Effect of viscosity on the propagation of nonlinear Alfvén waves along a tangential magnetohydrodynamic discontinuity in an incompressible fluid

It is proposed to consider the propagation of surface waves along a tangential magnetohydrodynamic discontinuity in the particular case where the fluid velocities on both sides of the interface are equal to zero. In [1] it was shown that waves called surface Alfvén waves may be propagated along the surface separating a semi-infinite region without a field from a region with a uniform magnetic field. The linear theory of surface Alfvén waves in a compressible medium was considered in [2]. In [3] the damping of surface Alfvén waves as a result of viscosity and heat conduction was investigated. The propagation of low-amplitude nonlinear surface Alfvén waves in an incompressible fluid in the absence of dissipative processes is described by the integrodifferential equation obtained in [4]. By means of a numerical solution of this equation it was shown that a perturbation initially in the form of a sinusoidal wave will break. The breaking time was determined. In this paper the equation derived in [4] is extended to the case of a viscous fluid. It is shown that the equation obtained does not have steady-state solutions. The propagation of periodic disturbances is investigated numerically. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 94–104, November–December, 1986. The author wishes to thank L. S. Fedorov for assisting with the calculations.     

Tangential discontinuities of parameters of a polar fluid under shear strain

Possible formation of tangential discontinuities of parameters of a deformable polar fluid is examined by the example of glycerin. It is experimentally established that glycerin under weak shear loads possesses the properties of a non-Newtonian elastoviscoplastic fluid, and formation of tangential discontinuities in viscosity is possible. In the discontinuity region, glycerin has the properties of a low-viscosity fluid, and the structure of the medium is reconstructed after unloading. A rheological equation of the examined fluid is derived, which allows one to analyze the behavior of the medium in different modes of its deformation, including the formation of a local region with reduced viscosity and a tensile stress field. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 41–49, May–June, 2005.     

The propagation of shock waves in viscous media

The question of the thickness of shock waves in a viscous gas was treated in papers [1, 2]. The present paper derives general equations for solving problems concerning the flow of a medium inside a shock wave layer, and the change of this layer in viscous media. By way of an example we consider a problem of this type for a Kelvin medium.     

Application of unsteady mixing equations to some aerodynamic problems

Tangential discontinuities [1] are introduced in solving several transient and steady-state problems of gas dynamics. These discontinuities are unstable [2] as a result of the effects of viscosity and thermal conductivity. Therefore it is advisable to replace the tangential discontinuity by a mixing region and account for its interaction with the inviscid flows, establishing on the boundaries of this region the conditions of vanishing friction stress and equality of the velocity and temperature components to the corresponding velocity and temperature components of the inviscid flows. This formulation improves the accuracy of the solution of such problems by posing them as problems with irregular reflection and intersection of shock waves [1].The consideration of the interaction of unsteady turbulent mixing regions with the inviscid flow also permits the formulation of several problems in which the effects of viscosity lead to complete rearrangement of the flow pattern (the lambda-configuration) with the interaction of the reflected shock wave with the boundary layer in the shock tube [3,4], the formation of zones of developed separation ahead of obstacles, etc.).In this connection, §1 presents an analysis of the self-similar solutions of the unsteady turbulent mixing equations (a corresponding analysis of the laminar mixing equations which coincide with the boundary layer equations is presented in [1]). It is shown that these self-similar solutions describe, along with the several problems noted above, the problems of the formation of steady jets and mixing zones in the base wake.As an example, §2 presents, within the framework of the proposed schematization, an approximate solution of the problem of the interaction of a shock wave reflected from a semi-infinite wall with the boundary layer on a horizontal plate behind the incident shock wave. The results obtained are applied to the analysis of reflection in a shock tube. Computational results are presented which are in qualitative agreement with experiment [3, 4].     

Investigation of surfaces of discontinuity in perfectly conducting magnetizing media

Relationships on discontinuities in magnetizing perfectly conducting media in a magnetic field are investigated. The magnetic permeabilities before and after the discontinuity are assumed to be constant, but unequal, quantities. It is shown that shocks of two kinds, fast and slow, are possible in the formulation under consideration in the hydrodynamics of magnetizing media, as in magnetic hydrodynamics: It is shown that the entropy decreases on the rarefaction shocks diminishing the magnetic permeability, but can grow on the rarefaction shocks increasing the magnetic permeability, but such waves are not evolutionary. The relationships on discontinuities in the mechanics of a continuous medium are written down in general form in [1] with the electromagnetic field, polarization, and magnetization effects taken into account. Relationships on discontinuities in the ferrohydrodynamic and elec trohydrodynamic approximations were written down in [2] and [3–5], respectively, for the cases when the magnetic permeability and dielectric permittivity of the medium ahead of and behind the discontinuity are arbitrary functions of their arguments and are identical. A system of relationships on discontinuities propagated into a magnetizing perfectly conducting medium is investigated in this paper. The method proposed in [6] is used in the investigation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 104–110, January–February, 1976.We are grateful to A. A. Barmin for discussing the paper and for valuable remarks.     

Quasitransverse shock waves in elastic media with an internal structure

We consider the structure of small-amplitude quasitransverse shock waves in a weakly anisotropic elastic medium which possesses an internal structure generating the wave dispersion. The dispersion is modeled by introducing terms with higher derivatives into the equations of the theory of elasticity, and the dissipation is represented by viscous terms. In one of the two possible cases treated below, the requirement that the discontinuity structure exist leads to a set of admissible discontinuities of complex structure. A considerable part of the shock adiabat consists of a set of short portions and separate points, the number of which increases as the viscosity decreases. This complex set of admissible discontinuities is the general case where the dispersion in the shock-wave structure is sufficiently strong. Steklov Institute of Mathematics, Russian Academy of Sciences, Moscow 117526. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 174–180, March–April, 1999.     

粘性金属中的正激波稳定性问题

G.H.Miller等把高压金属中的粘性激波作为强间断面处理,解析推论出:在大粘性系数条件下小扰动激波是不稳定的,物质粘性是导致失稳的因素。本文中针对平面正激波,认为高压金属中的粘性激波的物理量是连续变化的,利用线性稳定性理论,用数值解推论出:在有粘性条件下小扰动激波都是稳定的,物质粘性是致稳的因素。指出G.H.Miller等获得错误结论的原因在于:从无粘流动解推出的小扰动边界条件导致粘性激波小扰动增长。给出实验确定的小扰动速度梯度的边界条件,这样既可以把粘性正激波作为强间断面处理,也能够保证粘性正激波的稳定性。     

Propagation and Evolution of Wave Fronts in Two-Phase Porous Media

An investigation is made into the propagation and evolution of wave fronts in a porous medium which is intended to contain two phases: the porous solid, referred to as the skeleton, and the fluid within the interconnected pores formed by the skeleton. In particular, the microscopic density of each real material is assumed to be unchangeable, while the macroscopic density of each phase may change, associated with the volume fractions. A two-phase porous medium model is concisely introduced based on the work by de Boer. Propagation conditions and amplitude evolution of the discontinuity waves are presented by use of the idea of surfaces of discontinuity, where the wave front is treated as a surface of discontinuity. It is demonstrated that the saturation condition entails certain restrictions between the amplitudes of the longitudinal waves in the solid and fluid phases. Two propagation velocities are attained upon examining the existence of the discontinuity waves. It is found that a completely coupled longitudinal wave and a pure transverse wave are realizable in the two-phase porous medium. The discontinuity strength of the pore-pressure may be determined by the amplitude of the coupled longitudinal wave. In the case of homogeneous weak discontinuities, explicit evolution equations of the amplitudes for two types of discontinuity waves are derived.     

Evolution of weak shocks in one dimensional planar and non-planar gasdynamic flows

Asymptotic decay laws for planar and non-planar shock waves and the first order associated discontinuities that catch up with the shock from behind are obtained using four different approximation methods. The singular surface theory is used to derive a pair of transport equations for the shock strength and the associated first order discontinuity, which represents the effect of precursor disturbances that overtake the shock from behind. The asymptotic behaviour of both the discontinuities is completely analysed. It is noticed that the decay of a first order discontinuity is much faster than the decay of the shock; indeed, if the amplitude of the accompanying discontinuity is small then the shock decays faster as compared to the case when the amplitude of the first order discontinuity is finite (not necessarily small). It is shown that for a weak shock, the precursor disturbance evolves like an acceleration wave at the leading order. We show that the asymptotic decay laws for weak shocks and the accompanying first order discontinuity are exactly the ones obtained by using the theory of non-linear geometrical optics, the theory of simple waves using Riemann invariants, and the theory of relatively undistorted waves. It follows that the relatively undistorted wave approximation is a consequence of the simple wave formalism using Riemann invariants.     

Instability of a compound liquid jet

In the linear Rayleigh theory [1] the degree of stability of a jet is determined by the viscosity and inertia characteristics of the fluids and the interphase surface tension. The stability of a jet in an infinite medium increases with increase in the viscosity of both the jet and the medium [2, 3]. The presence of two interfaces is responsible for various features of the development of instability in a liquid layer on the surface of a cylinder, and in particular a layer on the inner surface of a cylinder is more unstable than one on the outer surface [4]. In [5, 6] the breakup of a hollow jet in an external medium was investigated. In this paper we examine, in the linear approximation, the stability of a compound jet of nonmiscible liquids with respect to small axisynmetric perturbations of the interfaces. The instability characteristics are given for jets with inviscid and very viscous outer shells. The conditions governing the suppression of rapidly growing instabilities of the inner part (core) of the jet by a viscous shell are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–8, July–August, 1985.     

Critical distances for the formation of strong discontinuities in fluid-saturated solids

An increase in the stiffness of a solid in compression is known to lead to the steepening of the profiles of compression waves and, as a consequence, to the formation of strong discontinuities from continuous waves propagating in the solid. In this paper, the critical distance required for a continuous wave to turn into a shock wave is calculated from the evolution equation for a weak discontinuity (acceleration wave) propagating into a quiescent region. Infinite growth of the amplitude of an acceleration wave in a finite time signifies the transition to a strong discontinuity. Relations between the critical distances for plane, cylindrical and spherical waves are established. Numerical examples are presented for a particular case of the pressure-dependent stiffness typical of granular solids such as sand or soil, with emphasis placed on the influence of a small amount of free gas in the pore fluid.     

Smoothing the discontinuities of an electrical charge in electrodynamics as a result of diffusion processes

Electrohydrodynamic flows in which there are zones of abrupt changes in the electric charge (while remaining bounded, by assumption) are investigated. In a diffusionless approximation such flows are characterized by a discontinuity in the electric charge q. Examples of such motions are nonstationary flows with moving electrical charge fronts [1], stationary flows in which the electrical charge is lumped in just part of the hydrodynamic stream [2, 3], flows with discontinuity in q [4–7], boundary layers near an electrode grid mounted perpendicularly to the electrohydrodynamic stream. Diffusion effects of charged particles should cause smoothing of the electrical charge discontinuities. The diffusion structure of such discontinuities is studied for high electrical Peclet numbers. The distribution of q in gasdynamic jumps is analyzed taking account of the viscous and diffusion structure of the discontinuities in the small parameter approximation of the electrogasdynamic interaction. Three problems about flows with charged particle diffusion are examined: the problem of scattering of a finite electric charge in a medium at rest, initially concentrated at a point on a line of unit length; the boundary layer on an electrode grid perpendicular to the direction of the charged fluid stream; electrogasdynamic flows with an abrupt change in velocity not accompanied by the appearance of a surface charge.     

Calculation of viscous and plastic properties by the solution of wave problems

Models of elastoplastic media are applied to soils and rocks [1, 2]. In conformity with experimental data [3–5] a model of soils and rocks as a viscoplastic medium has been proposed [6]. Below we give a solution, based on this model, of the problem on the propagation of a plane one-dimensional wave. As the basis of computer programs we propose a finite-difference representation of the equations of motion of a continuous medium in Lagrange coordinates and the differential equations governing the behavior of the medium. A direct calculation procedure with pseudoviscosity is applied. It is shown that the damping of plane waves is connected with two energy-dissipating mechanisms, determined by the viscous and plastic properties of the medium. The washing out of a discontinuity can occur in the absence of a segment of the dynamical compression curve that is concave to the strain axis. Under certain conditions the maximum strain is attained during the phase of decreasing stress. These results agree with the experimental data [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 114–120, March–April, 1973.The authors thank S. S. Grigoryan for his discussion of the work.     

Investigation of pressure-discontinuity system with one and two triplets,taking into account equilibrium physicochemical transformations of air

It is known that the interaction of pressure discontinuities preceding the projecting elements of a supersonic object considerably increase the pressure in the interaction region [1–3]. Existing methods of estimating this excess pressure at the leading edge of the projecting element are based on the calculation of the configuration of pressure-discontinuity intersections with two or one triple points for a perfect gas with a constant adiabatic modulus . The calculation reduces to the successive solution of two transcendental equations for the determination of the angles of slope of the discontinuities at the node points [2, 4]. The present paper states the formulation of the problem and results of flow calculations in pressure-discontinuity configurations with triple points, taking into account the equilibrium dissociation of air. The Predvoditelev approximation is used to calculate the thermodynamic function of the pressure p, as proposed in [5]. The formulation of the problem is considered for the calculation of the flow taking into account the equilibrium dissociation of air in the interference region of pressure discontinuities with two and one triple points — interactions of types I and II, according to the classification of [4]. Some results of the computer solution of the resulting system of equations are given both for a flow of cold unperturbed air (the interaction region w of the leading shock wave of an object with its projecting elements) and for a flow of hot dissociating air (the interaction region O with the boundary-layer breakaway region at the surface of the supersonic object). It is shown that, both in region w and in region O, the relative pressure is considerably affected not only by the velocity and the angle of the incident pressure discontinuity but also by the density of the incoming flow (the flight altitude of the object). Depending on this parameter, the relative pressure in the interaction region may be less or more than the pressure calculation for a perfect gas with = 1.4 to analogous flow conditions. The results obtained indicate the need to take account of the real properties of air in determining the mechanical and thermal loads in the interaction region of the pressure discontinuities at the surface of projecting elements of a hypersonic object.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 111–116, September–October, 1978.     

Non-homogeneous displacement jumps in strong embedded discontinuities

In this paper, strong discontinuities are embedded in finite elements to describe fracture in quasi-brittle materials. A new numerical formulation is introduced in which the displacement jumps do not need to be homogeneous within each finite element. Both the crack path and the displacement jumps are continuous across element boundaries. This formulation is compared with the discrete approach, in which interface elements are inserted to model the discontinuities, as well as with other embedded discontinuity approaches and with the partition of unity method. Numerical results have been obtained with relatively coarse meshes, which compare well with experimental results and with the results obtained from analyzes with interface elements.     

Flow of a supersonic electrogasdynamic stream,with formation of an electroconducting gaseous shock layer

Surfaces of strong discontinuity in electrogasdynamics were considered in [1, 2]. An investigation was done for the case when a gas has the properties of a unipolar charged medium on both sides of a surface of discontinuity. However, with sufficiently high supersonic gas flow over bodies the gas becomes electroconducting and acquires the properties of a low-temperature plasma in the compressed layer between the shock wave and the body, because of the temperature increase. Therefore, there is great interest in investigating type S* Shockwaves dividing a unipolar charged medium and a low-temperature plasma. The S* waves separating the uncharged medium and a gas with high electrical activity in the presence of an electrical field were studied in [3]. Below we examine the general properties of S* waves (physicalmodel, relations at the wave, conditions for development, shock adiabats, and polars). We formulate the problem of flow of a supersonic electrogasdynamic stream over bodies, with formation of S* waves. A perturbation method is proposed for solution of the problem, using a small parameter to describe electrogasdynamic interaction. By way of example a complete solution for flow over a wedge is constructed.     

Stability of surfaces of strong discontinuity in gases

The existence of solutions with surfaces of strong discontinuity is one of the principal features of the continua whose motions are described by systems of differential equations of hyperbolic type. Shock waves in gas dynamics, magnetohydrodynamics and in solids, detonation waves and combustion fronts, contact discontinuities, etc. are well-known examples of these surfaces. The discontinuities are usually investigated in accordance with the following scheme: 1) derivation of the boundary conditions on the discontinuity from the input system of differential equations in integral form; 2) verification of the fulfilment of the evolution conditions; 3) solution of the problem of the discontinuity structure and, when the occasion requires, obtaining supplementary boundary conditions; 4) investigation of the stability of the discontinuity. Only after obtaining positive results in all fours stages can we assert that the existence of the discontinuity is theoretically justified and that it can be used for constructing the solutions of particular boundary value problems. In the present paper attention will be concentrated on the problem of the stability of discontinuities, all the material, with the exception of the general results of Sec.1, being concerned with gas media and relating to discontinuities on whose surface the normal mass flow is nonzero. Having no way of exploring all the aspects of the problem of the stability of discontinuities in the same detail within the limited context of this paper, the authors hope to demonstrate the most general ideas and approaches which could subsequently be used to investigate the stability of discontinuities in various particular models of continua.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–22, March–April, 1996.     

Rayleigh Taylor instability of rotating compressible fluid through porous media

The hydromagnetic instability of a stratified horizontal layer of viscous compressible rotating fluid through porous media in the presence of vertical magnetic field is considered. The solution has been obtained through the use of variational principle. The dispersion relation is derived for a layer having exponential density stratification along the vertical direction. It is found that viscosity has stabilizing influence while permeability of porous medium, magnetic resistivity and coriolis forces have destabilizing influence on the system.