Instability of the quiescent state of an ideal conducting medium in a magnetic field

The linear stability of the quiescent states of an ideal compressible medium with infinite conductivity in a magnetic field is studied. It is shown by Lyapunov’s direct method that these quiescent states are unstable relative to small spatial perturbations, which decrease the potential energy (the sum of the internal energy of the medium and the energy of the magnetic field in this case). Two-sided exponential estimates of perturbation growth are obtained; the exponents in these estimates are calculated using the parameters of the quiescent states and the initial data for perturbations. A class of the most rapidly growing perturbations is separated and an exact formula to determine the rate of their increase is derived. An example is constructed of the quiescent states and the initial perturbations whose linear stage of evolution in time occurs in correspondence with the estimates. From the mathematical viewpoint, our results are preliminary, because the existence theorems for the solutions of the problems considered are not proved. Deceased. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 148–155, March–April, 1999.     

Energy analysis of development of kinematic perturbations in weakly inhomogeneous viscous fluids

The deformation stability relative to small perturbations is analyzed for weakly inhomogeneous viscous media on the assumption that both the main flow and perturbation field are three-dimensional. To test the damping or growth of initial perturbations, sufficient estimates based on the use of variational inequalities in different function spaces (energy estimates) are obtained. The choice of function space determines the measures of the parameter deviations, which may be different for the initial and current parameters. The unperturbed process chosen is a fairly arbitrary unsteady flow of homogeneous incompressible viscous fluid in a three-dimensional region of Eulerian space. At the initial instant, not only the kinematics of the motion but also the density and viscosity of the fluid are disturbed and the medium is therefore called weakly inhomogeneous. On the basis of the integral relation methods developed in recent years, sufficient integral estimates are obtained for lack of perturbation growth in the mean-square sense (in theL ₂ space measure). The rate of growth or damping of the kinematic perturbations depends linearly on the initial variations of the kinematics, density and viscosity. Illustrations of the general result are given. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 56–67, March–April, 2000. The work was supported by the Russian Foundation for Basic Research (projects No. 99-01-00125 and No. 99-01-00250) and by the Federal Special “Integration” Program (project No. 426).     

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.     

Rotationally symmetric spontaneous swirling in MHD flows

The stability of steady axisymmetricMHD flows of an inviscid, incompressible, perfectly conducting fluid with respect to swirling—perturbations of the azimuthal components of the velocity field—is studied in a linear approximation. It is shown that for flows similar to a magnetohydrodynamic Hill-Shafranov vortex, the problem reduces to a one-dimensional problem on a closed streamline of the unperturbed flow (the arc length of the streamline is the spatial coordinate). A spectral boundary-value eigenvalue problem is formulated for a system of two ordinary differential equations with periodic coefficients and periodic boundary conditions. Sufficient conditions under which swirling is impossible are obtained. Numerical solution of the characteristic equation shows that, under certain conditions, for each streamline there is a real eigenvalue that yields monotonic exponential growth of the initial perturbations. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 120–129, September–October, 2000.     

Linear Longwave Instability of a Single Class of Steady-State Jet Flows of an Ideal Fluid in the Field of a Self-Electric Current

A necessary and sufficient condition of linear stability of a certain two-parameter class of cylindrical steady-state shear jet MHD flows of an inviscid incompressible ideally-conducting fluid with a free boundary is obtained by the direct Lyapunov method. The magnetic field is induced by a direct current flowing along the jet so that the field linearly depends on the radius. The stability with respect to small axisymmetric longwave perturbations is considered. The perturbations conserve leave the ratio of the distance between a fluid particle and the jet axis to the azimuthal vorticity component unchanged in each fluid particle. Two-sided exponential estimates of the perturbation growth are derived in the case of violation of the stability condition obtained.     

Spontaneous swirling in axisymmetric MHD flows of an ideally conducting fluid with closed streamlines

The region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 40–50, May–June, 2007.     

Breaking of waves of limiting amplitude over an obstacle

Homogeneous heavy fluid flows over an uneven bottom are studied in a long-wave approximation. A mathematical model is proposed that takes into account both the dispersion effects and the formation of a turbulent upper layer due to the breaking of surface gravity waves. The asymptotic behavior of nonlinear perturbations at the wave front is studied, and the conditions of transition from smooth flows to breaking waves are obtained for steady-state supercritical flow over a local obstacle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 3–11, May–June, 2006.     

Thermal convection in an acoustic field

The linear stability of flow in a horizontal fluid layer is investigated within the framework of thermoacoustic convection. The flow is initiated by a longitudinal temperature gradient and the propagation of an acoustic wave in the fluid. Instability modes corresponding to perturbations of both plane and longitudinal roller and oblique wave type are detected. Using weakly nonlinear analysis, it is shown that these regimes develop softly; the stability of various secondary flows is investigated for small supercriticalities. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 10–21, May–June, 2000. The work was carried out with partial support from the Program of State Support for Leading Science Schools (grant No. 96-015-96084).     

Spontaneous swirling in MHD flows with circular streamlines

This paper considers the problem of the evolution of azimuthal perturbations in axisymmetric magnetohydrodynamic. flows of an ideally conducting inviscid fluid with circular streamlines. The fluid is. in a toroidal gap between two surfaces with constant values of the stream function. The equations of. fluid motion are derived in the approximation of infinitely a narrow gap. The parameters at which. spontaneous swirling is possible are determined numerically, and the properties of secondary swirling. flows resulting from instability of the initial steady-state poloidal flow are established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 89–97, March–April, 2009.     

Stability and nonlinear wavy regimes in downward film flows on a corrugated surface

The linear and nonlinear stability of downward viscous film flows on a corrugated surface to freesurface perturbations is analyzed theoretically. The study is performed with the use of an integral approach in ranges of parameters where the calculated results and the corresponding solutions of Navier-Stokes equations (downward wavy flow on a smooth wall and waveless flow along a corrugated surface) are in good agreement. It is demonstrated that, for moderate Reynolds numbers, there is a range of corrugation parameters (amplitude and period) where all linear perturbations of the free surface decay. For high Reynolds numbers, the waveless downward flow is unstable. Various nonlinear wavy regimes induced by varying the corrugation amplitude are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 110–120, January–February, 2007.     

Experimental study of vortex structures in a hypersonic shock layer on a flat plate

The characteristics of travelling perturbations of density in a hypersonic shock layer on a flat plate for the Mach number M_∞=21 and unit Reynolds numberRe ₁=6·10⁵ m⁻¹ were experimentally studied by the method of electron-beam fluorescence. The perturbations were generated by interaction of the shock layer behind an oblique gas-dynamic whistle and the leading edge of the plate. The cases of unsteady and quasi-steady interaction were considered. In both cases, vortex disturbances of finite amplitude were generated. The measurements were performed at the fundamental frequency F=0.6·10⁻⁴ and at the harmonic; the streamwise phase velocities, the growth rates of the disturbances, and the angles of wave propagation were obtained. The measurement results are compared with some experimental data for subsonic flows, some particular results of the linear stability theory for compressible flows, and the results obtained on the basis of a simple model of the nonlinear stage of disturbance evolution in a hypersonic boundary layer. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 41–47, November–December, 1999.     

Three-dimensional conical viscous incompressible fluid flows

New exact solutions of the Navier-Stokes equations are obtained for steady-state three-dimensional conical flows. In this class of flows the velocity decreases in inverse proportion to the distance from the source and the input equations reduce to two-dimensional ones. It is shown that in the spherical coordinate system the equations of motion reduce to a single nonlinear equation with respect to a scalar function which depends on the polar angles. The case in which this equation reduces to the integrable Liouville equation is discussed. This makes it possible to obtain a wide class of three-dimensional solutions in analytic form. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 144–148, November–December, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 97-01-00063).     

Effect of fluid compressibility on the linear stability of Poiseuille flow in a circular tube

The linear stability of a barotropic fluid flow in a circular absolutely rigid tube is considered. The behavior of perturbations in the form of monoharmonic waves resulting from the viscosity as well as the compressibility of the fluid is investigated. It is shown that the compressibility of the fluid affects the first type of perturbations only slightly and the second significantly and that the latter can be more dangerous from the standpoint of initiating instability, even for weakly compressible flows. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 5–10, March–April, 1994.     

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.     

The torsional stability of a cylinder subject to finite perturbations

The problem of torsional stability of a circular cylinder made from a compressible nonlinearly elastic material is solved for finite perturbations. In contrast to the classical theory of bifurcation, an infinite sequence of steady states that bounds the domain of allowed initial perturbations is constructed. The applicability of the classical three-dimensional linear theory of stability is evaluated. Voronezh University, Russia. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 133–136, March, 2000.     

Pulse testing of wells in reservoirs with a nonlinear seepage law

Non-steady-state flows through porous media generated by pulse excitation are studied for flows following obeying a law with a limiting pressure gradient. The phenomena of approach to the steady-state and steady-state hysteresis due to the limiting pressure gradient are investigated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 104–110, May–June, 1998. The study was supported by the Russian Foundation for Basic Research (project No. 96-01-0748).     

Non-linear characteristics of Rayleigh–Taylor instable perturbations

The direct numerical simulation method is adopted to study the non-linear characteristics of Rayleigh-Taylor instable perturbations at the ablation front of a 200 μm planar CH ablation target. In the simulation, the classical electrical thermal conductivity is included, and NND difference scheme is used. The linear growth rates obtained from the simulation agree with the Takabe formula. The ampli- tude distribution of the density perturbation at the ablation front is obtained for the linear growth case. The non-linear characteristics of Rayleigh-Taylor instable perturbations are analyzed and the numerical results show that the amplitude distributions of the compulsive harmonics are very different from that of the fundamental perturbation. The characteristics of the amplitude distributions of the harmonics and their fast growth explain why spikes occur at the ablation front. The numerical results also show that non-linear effects have relations with the phase differences of double mode initial perturbations, and different phase differences lead to varied spikes.     

Implicit difference methods for parabolic functional differential problems of the Neumann type

Nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type are considered. A general class of difference methods for the problem is constructed. Theorems on the convergence of difference schemes and error estimates for approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison technique. Nonlinear estimates of the Perron type with respect to the functional variable for given functions are used. Numerical examples are given. Published in Neliniini Kolyvannya, Vol. 11, No. 3, pp. 329–347, July–September, 2008.     

Construction of the solution of the linear problem of shock wave stability

Within the framework of the linear theory a solution is obtained in explicit form for a solitary plane shock using Fourier and Laplace transforms and assuming only the finiteness of the small perturbations. In the case of three-dimensional flows the small deformations of the shock wave surface are represented in the form of integral functionals, with Poisson kernels, of the initial perturbations of both the shape of the shock wave and the parameters of the flow field beyond it. The solution for plane flows is then constructed by the method of descent. From the equations obtained it follows that: for the region of stability and the intermediate region the solution has a finite domain of dependence on the initial perturbations; despite the fact that the structure of the domain of dependence in these regions is different, at large times the damping of the perturbations proceeds in accordance with a single law at a rate that depends on the dimensionality of the shock front.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. A, pp. 130–138, July–August, 1988.     

Effect of high-frequency vibrations on the stability of advective flow

The stability of plane convective flow in a horizontal layer with a longitudinal temperature gradient under the action of longitudinal vibrations is considered. The behavior of small normal plane and spiral perturbations is investigated. It is shown that the vibrations enhance the stability with respect to almost all types of perturbations. The sole exception is plane thermal waves whose existence domain extends toward low Prandtl numbers. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–22, January–February, 1998. The work was supported by the Russian Foundation for Fundamental Research (project No. 94-01-01730).