Finite-amplitude self-oscillations in a rotating Hagen-poiseuille flow

The nonlinear stability of a viscous incompressible flow in a circular pipe rotating about its own axis is investigated. A solution of the initial—boundary value problem for the unsteady three-dimensional Navier—Stokes equations is found by means of the Bubnov—Galerkin method [1–5]. A series of methodological investigations were made. The nonlinear evolution of the periodic self-oscillating regimes is studied, and their characteristic stabilization times, amplitudes, and other integral and fluctuational characteristics are found. The secondary instability of these finite-amplitude wave motions is examined. It is established that the secondary instability is initially weak and linear in character; the corresponding growth times are approximately an order greater than for the primary perturbations. There is the possibility of a sharp, explosive restructuring of the motion when the secondary perturbations reach a certain critical amplitude. A survival curve [5] is constructed, which makes it possible to determine the preferred perturbation, distinguishable from the rest if the initial perturbation amplitudes are equal, and the critical amplitude values starting from which the other perturbations may prevail even over the preferred one. The range of these surviving perturbations is obtained. It is shown that as a result of the non-linear interaction of several perturbations at low levels of supercritlcality periodic motion in the form of a single traveling wave is generated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 22–28, July–August, 1985.     

Propagation of long nonlinear waves in a ponderable fluid beneath an ice sheet

In Sec. 1 the stability of small-amplitude steady-state periodic solutions of Eq. (0.1) in the neighborhood of k=k_n are investigated. The results of the investigations are consistent with those of [1]. In Sec. 2 the stability of periodic waves not lying in the neighborhood of resonance is considered. It is shown that in the region of instability when =1 steady-state solutions of the soliton type with oscillatory structure may exist. In Sec. 3 the properties of certain exact solutions — periodic waves and solitons — are studied in relation to the nature of the singular points of the dynamical system derived from (0.1). In Sec. 4 the evolution of rapidly decreasing Cauchy data is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 88–95, January–February, 1989.     

Nonlinear Oscillations in a System with Dry Friction

The case is examined where the right-hand side of the equations of motion is discontinuous. Attraction only in the stick domain ensures existence of periodic oscillations. Sufficient stability conditions for the periodic solution of a nonlinear system with dry friction are established__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 110–116, April 2005.     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.     

Autowave flows of weakly conducting gas in rough channels in the presence of strong electromagnetic fields

The steady-state flow of a gas in a rough channel in the presence of strong transverse magnetic and electric fields is considered. It is shown that in this case there also exists a periodic regime consisting of a sequence of deceleration waves [5–7]. The periodic regime results from the action of the forces of resistance to the motion of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 152–158, January–February, 1989.The authors wish to thank A. A. Barmin and A. G. Kulikovskii for valuable discussions.     

Transition from Planar to Whirling Oscillations in a Certain Nonlinear System

A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the transmitted Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.     

Influence of an elastic plate on the motion of an inhomogenfous fluid

The influence of a thin elastic isotropic plate on the wave motion of an inhomogeneous fluid originating under the effect of external periodic perturbations is investigated. The fluid density increases constantly with depth. Analogous problems have been examined for an inhomogeneous fluid without a plate in [1, 2] and with a plate on the surface of a homogeneous fluid in [3–5].Sevastopol'. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 60–67, January–February, 1972.     

The nonlinear evolution of two-dimensional and three-dimensional waves in mixing layers

The authors consider problems connected with stability [1–3] and the nonlinear development of perturbations in a plane mixing layer [4–7]. Attention is principally given to the problem of the nonlinear interaction of two-dimensional and three-dimensional perturbations [6, 7], and also to developing the corresponding method of numerical analysis based on the application to problems in the theory of hydrodynamic stability of the Bubnov—Galerkin method [8–14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza, No. 1, pp. 10–18, January–February, 1985.     

Mechanism of energy separation in a gas ejector

A new phenomenon is revealed — the rotation of an ejecting jet, discharging from a nozzle and adhering to the wall of the mixing chamber, in an axisymmetric gas ejector in modes with zero and negative ejection coefficients — and a possible mechanism for its origin is discussed. It is suggested that the rotation of an adhering jet, which induces axisymmetric vortex motion of the gas in the injector, is responsible for the inverse separation of the initially energetically homogeneous stream into heated and cooled sections.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 145–151, November–December, 1977.     

Electroconvective stability of a weakly conducting liquid

In inhomogeneous electric fields, at sufficiently high field strengths, a weakly conducting liquid becomes unstable and is set in motion [1–4]. The cause of the loss of stability and the motion is the Coulomb force acting on the space charge formed by virtue of the inhomogeneity of the electrical conductivity of the liquid [4–13]. This inhomogeneity may be due to external heating [4–6], a local raising of the temperature by Joule heating [2, 7, 8], and nonlinearity of Ohm's law [9–13]. In the present paper, in the absence of a temperature gradient produced by an external source, a condition is found whose fulfillment ensures that the influence of Joule heating on the stability can be ignored. Under the assumption that this condition is satisfied, a criterion for stability of a weakly conducting liquid between spherical electrodes is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–142, July–August, 1979.     

Stability of flow of a viscous liquid down an inclined plane

Using the Navier-Stokes equation the stability of a layer of viscous liquid flowing down a solid surface under gravity is studied in the linear formulation. The effect of surface tension and the inclination of the solid surface on the limits of stability are examined also. Curves are calculated for the neutral stability with respect to two types of perturbations — surface waves and shear waves.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskol Fiziki, No. 2, pp. 172–176, March–April, 1975.     

Evolution of three-dimensional gravitationally warped waves during the movement of a pressure zone of variable intensity

Three-dimensional, unestablished, gravitationally warped waves arising due to the motion of a harmonically time-varying pressure zone over a solid, thin plate floating on the surface of a homogeneous liquid of finite depth have been studied in the linear formulation. In the absence of a plate, three-dimensional waves are generated by the movement of a region of periodic perturbations, where established waves have been studied in [1, 2], and unestablished waves have been investigated in [3–5]. The evolution of three-dimensional, gravitationally warped waves formed during the motion of a constant load over a plate has been considered in [6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 54–60, September–October, 1986.     

Periodic dislocation motion in a relaxing medium

We consider periodic helical dislocation motion in a para-elastic medium under variable external stresses. The para-elastic properties of the medium are determined by the short-range-order parameters between atoms of the different components of the alloy. The solution of the nonlinear dislocation equation of motion is obtained in four different regions of the amplitude-frequency space. The conditions are indicated under which dislocation motion is viscous and is in the nature of breakaway from the polarization atmosphere.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 79–85, November–December, 1970.     

DYNAMICS AND STABILITY OF PINNED–CLAMPED AND CLAMPED–PINNED CYLINDRICAL SHELLS CONVEYING FLUID

The paper examines the dynamics and stability of fluid-conveying cylindrical shells having pinned–clamped or clamped–pinned boundary conditions, where “pinned” is an abbreviation for “simply supported”. Flügge's equations are used to describe the shell motion, while the fluid-dynamic perturbation pressure is obtained utilizing the linearized potential flow theory. The solution is obtained using two methods — the travelling wave method and the Fourier-transform approach. The results obtained by both methods suggest that the negative damping of the clamped–pinned systems and positive damping of the pinned–clamped systems, observed by previous investigators for any arbitrarily small flow velocity, are simply numerical artefacts; this is reinforced by energy considerations, in which the work done by the fluid on the shell is shown to be zero. Hence, it is concluded that both systems are conservative.     

Optimization of body shape at small reynolds numbers

The problem of the optimization of the shape of a body in a stream of viscous liquid or gas was treated in [1–5]. The necessary conditions for a body to offer minimum resistance to the flow of a viscous gas past it were derived in [1], The necessary optimality conditions when the motion of the fluid is described by the approximate Stokes equations were derived in [2], The shape of a body of minimum resistance was found numerically in [3] in the Stokes approximation. The optimality conditions when the motion of the fluid is described by the Navier—Stokes equations were derived in [4, 5], and in [4] these conditions were extended to the case of a fluid whose motion is described in the boundary-layer approximation. The necessary optimality conditions when the motion of the fluid is described by the approximate Oseen equations were derived in [5] and an asymptotic analysis of the behavior of the optimum shape near the critical points was performed for arbitrary Reynolds numbers.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp, 87–93, January–February, 1978.     

Instability of a spherical gas bubble in a variable-pressure field

Only a few studies, of which we mention [1–5], have been addressed to the problem of the stability of the accelerated motion of a spherical interface of two fluids. In the present paper we consider the problem of the stability of radial motion of the spherical boundary of a gas bubble in an incompressible inviscid liquid under the action of variable external pressure. Surface tension is not taken into account. We study the possibility of the existence of stable motions for broad classes of time dependence of the external pressure, namely for monotonic and periodic dependences. It is shown that stability is possible only for infinitely large bubble radii or for very specific assumptions concerning the initial conditions and the pressure-time dependence law.     

Stability of Hartmann flows

The stability of Hartmann flows for arbitrary magnetic Reynolds numbers is investigated in the framework of linear theory. The initial three-dimensional problem reduces to the equivalent two-dimensional problem. Perturbation theory is used to find asymptotic expressions for the eigenvalues. Distinguishing two types of disturbances — magnetic and hydrodynamic — is shown to be advantageous in a number of cases. Simple features of the stability are considered for particular cases. The well-know Lundquist result is generalized. An energy approach is applied to the problem of stability. The results of simulations involving the solution of the linear stability problem are described. A distinctive picture of stability is developed. There are several types of instability and they can develop simultaneously. The hydrodynamic and magnetic phenomena interact with each other in a very complex fashion. The magnetic field can either enhance flow stability or reduce it.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–31, November–December, 1972.     

Self-oscillational motion of a cylinder towed in weak aqueous polymer solution

A study is made of the effect of polymer additions on the nonstationary vibrational motion of a cylinder towed in a weak aqueous solution of the polymer and capable of making transverse displacements under the action of the force arising from asymmetry of the periodic detachment of the boundary layer. The Reynolds number was 3·10³–10⁴.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 162–164, 1973.The authors wish to thank V. P. Borovikov for his assistance in organizing the experiments.     

Propagation of solitary waves over underwater ridges

The propagation of solitary waves is investigated on the basis of a nonlinear system of equations of hyperbolic type describing the motion of the crest of a solitary wave over the surface of a liquid of variable depth [1]. The existence of solutions with discontinuities, the boundary conditions at which are introduced on the basis of [2, 3], is assumed. In the case of an infinite cylindrical ridge both solitary and periodic captured waves are found. Depending upon the height of the ridge and the parameters of the wave, the encounter between a uniform wave and a semi-infinite ridge yields qualitatively different solutions — continuous and discontinuous, where the primary wave is broken down by the ridge into several solitary waves. The amplitude of the wave may either increase or decrease over the ridge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–93, January–February, 1985.The author is grateful to A. G. Kulikovskii and A. A. Barmin for their interest in his work, useful discussions and valuable comments offered during the preparation of the article for the press.     

Stability of Filtration of a Gas—Liquid Mixture

The stability of steady regimes of filtration of a gas—liquid mixture at pressure lower than the saturation pressure is studied for the case of a nonmonotonic dependence of the relative phase permeability of the liquid on the gas saturation. It is shown that periodic self–oscillations can appear, and their evolution leads to deterministic chaos due to the appearance and destruction of quasiperiodic motions.