Instability of axially symmetric MHD flows

The problem of linear stability of axially symmetric steady-state flows of an ideal incompressible fluid in a magnetic field is studied. A necessary and sufficient condition of stability of these flows with respect to perturbations of the same symmetry type is obtained by the direct Lyapunov method. This condition represents a generalization of the well-known Rayleigh criterion [3, 4] of centrifugal stability of rotating streams to the magnetohydrodynamic case. Two-sided exponential estimates of the perturbation growth are derived. A class of the most rapidly growing perturbations is identified and exact formulas for determining their growth rate are obtained. The corresponding exponents are calculated using the steady flow parameters and initial data for the perturbation field. From the mathematical point of view, the results of the present paper are preliminary in character, since the theorems of existence of the solutions of the problem in question have not been proved.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 19–25, November–December, 1995.     

Convection in a horizontal fluid layer with specific-volume inversion

The effect of the position of the inversion point within the layer on the critical values of the Rayleigh number and the amplitudes of the rectangular-cell convective flows is numerically investigated. The monotonic instability of the mechanical equilibrium of the fluid with respect to small perturbations periodic along the layer is studied by the linearization method. The Lyapunov-Schmidt method is used to construct the secondary steady convective flows. The applicability of these methods in incompressible fluid stability problems was demonstrated in [8–10]. The calculations show that, starting from a certain value of the parameter , the branching is subcritical for any cell side ratio and a fixed wave vector modulus. For smaller values of the nature of the branching depends on the cell side ratio. This points to subcritical branching and hysteresis effects in those cases in which the periodicity of the perturbations is determined by external factors (corrugation of the boundary, spatially periodic temperature modulation, etc.). It is noted that the rectangular convection amplitude tends to zero when the cell side ratio tends to 3, the value at which hexagonal cellular convection is possible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 43–49, January–February, 1989.The author wishes to thank V. I. Yudovich for his interest and useful advice and the participants in the Rostov State University Computational Mathematics Department's Scientific Seminar for discussing the results.     

Development of perturbations in plane shock waves

Investigation of the stability of plane shock waves as regards nonuniform perturbations was first performed by D'yakov [1]. He obtained criteria for stability, and showed that perturbations grow exponentially with time in the case of instability. Iordanskii [2] has shown that in the case of stability, the perturbations are attenuated according to a power law. However, the stability criteria of [2] do not agree with the results of [1], Kontorovich [3] has explained the cause of the apparent discrepancies, and asserts the correctness of the criteria of [2]. A power law for the attenuation of perturbations has also been obtained in [4,5] under a somewhat different formulation of the boundary conditions.The Cauchy problem with perturbations is examined in §1 of this paper, results are obtained for cases of practical interest, and the asymptotic behavior is investigated.In §2 the effect of a low viscosity on the development of perturbations is examined. It is shown that when t the amplitude of perturbations is attenuated mainly as exp(-t), where >0 does not depend on the form of the boundary conditions at the shock wave front. The results of §2 were used in processing the experimental data of [6], which made it possible to determine the viscosity of a number of substances at high pressure.In conclusion, the author expresses his gratitude to A. D. Sakharov for valuable advice, and to A. G. Oleinik and V. N. Mincer for useful discussions. The author also thanks G. I. Barenblatt, L. A. Galin, and others who took part in a seminar at the Institute for Problems in Mechanics, for their interesting discussion and valuable comments.     

Stability of unsteady rectilinear plane-parallel ideal fluid flow

The stability of unsteady rectilinear plane-parallel ideal fluid flow is solved by the modified Rayleigh method [1, 2]. The numerical results apply to the so-called shear layers that form in the boundary layer prior to breakdown. The corresponding amplification factors and the most hazardous wavenumbers are found. It is shown that an analog of the Squire theorem is valid for the shear layers. In justifying the crude approximation of the initial profile, the Rayleigh method yields the exact solution for the limiting problem. A strong contraction of the class of possible initial values is not essential for finding the critical characteristics.The author thanks G. I. Petrov for his continued interest and guidance in this study.     

Instability of helical magnetohydrodynamic flows

The problem of the linear stability of a single particular class of helical steady-state flows of an ideal incompressible infinitely-conducting fluid in a magnetic field is studied. A necessary and sufficient condition of stability of this class of flows with respect to perturbations of the same symmetry type is obtained by the direct Lyapunov method [1, 2]. A priori two-sided exponential estimates of the perturbation growth are derived, the corresponding exponents being calculated using the steady flow parameters and the initial data for the perturbations. A class of the most rapidly growing perturbations is identified and an exact formula for determining their growth rate is obtained. An example of steady-state flows and initial perturbations whose linear stage of development with time can be described by means of the estimates obtained is constructed. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–156, January–February, 1999. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-01771).     

Spatial Simple Waves on a Shear Flow

The paper studies simple waves of the shallowwater equations describing threedimensional wave motions of a rotational liquid in a freeboundary layer. Simple wave equations are derived for the general case. The existence of unsteady or steady simple waves adjacent continuously to a given steady shear flow along a characteristic surface is proved. Exact solutions of the equations describing steady simple waves were found. These solutions can be treated as extension of Prandtl–Mayer waves for sheared flows. For shearless flows, a general solution of the system of equations describing unsteady spatial simple waves was found.     

Small perturbation spectrum for plane-parallel flows of viscoelastic fluids

The investigation of the occurrence of a transition from the laminar to the turbulent flow regime in weak polymer solutions is of great practical interest. Experimental data indicate both an increase in flow stability and an occurrence of early turbulence [1]. Paper [2] explains the discrepancy in the experimental data for the numerical investigation of the first-mode symmetric perturbations, which are unstable for a Newtonian fluid. Paper [3] shows that other modes also become unstable in the case of the flow of a viscoelastic Maxwellian fluid in a channel. These features of the hydrodynamic stability of viscoelastic fluids indicate a significant rearrangement of the small perturbation spectrum. In the present paper, the perturbation spectrum for plane-parallel flows of viscoelastic Oldroyd and Maxwellian fluids is investigated at small Reynolds numbers, and at large and small wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 164–167, November–December, 1984.     

Dissipative instability of a nonisothermal electrically conducting flow between parallel plates in a transverse magnetic field

Problems of dissipative instability (in particular, overheating) in magnetohydrodynamies has been studied in [1–6]. The Leontovieh mechanism of overheating instability is explained in [I] by the example of a stationary homogeneous plasma in a strong magnetic field along which current flows. The rate of buildup of perttbations is estimated in [2] to explain the effect of overheating instability on the operation of an MHD generator. The effect of inhomogeneity in the temperature field and in the boundaries of the region on the formarion of this instability has been studied by the example of discharge in a stationary medium in the absence of a magnetic field [3], Certain cases of overheating instability in magnetohydrodynamies are considered in [4, 6], where it is shown that it can be aperiodic as well as oseillatery (Alfven and acoustic waves). Finally, the hydro-dynamic and overheating branches of instability in the ease of non-isothermal plasma flow in a plane MHD channel was investigated in [6]. But the overheating instability was examined without allowance for the dependence of the viscosity and thermal-conductivity coefficients on temperature in the limiting case S R_m 1 and only for small perturbation wavelengths. The development of shortwave perturbations is studied below with allowance for viscosity and thermal conductivity and for a wider range of conditions A 1. Overheating instability over the entire range of wavelengths for the ease considered in [6] is also studied.The author thanks Yu. M. Zolotaikin for programming and performing the calculations.     

Stability of the equilibrium of a flat layer in a microconvection model

The stability of the equilibrium state of a flat layer bounded by rigid walls is studied using a microconvection model. The behavior of the complex decrement for longwave perturbations has an asymptotic character. Calculations of the full spectral problem were performed for melted silicon. Unlike in the classical Oberbeck–Boussinesq model, the perturbations in the microconvection model are not monotonic. It is shown that for small Boussinesq parameters, the spectrum of this problem approximates the spectra of the corresponding problems for a heatconducting viscous fluid or thermal gravitational convection when the Rayleigh number is finite.     

Bifurcation and transition to turbulence in the gap between rotating and stationary parallel disks

The flow in the gap between rotating and stationary parallel disks is an attractive object for studying the transition characteristics in three-dimensional internal flows. Firstly, in this case a large region of the basic motion is satisfactorily described by a self-similar solution to the Navier-Stokes equations [1]; secondly, as the parameter = h²/v ( is the. angular velocity of rotation of one of the disks and h is the gap width) varies, there is an evolution of the basic motion, so that it is easy to produce different types of initial and subsequent instabilities. The basic steady regime for axially symmetric flow has been studied by many authors (see [1, 2]). Questions of the transition in the gap between disks have been considered [3, 4]. This paper presents a methodology and the results of experimental investigations for different types of initial and subsequent instabilities in the gap between disks enclosed by a cylindrical cover. It was found that as a result of the loss of stability of the basic regime one of two steady vortex regimes is developed depending on the value of the relative gap width. The subsequent stages of soft excitation of the turbulent regime are described and the corresponding boundaries established. It is shown that in very narrow gaps the excitation of turbulence has a hard nature of the type realized in Couette flow. The stability limit for a laminarized boundary layer on a rotating disk and the boundary for complete turbulence of the layer were determined for relatively wide gaps. A comparison was made with known data for an unenclosed rotating disk.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 28–36, September–October, 1984.     

Some properties of homogeneous flows of rarefied gases

A generalization of the existence conditions for homogeneous flows of a rarefied monatomic gas mixture [2, 3] to the case where external forces are present is presented in [1]. Below we obtain for this case the solution of the Cauchy problem for the Boltzmann equation under free molecular (collisionless) conditions, when the collision integrals may be neglected (Knudsen number K 1). On the basis of this solution we construct a general solution for the equations of the kinetic moments of a Maxwellian monatomic gas mixture in the form of a series in inverse powers of K. Some additional remarks are made concerning the properties of the solutions of the second-order kinetic moment equations, and on the applicability of the Grad 13-moment equations and the Chapman-Enskog method [in particular, for the calculation of slow (Stokesian) motions of a gas mixture].The authors wish to thank M. N. Kogan and A. A. Nikol'skii for their comments.     

Stability of Steady‐State Shear Jet Flows of an Ideal Fluid with a Free Boundary in an Azimuthal Magnetic Field Against Small Long‐Wave Perturbations

The problem of the linear stability of steadystate axisymmetric shear jet flows of a perfectly conducting inviscid incompressible fluid with a free surface in an azimuthal magnetic field is studied. The necessary and sufficient condition for the stability of these flows against small axisymmetric longwave perturbations of special form is obtained by the direct Lyapunov method. It is shown that if this stability condition is not satisfied, the steadystate flows considered are unstable to arbitrary small axisymmetric longwave perturbations. A priori exponential estimates are obtained for the growth of small perturbations. Examples are given of the steadystate flows and small perturbations imposed on them which evolve in time according to the estimates obtained.     

On various variational principles in nonlinear theory of elasticity

In the present paper functionals for the various possible main variational principles in the nonlinear theory of e-lasticity are derived from the "full energy principle" and several of them are not found yet in the literatures available. Through the derivation of this paper we suggest a conjecture on the nonexistence of the eleventh and the sixth classes for the variational principles in Table 6.1 of H.C. Hu’s monograph [1].     

The axisymmetric problem with initial conditions for the equations of motion of an ideal incompressible fluid

The present paper presents a proof of the existence and uniqueness theorem for the solution of the axisymmetric problem with initial conditions for the Euler equations in the case of an incompressible fluid. We consider the case of the nonporous wall, and also the transpiration problem in the formulation given in [1]. Global unique solvability is proved for assumptions only on the smoothness of the conditions and for all values of the time t. The existence theorem for a small time segment in the case of a nonporous wall has been proved for the general three-dimensional problem in [2, 3]. For the proof we use a method analogous to that developed in [1] for planar flows. The a priori estimate of the vorticity which is used in the present study was obtained previously in [4],The author wishes to thank V. I. Yudovich for continued interest in the study and many valuable suggestions.     

Small perturbations spectrum in boundary-layer flow on a flat plate

The small perturbation spectrum of a number of flows has recently been analyzed carefully [1–3]. At the same time, investigations for the boundary layer have been limited within the framework of linear perturbation theory to the neighborhood of the neutral curve although a spectrum analysis is of indubitable interest not only to find the stability criterion of a laminar stream, but also to solve a problem with initial data about the time development of an arbitrary small perturbation. In particular, the possibility of representing an arbitrary perturbation in terms of a system of basis functions is related to the question of the completeness of the system. The finiteness was proved [4] and an estimate was obtained of the domain of eigenvalue existence in an investigation of the boundary-layer stability and a deduction has been made about the finiteness of the small perturbations spectrum for boundary-layer flow on this basis. A sufficiently complete survey of the investigation of the neutral stability of a laminar boundary layer can be found in the monograph [5]. The small perturbations spectrum in a boundary layer flow is obtained in this paper by methods of the linear theory of hydrodynamic stability by using the complete boundary conditions on the outer boundary. It is shown that the small perturbations spectrum is finite for each fixed value of the wave number . Singularities in the spectrum behavior are investigated for sufficiently small .Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 112–115, July–August, 1975.The author is grateful to M. A. Gol'dshtik and V. N. Shtern for useful discussions of the results of the research.     

Stability of subsonic gasdynamic flows

A system of differential equations describing small perturbations of the steady flow of a non-viscous ideal gas in a channel of variable cross section is analyzed in this paper. The equations of nonsteady flow and the boundary conditions are linearized, and the solution of the linearized equations is sought in the form v(x)expt t, where v(x) is an eigenfunction while is the natural frequency for the boundary problem being studied. With such an approach the problem is reduced to finding the solutions to ordinary differential equations with variable coefficients which depend on the parameter . Analytical solutions of this system are obtained for small values of and for values of ¦¦1. The results can be used to calculate the growth of high-frequency and low-frequency perturbations imposed on subsonic, supersonic, and mixed (i.e., with transitions through the velocity of sound) gasdynamic flows, to analyze the stability of subsonic sections, and to verify and supplement various numerical methods for calculating unsteady flows and numerical methods for studying stability in gasdynamics. The application of the solutions found for small and large is demonstrated on a study of flow stability behind a shock wave (a direct compression shock in the present formulation). Analytical expressions are obtained for the determination of from which it follows that the flow stability behind a shock essentially depends on the shape of the channel at the place where the shock is located in the steady flow, which was noted earlier in [1], and on the conditions of the reflection of small perturbations in the exit cross section of the channel, which was first pointed out in [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 90–97, January–February, 1978.In conclusion, the author thanks A. G. Kulikovskii and A. N. Sekundov for helpful discussions of the work.     

Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

Superposition of low-amplitude shear vibrations on steady shear flow of an elastic fluid

The previously proposed theory of viscoelastic behavior of polymer fluids is compared with experiments on the superposition of low-amplitude shear vibrations on a steady flow. It is shown that the theory agrees satisfactorily with experiments on a single polymer solution. The superposition of a steady shear flow and low-amplitude vibrations can be used to investigate some nonlinear effects characteristic of elastic fluids by relatively simple methods. The literature devoted to this question is fairly extensive; we cite only investigations in which the main results have been obtained [1–3]. The most common experimental scheme is one-dimensional (parallel superposition), although there is also a two-dimensional scheme of orthogonal superposition of shear vibrations on steady flows. Since almost all the effects in the second scheme are qualitatively similar to the first [3], but are not so clearly manifested, we give the theoretical and experimental results relating to the parallel scheme in this paper.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–7, January–February, 1976.We are grateful to É. Kh. Lipkina for help in the calculations.     

Method of calculating supersonic three-dimensional flow past smooth bodies

A method is suggested in [1] for calculating supersonic flow past smooth bodies that uses an analytic approximation of the gasdynamic functions on layers and the method of characteristics for calculating the flow parameters at the nodes of a fixed grid. In the present paper this method is discussed for three-dimensional flows of a perfect gas in general form for cylindrical and spherical coordinate systems; relations are presented for calculating the flow parameters at the layer nodes, results are given for the calculation of the flow for specific bodies, and results are shown for a numerical analysis of the suggested method. Three-dimensional steady flows with plane symmetry are considered. In the relations presented in the article all geometric quantities are referred to the characteristic dimension L, the velocity components u, v, w and the sonic velocitya are referred to the characteristic velocity W, the density is referred to the density of the free stream, and the pressure p is referred to w².     

The stability of the flow of a layer of a viscous liquid with moment stresses along an inclined plane

The stability of the flow of a layer of liquid over an inclined plane, taking account of the spin of the molecules and the internal moment stresses, was discussed in [1], However, in [1], a number of errors were allowed to creep in, which led the authors to untrue qualitative and quantitative results. In the present work, the stability of the flow of a layer with respect to long-wave perturbations is investigated by the method of successive approximations [2, 3] under the assumption that the coefficient of rotational viscosity nr is considerably less than the coefficient of Newtonian viscosity . It is shown that, in a first approximation, internal moment stresses do not affect the stability of the motion, and that the rotation of the particles exerts a destabilizing effect on the flow of the layer with respect to three-dimensional periodic perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, pp. 149–151, September–October, 1977.