Simple waves in a plasma with anisotropic pressure

Fast and slow simple waves are studied in the framework of the anisotropic magnetohydrodynamics of Chew, Goldberger, and Low [1]. Baranov [2] has constructed fields of integral curves for fast and slow waves and in two special cases has shown that such waves break in the compression section. The possibility of breaking of a slow wave in a rarefaction section was noted by Akhiezer et al. [3]. However, their general relations in simple waves [3] have been shown to be incorrect [2, 4]. In the present paper the nature of the variation of the longitudinal and transverse plasma pressures is determined, and the problem of the breaking of fast and slow waves is completely solved. Conditions under which a slow wave breaks in a rarefaction section are found. A fast wave always breaks in a rarefaction section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 181–183, July–August, 1988.     

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.     

Flow of a liquid with an anisotropic viscosity tensor: some axisymmetric flows

Of a class of idealized anisotropic liquids presented earlier [1,2], two particular cases, referred to as liquids D and F, are now analysed in some axially symmetric flows generated by relative motion of the boundaries. The liquids are locally transversely isotropic at each point at some initial instant, and the different responses associated with some different initial directions of orientation are considered, in torsional flow, in Couette flow, and in longitudinal flow between concentric circular cylinders.As in [1,2], it is found that only in special circumstances can the liquids behave in a Newtonian fashion, without change of orientation pattern. In general, even when the motion of boundaries is steady, the flow is unsteady, stresses are time-dependent, and initial transverse isotropy does not persist.     

Interaction of an elastic shell with two-dimensional flows of an ideal fluid

In [1, 2], Kiselev and Rapoport investigated the flow of a jet over an elastic plate and shell. In the present paper, the problem of two-sided flow past an elastic shell is investigated in the exact nonlinear formulation. At a sufficiently high rigidity and small curvature of the shell in undeformed state it is shown that the problem has a unique solution, and a method is proposed for finding it. Some results of calculations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 139–143, September–October, 1981.     

Coupled fields of two-dimensional anisotropic magneto-electro-elastic solids with an elliptical inclusion

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Simulation of anisotropic diffusion processes in fluids with smoothed particle hydrodynamics

Anisotropic diffusion phenomenon in fluids is simulated using smoothed particle hydrodynamics (SPH). A new SPH approximation for diffusion operator, named anisotropic SPH approximation for anisotropic diffusion (ASPHAD), is derived. Basic idea of the derivation is that anisotropic diffusion operator is first approximated by an integral in a coordinate system in which it is isotropic. The coordinate transformation is a combination of a coordinate rotation and a scaling in accordance with diffusion tensor. Then, inverse coordinate transformation and particle discretization are applied to the integral to achieve ASPHAD. Noting that weight function used in the integral approximation has anisotropic smoothing length, which becomes isotropic under the inverse transformation. ASPHAD is general and unique for both isotropic and anisotropic diffusions with either constant or variable diffusing coefficients. ASPHAD was numerically examined in some cases of isotropic and anisotropic diffusions of a contaminant in fluid, and the simulation results are very consistent with corresponding analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

GREEN＇S FUNCTIONS OF TWO－DIMENSIONAL ANISOTROPIC BODY WITH A PARABOLIC BOUNDARY

ＧＲＥＥＮ＇ＳＦＵＮＣＴＩＯＮＳＯＦＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＡＮＩＳＯＴＲＯＰＩＣ ＢＯＤＹ ＷＩＴＨ Ａ ＰＡＲＡＢＯＬＩＣ ＢＯＵＮＤＡＲＹ（胡元太）（赵兴华）ＧＲＥＥＮ＇ＳＦＵＮＣＴＩＯＮＳＯＦＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＡＮＩＳＯＴＲＯＰ?..     

Stress distribution in a two-dimensional infinite anisotropic medium with collinear cracks

The plane problem of an anisotropic material with cracks, whose surfaces are subject to surface tractions of a general kind, is studied. The medium considered if of infinite extent and the cracks are located on a single line. The Fourier transform method is employed to derive the stress and displacement components at an arbitrary point of the medium in terms of the dislocation densities and the stress discontinuities on the crack line.These formulae for stress and displacement components involve the roots of a quartic equation whose coefficients are the material constants. The cases of different roots and pairwise coincident roots are examined separately. An orthotropic medium is an important example for the case of different roots while an isotropic medium is that for the case of pairwise coincident roots. These examples are discussed in detail.As an illustration of the use of these formulae the problem of a single crack in an infinite anisotropic medium is examined in detail.Work supported in part by a grant from Council of Scientific and Industrial Research, New Delhi, India.     

Electrode layers in flows of an inviscid plasma with good conductivity

The structure of the electromagnetic electrode layers that are produced in flows across a magnetic field by a completely ionized and inviscid plasma with good conductivity and a high magnetic Reynolds number is examined in a linear approximation. Flow past a corrugated wall and flow in a plane channel of slowly varying cross section with segmented electrodes are taken as specific examples. The possibility is demonstrated of the formation of nondissipative electrode layers with thicknesses on the order of the Debye distance or electron Larmor radius and of dissipative layers with thicknesses on the order of the skin thickness, as calculated from the diffusion rate in a magnetic field [2].In plasma flow in a transverse magnetic field, near the walls, along with the gasdynamie boundary layers, which owe their formation to viscosity, thermal conductivity, etc. (because of the presence of electromagnetic fields, their structures may vary considerably from that of ordinary gasdynamic layers), proper electromagnetic boundary layers may also be produced. An example of such layers is the Debye layer in which the quasi-neutrality of the plasma is upset. No less important, in a number of cases, is the quasi-neutral electromagnetic boundary layer, in which there is an abrupt change in the frozen-in parameter k=B/p (B is the magnetic field and p is the density of the medium). This layer plays a special role when we must explicitly allow for the Hall effect and the related formation of a longitudinal electric field (in the direction of the veloeiryv of the medium). We will call this the magnetic layer. The magnetic boundary layer can be dissipative as well as noudissipative (see below). The dissipative magnetic layer has been examined in a number of papers: for an incompressible medium with a given motion law in [1], for a compressible medium with good conductivity in [2], and with poor conductivity in [3]. In the present paper, particular attention will be devoted to nondissipative magnetic boundary layers.     

Some two-dimensional flows of a heavy liquid

The solution is presented of the problem of a two-dimensional fountain of a heavy liquid above a horizontal bottom. The approximate method of conformal mapping of similar regions developed by Lavrent'ev [1] and Moiseev [2] is used.Many or the questions considered above were discussed with M. A. Gol'dshtik, to whom the author is very grateful.     

Viscometric flows of anisotropic fluids

Summary A brief introduction to the continuum theory for nematic liquid crystals is given. The problem of simple shearing flow is examined in some detail using this theory and some of the difficulties that arise are discussed. It is shown that under reasonable simplifying assumptions, useful asymptotic formulae can be found for the apparent viscosity in many viscometric flows.

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Zusammenfassung Es wird eine kurze Einführung in die Kontinuumstheorie nematischer Flüssigkristalle gegeben. Die Theorie wird benutzt, um das Problem der einfachen Scherströmung ausführlich zu behandeln. Einige der auftretenden Schwierigkeiten werden erörtert. Es wird gezeigt, daß unter vernünftigen vereinfachenden Annahmen nützliche asymptotische Formeln für die scheinbare Viskosität in vielen viskometrischen Strömungen gefunden werden können.

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With 4 figures and 1 table     

Some classes of two-dimensional vortex flows of an ideal fluid

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 109–117, January–February, 1989.     

The stability of two-dimensional linear flows of an Oldroyd-type fluid

A linear stability analysis is made for an Oldroyd-type fluid undergoing steady two-dimensional flows in which the velocity field is a linear function of position throughout an unbounded region. This class of basic flows is characterized by a parameter λ which ranges from λ = 0 for simple shear flow to λ = 1 for pure extensional flow. The time derivatives in the constitutive equation can be varied continuously from co-rotational to co-deformational as a parameter β varies from 0 to 1. The linearized disturbance equations are analyzed to determine the asymptotic behavior as time t → ∞ of a spatially periodic initial disturbance. It is found that unbounded flows in the range 0 < λ ? 1 are unconditionally unstable with respect to periodic initial disturbances which have lines of constant phase parallel to the inlet streamline in the plane of the basic flow. When the Weissenberg number is sufficiently small, only disturbances with sufficiently small wavenumber α₃ in the direction normal to the basic flow plane are unstable. However, for certain values of β, critical Weissenberg numbers are found above which flows are unstable for all values of the wavenumber α₃.     

Stability of two-dimensional curvilinear flows of an ideal barotropic fluid

It was established by Arnol'd [1] that the conservation laws for the energy and vorticity can be used to establish sufficient conditions of stability of two-dimensional curvilinear flows of an ideal incompressible fluid in the exact nonlinear formulation. It is shown below that one can obtain similarly conditions of stability of two-dimensional curvilinear steady flows of an ideal barotropic fluid in the linear approximation. One of the conditions has a significance similar to Rayleigh's criterion and its generalization by Arnol'd [1]; the other is the condition of subsonic flow. In addition, a variational principle is established and an expression found for the second variation of the corresponding functional; these can be used to prove the stability of these flows in the exact nonlinear formulation.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–25, September–October, 1981.I am sincerely grateful to V. L. Berdichevskii and A. G. Kulikovskii for constructive advice.     

Comparison of results from DNS of bubbly flows with a two-fluid model for two-dimensional laminar flows

Results from direct numerical simulations of laminar bubbly flow in a vertical channel are compared with predictions of a two-fluid model for steady-state flow. The simulations are done assuming a two-dimensional system and the model coefficients are adjusted slightly to match the data for upflow. The model is then tested by comparisons with different values of flow rate and gravity, as well as downflow. In all cases the results agree reasonably well, even though the simulated void fraction is considerably higher than what is assumed in the derivation of the model. The results do, however, suggest a need to understand the lift and the wall repulsion force on bubbles better, particularly in dense flows.