Plane-parallel flow around a gas cavity

The stationary motion of a gas cavity in an ideal incompressible fluid is studied taking account of surface tension by using a variational equation. Approximate analytical dependences of the dimensionless parameters on the degree of cavity deformation are obtained. It is shown that the variational equation admits of an exact analytical solution. The stability of motion corresponding to the exact solution is proved relative to arbitrary perturbations in the cavity shape. A solution is given for the problem of stationary motion of an elliptical cavity in a gravity viscous fluid and the stability problem is investigated. Dependences are found for the velocity of cavity rise, the Reynolds number, and the Froude number as a function of the cavity size.     

Stability of heterogeneous equilibrium in a system with liquid phases

Gibbs's method is used to study the equilibrium and stability in a heterogeneous system consisting of single-component liquid phases. The coefficient of surface tension is assumed to be a constant independent of the state of the phases. An expression is obtained for the second variation of the corresponding functional, and this expression can be used to analyze the stability of nucleating centers. It is shown that in the absence of external force fields and surface tension forces the equilibrium state of a closed thermodynamic system consisting of single-component liquid phases satisfying the classical Gibbs inequalities is always stable.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 88–94, March–April, 1982.     

Oscillations of a gas bubble in a non-Newtonian liquid under the action of an acoustic field

The evolution of the radius of a spherical cavitation bubble in an incompressible non-Newtonian liquid under the action of an external acoustic field is investigated. Non-Newtonian liquids having relaxation properties and also pseudoplastic and dilatant liquids with powerlaw equation of state are studied. The equations for the oscillation of the gas bubble are derived, the stability of its radial oscillation and its spherical form are investigated, and formulas are given for the characteristic frequency of oscillations of the cavitation hollow in a relaxing liquid. The equations are integrated numerically. It is shown that in a relaxing non-Newtonian liquid the viscosity may lead to the instability of the radial oscillations and the spherical form of the bubble. The results obtained here are compared with the behavior of a gas bubble in a Newtonian liquid.     

Theory of gravitational stability of a rotating cylinder

The stability of a rotating dust cylinder against perturbations located in the plane perpendicular to the axis of rotation is investigated. It is shown that a homogeneous rotating cylinder containing a weak inhomogeneity is stable against such perturbations. A weakly inhomogeneous cylinder with opposite streams of equal density is unstable for thel=2 mode in the case of a perturbation of the form_ei(l–t), when the density increases radially. The instability of a system consisting of a homogeneous rotating dust cylinder in a hot homogeneous medium is determined. It is shown that the maximum growth rate corresponds tol = 2 when the density of a cold cylinder is not negligible in comparison with the density of the medium. In the opposite case, the maximum growth rate shifts toward l=3. An attempt is made to associate the existence of the maximum growth rate for l=2 with the presence of two spiral arms in most galaxies. It is shown that, when the longitudinal temperature is high enough, a rotating cylinder which is bounded in the radial direction is stable against arbitrary perturbations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol.10, No. 3, pp. 3–11, May–June, 1969.     

Shape and stability of a liquid drop moving at low Weber numbers

The equilibrium cross sectional shape and stability of a liquid drop moving in a gaseous medium is studied analytically. Such liquid drops appear as the final product in numerous industrial spraying and atomisation processes. Raindrops falling at their terminal speed can also be described by the present model. The equilibrium shape is formed by the interaction of two main factors; the dynamic pressure distribution in the gaseous medium which tends to deform the liquid drop into an oblate shape and the surface tension which tends to restore the spherical shape. The meridional shape of the liquid drop is obtained as a power series in the Weber number. The linear stability of the deformed shapes described above to small surface disturbances is studied. The stability analysis shows the effect of the surrounding gas flow on the natural frequencies of oscillation (vibration) of the liquid drop. The liquid drop is found to be stable in the region of low Weber numbers studied with a decrease in oscillation frequency proportional to the Weber number. This is in agreement with existing experimental data. Extrapolation of the results here lead to a Weber number of W=5.33 for breakup, again in agreement with experimental correlations.     

Studying the stability of equilibrium solutions in a planar circular restricted four-body problem

The Newtonian circular restricted four-body problem is considered. We obtain nonlinear algebraic equations determining equilibrium solutions in the rotating frame and find six possible equilibrium configurations of the system. Studying the stability of equilibrium solutions, we prove that the radial equilibrium solutions are unstable, while the bisector equilibrium solutions are stable in Lyapunov’s sense if the mass parameter satisfies the conditions μ ∈ (0, μ₀, where μ₀ is a sufficiently small number, and μ ≠ μ_j, j = 1, 2, 3. We also prove that, for μ = μ₁ and μ = μ₃, the resonance conditions of the third order and the fourth order, respectively, are satisfied and, for these values of μ, the bisector equilibrium solutions are unstable and stable in Lyapunov’s sense, respectively. All symbolic and numerical calculations are done with the Mathematica computer algebra system. Published in Neliniini Kolyvannya, Vol. 10, No. 1, pp. 66–82, January–March, 2007.     

Hydrodynamic stability of dense plasma in a corrugated magnetic field

The hydrodynamic stability of plasma in a corrugated magnetic field is considered. A stability criterion is established for flute oscillations; it is valid for arbitrary values of β = 8πp/B². In a fairly long system unstable flute perturbations with a wavelength much greater that the period of corrugation always exist. The equations of motion are solved for these most dangerous perturbations, and the instability increments are derived for the case of an ideal plasma and also with due allowance for viscosity. The viscosity is considerable for large β and may lead to a reduction by a factor of ～ √β in the increments.     

Stability of axisymmetric motions in a rotating inviscid atmosphere

Summary We study the set of ordinary differential equations representing the truncation of the spectral vorticity equation in spherical geometry for two-dimensional, non divergent motions in a homogeneous, inviscid rotating atmosphere. The set of included spectral components represent an arbitrary axisymmetric field of motion (zonal flow) interacting with a couple of non zonal, wave-like perturbations. We analize the stability of the equilibrium configuration described by purely zonal flow, by an esplicit representation of the linear and non-linear parameters which characterize the unstable oscillations induced by perturbations of the equilibrium.

---

Sommario Si studiano le soluzioni di un sistema di equazioni differenziali ordinarie che rappresentano il troncamento a poche componenti di Fourier dell'equazione di conservazione della vorticità per moti bidimensionali, non divergenti in una atmosfera rotante, inviscida e omogenea. Tale troncamento include tutte le componenti spettrali che rappresentano flussi a simmetria assiale (moti zonali) in interazione non lineare con una coppia di componenti asimmetriche che rappresentano flussi ondulatori. Si studia la stabilità della configurazione di equilibrio caratterizzata da moto zonale stazionario, analizzando i parametri lineari e non lineari che descrivono le oscillazioni instabili di ampiezza finita del campo di flusso che nascono da condizioni iniziali nell'intorno dell'equilibrio.

---

---

This work was supported by the C.N.R. through the Gruppo Nazionale per la Fisica Matematica.     

Secondary thermocapillary motions of soliton type

An exact analytic solution is obtained for the problem of the stability of the axisymmetric thermocapillary motion due to a point heat source of constant power located on the horizontal free surface of a viscous fluid. Analytic expressions are found for monotonic neutral disturbances of hydrodynamic and thermal type. The critical values of the dimensionless source power for disturbances with arbitrary quantum numbersl andm are determined, together with the secondary motions near the stability threshold. An exact solution of the problem of the axisymmetric thermocapillary motion due to a spherical heat source is presented and its stability is investigated. It is shown that it is always possible to select physical heater properties such that for arbitrarily small source power, the axisymmetric motion is unstable relative to the vortex motion. A comparison is made with experiment.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 20–27, July–August, 1992.     

On the stability of liquid droplets with line tension

Prompted by recent works on line tension effects upon the stability of liquid droplets laid on a rigid substrate, we prove existence and stability of equilibrium configurations when the substrate is either planar or spherical. For positive line tension our argument involves only set symmetrization, while for negative line tension a stability criterion is proposed which makes variational problems well-posed in the framework of sets with finite perimeter. Several applications illustrate the role of the criterion in providing a natural cutoff length that selects destabilizing modes. The work of M. Negri has been partially supported by INdAM through the grant “Mathematical challenges in nanomechanics at the interface between atomistic and continuum models”.     

Convective instability of a horizontal layer of fluid between permeable membranes

The equilibrium stability is investigated of a system consisting of two semi-infinite isothermal masses of fluid divided by a horizontal layer of finite thickness of the same fluid with a vertical temperature gradient directed downwards. The transition layer is separated by thin permeable membranes. Neutral stability curves are constructed for different membrane resistances. In the case of high permeability, the equilibrium is absolutely unstable with respect to monotonic-type longwave perturbations. For low permeability membranes, instability with respect to monotonic finite-wavelength perturbations is characteristic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 171–173, July–August, 1985.     

Spherically symmetric two-phase deformations and phase transition zones

This paper is concerned with characterization and stability assessment of two-phase spherically symmetric deformations that can be supported by a nonlinear elastic isotropic material. We study general properties of equilibrium two-phase spherically symmetric deformations. Then we specialize to phase transformations of a solid sphere that is subjected to an all-round tension/pressure. Two material models are used to demonstrate a variety of transformation behaviours and some common features. For both materials we construct phase transition zones (PTZs) formed in the space of principal stretches by those which can exist adjacently to an equilibrium interface. Then we demonstrate how the PTZ can be used for the prediction of the number of two-phase spherically symmetric solutions and study how the deformation field associated with each solution is related to the PTZ. We show that even in the simplest case of one interface the solution is not unique: two equilibrium two-phase solutions as well as one uniform one-phase solution are found under the same boundary conditions. For the three solutions we construct their load-deformation diagrams and compare the associated total energies. The stability of the two-phase states with respect to radial and small-wavelength perturbations is also examined. We observe how unstable solutions are related with the PTZ.     

Dynamic Elastoplastic Interaction between an Impactor and a Spherical Shell

Dynamic axisymmetric elastoplastic interaction between a massive body and a simply supported, circular segment of a spherical shell is studied. The problem of determining the contactinteraction force is formulated for the case of spherical and conical bodies. A nonlinear integral equation is derived for various models of local plastic compression using the equations of equilibrium of a membrane spherical shell written in terms of radial displacement of the shell. Numerical results are presented graphically.     

On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line. We study the stability of these relative equilibria. For arbitrary physical pendulums, we obtain stability conditions in the linear approximation. For a system consisting of two identical rods, we solve the stability problem the in nonlinear setting. For the same system, we study the existence, bifurcations, and stability of high-frequency periodic motions of small amplitude other than the relative equilibria on the vertical line. The studies of dynamic stability augmentation in mechanical systems under the action of high-frequency perturbations was initiated in the paper [1], where it was shown that the unstable inverted equilibrium of a pendulum may become stable if the suspension point vibrates rapidly. This idea was developed in [2–10] and other papers, where several aspects of motion of a mathematical pendulum in the case of rapid small-amplitude vibrations of the suspension point were studied in the linear setting and also (without full mathematical rigor) in the nonlinear setting. The motions of the suspension point along an arbitrary oblique straight line [2, 4, 7, 8], along the vertical [3, 5, 6], along the horizontal [9], and in the case of damping [8] were considered. The monograph [10] deals with the stabilization of a pendulum or a system of pendulums under periodic and conditionally periodic vibrations of the suspension point along the vertical, along an oblique straight line, and along an ellipse. A rigorous nonlinear analysis of the existence and stability of periodic motions of the mathematical pendulum under horizontal and oblique vibrations of the suspension point at arbitrary frequencies and amplitudes can be found in [11, 12]. For the case of vertical vibrations of the suspension point at an arbitrary frequency and amplitude, a rigorous stability analysis of the relative equilibria of the pendulum on the vertical was carried out in [13].     

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.     

Stability of pretwisted beams in cross bracings

New systems of pretwisted bars are suggested, and their stability under a conservative load is investigated. In this paper, closedform relationships are obtained for the direct evaluation of critical loads of pretwisted cross bracings with different end connections and an arbitrary ratio of the dimensionless parameters of tension and compression elements. The equation for the critical loads is derived for tension and compression braces with different section properties, lengths, and axial loading. It is found that the critical load of a twisted brace is higher than that of a nontwisted brace. Parametric solutions are graphically displayed to clarify the distinct behavior, including the boundary separating symmetric and antisymmetric modes of buckling.     

Hydrodynamic stability of the spherical front of a reaction accompanied by a strong increase in viscosity

The hydrodynamic stability of the steady spherical front of a reaction accompanied by a strong increase in viscosity taking place in a radial, centrally symmetric flow of condensed reaction medium is considered. The flow is assumed to be bounded by two permeable concentric spherical surfaces. The stability is investigated in the linear approximation with respect to small perturbations. A relation is obtained for the dependence of the perturbation growth rate on the parameters of the problem: the viscosity ratio of the medium at the front, the ratio of the boundary surface radii to the radius of the front, and the dimensionless hydrodynamic resistances of the boundary surfaces. It is shown that the front is stable over almost the entire physical region of variation of the parameters and that instability occurs only in the case of a low hydrodynamic resistance of the outer boundary surface as the front approaches.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 46–53, May–June, 1988.     

Electrohydrodynamic instability of two superposed Walters B´ viscoelastic fluids in relative motion through porous medium

Summary  The electrohydrodynamic Kelvin–Helmholtz instability of the interface between two uniform superposed viscoelastic (B′ model) dielectric fluids streaming through a porous medium is investigated. The considered system is influenced by applied electric fields acting normally to the interface between the two media, at which there are no surface charges present. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by either streaming and applied electric fields for the potentially unstable configuration, or streaming only for the potentially stable configuration, as long as perturbations in the direction of streaming are ignored. For perturbations in all other directions, there exists instability for a certain wavenumber range. The instability of this system can be enhanced (increased) by normal electric fields. In the presence of surface tension, it is found also that the normal electric fields have destabilizing effects, and that the surface tension is able to suppress the Kelvin–Helmholtz instability for small wavelength perturbations, and the medium porosity reduces the stability range given in terms of the velocities difference and the electric fields effect. Finally, it is shown that the presence of surface tension enhances the stabilizing effect played by the fluid velocities, and that the kinematic viscoelasticity has a stabilizing as well as a destabilizing effect on the considered system under certain conditions. Graphics have been plotted by giving numerical values to the parameters, to depict the stability characteristics. Received 27 March 2000; accepted for publication 3 May 2001     

跳跃振子平衡及其分岔的能量分析

用能量方法研究了跳跃振子的平衡与分岔.用势能驻值条件确定了平衡位置所满足的方程,通过势能极值判断平衡的稳定性.在不同的弹簧构型下,数值计算了平衡随系统弹性刚度和质量比变化的分岔图.结果表明,弹性刚度和质量比较小时,系统只有一个稳定平衡点和一个不稳定平衡点;刚度和质量比充分大时,系统分岔出一个新的稳定平衡点和一个新的不稳定平衡点.     

Closed polymerization front dynamics in a plane radial flow

The behavior of perturbations of a steady cylindrical front of arbitrary amplitude is investigated numerically. A calculation algorithm is developed on the basis of the well-known boundary integral equation method [7]. The conclusion [5] concerning the stability of the front in the case of an internal supply of medium for perturbations of arbitrary amplitude is confirmed. In the case of an external supply of medium (in the direction of decreasing radial coordinate) the instability of the front and the formation of radial low-viscosity fingers is demonstrated. The method proposed can easily be modified to permit the calculation of the motion of free surfaces and interfaces in chemically inert high-viscosity media. In this connection, it can be used for numerically modeling the flow of plastic and glass melts.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–10, September–October, 1991.