodynamics.     

SCATTERING OF ANTI-PLANE SHEAR WAVES BY A SINGLE CRACK IN AN UNBOUNDED TRANSVERSELY ISOTROPIC ELECTRO-MAGNETO-ELASTIC MEDIUM

A theoretical treatment of the scattering of anti-plane shear (SH) waves is provided by a single crack in an unbounded transversely isotropic electro-magneto-elastic medium. Based on the differential equations of equilibrium, electric displacement and magnetic induction intensity differential equations, the governing equations for SH waves were obtained. By means of a linear transform, the governing equations were reduced to one Helmholtz and two Laplace equations. The Cauchy singular integral equations were gained by making use of Fourier transform and adopting electro-magneto imperme ableboundary conditions. The closed form expression for the resulting stress intensity factor at the crack was achieved by solving the appropriate singular integral equations using Chebyshev polynomial. Typical examples are provided to show the loading frequency upon the local stress fields around the crack tips. The study reveals the importance of the electro-magneto-mechanical coupling terms upon the resulting dynamic stress intensity factor.     

Simple stationary waves in an inclined magnetic field

One component of the solution to the problem of flow around a corner within the scope of magnetohydrodynamics, with the interception or stationary reflection of magnetohydrodynamic shock waves, and also steady-state problems comprising an ionizing shock wave, is the steady-state solution of the equations of magnetohydrodynamics, independent of length but depending on a combination of space variables, for example, on the angle. The flows described by these solutions are called stationary simple waves; they were considered for the first time in [1], where the behavior of the flow was investigated in stationary rotary simple waves, in which no change of density occurs. For a magnetic wave, of parallel velocity, the first integrals were found and the solution was reduced to a quadrature. The investigations and the applications of the solutions obtained for a qualitative construction of the problems of streamline flow were continued in [2–8]. In particular, problems were solved concerning flow around thin bodies of a conducting ideal gas. The general solution of the problem of streamline flow or the intersection of shock waves was not found because stationary simple waves with the magnetic field not parallel to the flow velocity were not investigated. The necessity for the calculation of such a flow may arise during the interpretation of the experimental results [9] in relation to the flow of an ionized gas. In the present paper, we consider stationary simple waves with the magnetic field not parallel to the flow velocity. A system of three nonlinear differential equations, describing fast and slow simple waves, is investigated qualitatively. On the basis of the pattern constructed of the behavior of the integral curves, the change of density, magnetic field, and velocity are found and a classification of the waves is undertaken, according to the nature of the change in their physical quantities. The relation between waves with outgoing and incoming characteristics is explained. A qualitative difference is discovered for the flow investigated from the flow in a magnetic field parallel to the flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1976.The author thanks A. A. Barmin and A. G. Kulikovskii for constant interest in the work and for valuable advice.     

Long waves in two superposed layers of magnetic fluid

The equations and boundary conditions describing plane-parallel potential motions of two superposed layers of stably stratified magnetic fluid are formulated. The fluid is assumed to fill entirely a horizontal plane channel in the presence of a uniform longitudinal magnetic field induced by external sources. With reference to the case of long waves propagating over the interface between the upper and lower layers, it is shown that the action of the field may be interpreted as the result of an increase in the nondimensional surface tension by an amount proportional to the square of the undisturbed field. In the linear formulation the effect of the field on the evolution of a long-wave perturbation of the initially plane interface is investigated. Korteweg-de Vries equations with quadratic and cubic nonlinearities are derived and the action of the field on the internal solitary waves is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 126–133, May–June, 1993.     

Dispersion of radial vibrations of an orthotropic cylindrical tube placed in a magnetic field

The present paper studies the dispersion relation of the radial vibrations of an orthotropic cylindrical tube. The effects of the magnetoelastic interaction on the problem are investigated. The problem is represented by the equations of elasticity taking into account the effect of the magnetic field as given by Maxwell's equations in the quasi-static approximation. The stress free conditions on the inner and outer surfaces of the hollow cylindrical cube are satisfied to form a dispersion relation in terms of the wavelength, the cylinder radii and the material constants. This study shows that waves in a solid body propagating under the influence of a superimposed magnetic field can differ significantly from those propagating in the absence of a magnetic field. The results have been verified numerically and represented graphically.     

Characteristics of shear horizontal waves in magnetically affected ultra-thin films accounting for surface effect

Consider a nano-scaled film which is made from highly conductive materials and is subjected to a uniform and constant magnetic field at the vicinity of its surfaces. The characteristics of the propagated shear horizontal (SH) waves within such a nanostructure are of particular interest. By decomposing the magnetically affected nanofilm into the surface layers and the bulk, surface elasticity is adopted and their equations of motion for the SH waves are constructed. The demonstrated dispersion curves reveal that the SH waves can propagate or be damped within the nanofilm in two manners: symmetric and asymmetric. Thereafter, the roles of the magnetic field strength and the thickness on the dispersion curves and phase velocities of both symmetric and asymmetric SH waves are addressed. Additionally, the limitations of the classical continuum theory in predicting the characteristics of SH waves are displayed and discussed.     

Stationary simple waves in a plasma with anisotropic pressure

Stationary simple waves in a plasma with anisotropic pressure are investigated on the basis of the hydrodynamic equations of Chew, Goldberger, and Low. In Sec. 1, for the case where the vectors of the average flow velocity and the magnetic field intensity are parallel, the system of equations is reduced to two quasilinear equations for the velocity components. In Sec. 2 the equations for the characteristics are obtained, the system being assumed to be hyperbolic. For the special case of irrotational flow the character of simple waves in flows adjacent to various contours is studied. Section 3 contains a qualitative investigation of changes in the flow parameters in simple waves. In Sec. 4 the possibility of a transition to an unstable state of the plasma is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 12–19, March–April, 1971.The author thanks V. B. Baranov for the formulation of the problem and for his advice and constant attention to the work and also A. G. Kulikovskii for discussion of the results.     

两微极磁热弹性固体分界面上波的反射与折射

给出了磁场、热场和弹性场多场耦合作用下微极广义热弹性固体的一般控制方程.该方 程既包含了磁场、热场和弹性场的耦合作用,又在其广义热传导方程中涵盖了耦合热弹理论 (C-D)及其5类推广(L-S理论,G-L理论,G-N(II,III)理论和C-T理论).运用该微极广义磁热 弹性控制方程,研究了在定常磁场作用下, 具有均匀初始温度的两理想接触微极弹性介质平面分界面上磁热弹性波的反射和折射现象.给出了分别在缺少磁场、热场作用或不同广义热传 导理论下反射或折射热波、纵向位移波、耦合横向和微旋转波与入射纵向位移波的振幅比随 入射角变化的关系曲线.对缺少磁、热和微极性以及热松弛时间时对应的反射、折射系数进 行了对比.结果表明磁、热和微极性以及热松弛时间对振幅比均有不同程度的影 响,与磁、热和微极性一样,热松弛时间对不同类型波的影响能力差别明显,但对同 一类型的反射波和折射波的影响相似.     

Specific features of propagation of unsteady waves in a rotating spherical layer of an ideal incompressible stratified electroconducting fluid in the equatorial latitude belt

Three-dimensional large-scale motions of a rotating inviscid incompressible stratified ideal electroconducting fluid in a spherical equatorial latitude belt are studied. The mathematical model of this physical process is a closed system of partial differential equations consisting of hydrodynamic equations, which take into account the Earth rotation and the Lorentz force, and corresponding equations of magnetic dynamics with appropriate boundary conditions. An analytical solution of the system is constructed in the approximation of an equatorial β-plane, which describes propagation of lowamplitude waves.     

Magnetoelastic vibration of a plate

Summary The propagation of plane harmonic waves in a thin flat homogeneous isotropic plate of finite width and infinite length permeated by a constant magnetic field is examined. The frequency equations corresponding to the symmetric and antisymmetric modes of vibration of the plate are obtained, and some limiting cases of the frequency equations are then discussed.     

Non-linear temporal dynamics of two-mode interactions of magnetized flow

This paper considers, in the frame work of the model of two superposed layers of viscous-potential incompressible magnetic fluids, the problem on formation of resonant waves of two modes on the interface between fluids that arisen as a result of second-harmonic resonance. The fluids moving with uniform velocities parallel to their interface are stressed by a tangential magnetic field. The analysis includes the linear, as well as the non-linear effects where the analytical solutions are constructed using the method of multiple scales, in both space and time, and hence the solvability conditions correspond to the uniform (convergent) solutions are obtained. The solvability conditions are then exploited to derive a more general system of non-linear partial differential equations with complex coefficients governing the amplitudes of the resonant waves. These equations are examined for solutions corresponding to sinusoidal wavetrains consequently different kinds of instabilities are demonstrated. The stability criterion in each case is derived and discussed both analytically and graphically.     

On the fundamental equations of electromagnetoelastic media in variational form with an application to shell/laminae equations

The fundamental equations of elasticity with extensions to electromagnetic effects are expressed in differential form for a regular region of materials, and the uniqueness of solutions is examined. Alternatively, the fundamental equations are stated as the Euler–Lagrange equations of a unified variational principle, which operates on all the field variables. The variational principle is deduced from a general principle of physics by modifying it through an involutory transformation. Then, a system of two-dimensional shear deformation equations is derived in differential and fully variational forms for the high frequency waves and vibrations of a functionally graded shell. Also, a theorem is given, which states the conditions sufficient for the uniqueness in solutions of the shell equations. On the basis of a discrete layer modeling, the governing equations are obtained for the motions of a curved laminae made of any numbers of functionally graded distinct layers, whenever the displacements and the electric and magnetic potentials of a layer are taken to vary linearly across its thickness. The resulting equations in differential and fully variational, invariant forms account for various types of waves and coupled vibrations of one and two dimensional structural elements as well. The invariant form makes it possible to express the equations in a particular coordinate system most suitable to the geometry of shell (plate) or laminae. The results are shown to be compatible with and to recover some of earlier equations of plane and curved elements for special material, geometry and/or effects.     

Scattering of anti-plane shear waves by a single crack in an unbounded trasversely isotropic electro-magneto-elastic medium

A theoretical treatment of the scattering of anti-plane shear (SH) waves is provided by a single crack in an unbounded transversely isotropic electro-magneto-elastic medium. Based on the differential equations of equilibrium, electric displacement and magnetic induction intensity differential equations, the governing equations for SH waves were obtained. By means of a linear transform, the governing equations were reduced to one Helmholtz and two Laplace equations. The Cauchy singular integral equations were gained by making use of Fourier transform and adopting electro-magneto impermeable boundary conditions. The closed form expression for the resulting stress intensity factor at the crack was achieved by solving the appropriate singular integral equations using Chebyshev polynomial. Typical examples are provided to show the loading frequency upon the local stress fields around the crack tips. The study reveals the importance of the electro-magneto-mechanical coupling terms upon the resulting dynamic stress intensity factor. Contributed by SHEN Ya-peng Foundation item: the National Natural Science Foundation of China (10132010, 50135030) Biographies: DU Jian-ke (1970∼)     

Coupled magnetoacoustic waves in a conducting paramagnetic fluid

We consider the propagation of small disturbances in a paramagnetic conducting fluid in a uniform constant magnetic field. Because of coupling of the mechanical and magnetic effects, coupled magnetoacoustic oscillations of a wave nature develop in a certain (resonant) frequency region. The usual MHD waves and uniform magnetization oscillations occur far from resonance. Dissipative processes are accounted for.The equations of motion for a conducting paramagnetic fluid in which interaction of the hydrodynamic velocity with the magnetization and the magnetic field was taken into account phenomenologically were obtained in [1], One of the consequences of this interaction is the propagation of coupled magnetoelastic waves in the fluid; this phenomenon is examined in the present paper.     

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials subjected to harmonic antiplane shear waves is investigated using the Schmidt method. The present problem can be solved using the Fourier transform and the technique of dual integral equations, in which the unknown variables are jumps of displacements across the crack surfaces, not dislocation density functions. To solve the dual integral equations, the jumps of displacements across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the electric, magnetic flux, and dynamic stress fields near crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter, the distance between the two parallel cracks, and the circular frequency of the incident waves upon the stress, electric displacement, and magnetic flux intensity factors at crack tips.