Stability of a disperse-mixture flow in a boundary layer

The hydrodynamic stability of a dilute disperse mixture flow in a quasi-equilibrium region of a boundary layer with a significantly nonuniform particle concentration profile is investigated. The mixture is described by a two-fluid model with an incompressible viscous carrier phase. In addition to the Stokes drag, the Saffman lifting force is taken into account in the interphase momentum exchange. On the basis of a numerical solution of the boundary-value problem for a modified Orr-Sommerfeld equation, neutral stability curves are analyzed and the dependence of the critical Reynolds number on the governing parameters is studied. It is shown that taking into account the particle concentration nonuniformity in the main flow and the Saffman lifting force significantly changes the stability limits of the two-phase laminar boundary layer flow. The effect of these factors on the boundary layer stability is considered for the first time.     

Stability of the equilibrium of a capillary fluid in a horizontal slit

The stability of axisymmetric shapes of equilibrium of a capillary fluid between two horizontal plates is investigated. It is shown that stability is lost with respect to axisymmetric perturbations. The boundaries of the stability region are calculated in the case when the wetting angles on the lower and upper plates are the same.     

Instability of a horizontal plane-channel flow of a dilute suspension

The stability of a horizontal plane-channel flow of a dilute suspension is studied theoretically. It is shown that the mechanism of action of the sedimenting particles on the flow stability parameters is equivalent to the effect of a distributed flow stratification and is attributable to the vertical nonuniformity of the body force induced by the excess weight of the sedimenting particles. A strong dependence of the disturbance growth rate on the location of the interface between the suspension and the pure liquid is detected.     

Transition to instability of a phase interface in a porous medium in the isothermal approximation

The stability of the phase interface in geothermal systems is considered in the isothermal approximation with allowance for capillary effects. The dispersion relation is obtained and the domains of stability and instability of steady-state vertical flows are found. Possible types of transition to instability, namely, transitions with the most unstable mode corresponding to zero and infinite wavenumbers or to all wavenumbers simultaneously, are described. In the first case the nonlinear Kolmogorov-Petrovskii-Piskunov equation describing the evolution of a narrow strip of weakly unstable modes on the stability threshold is derived. The effect of the parameters of the system on its stability is investigated.     

Stability of a plane ionizing shock in a longitudinal magnetic field

The influence of a permanent transverse magnetic field on the stability of a plane shock relative to small displacements of its front from the equilibrium position was examined in [1, 2]. Under the same simplifying assumptions, the stability of a shock in a longitudinal magnetic field (the induction vector is directed along the normal to the discontinuity) is investigated in this paper. The boundaries of the stability domain are determined. It is shown that the whole domain of neutral oscillations which exist in the gasdynamic case makes the transition into the stability domain in the presence of a longitudinal field. The boundaries of the stability domain are independent of the interaction parameter in contrast to the case of shock motion in a transverse field [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 195–198, March–April, 1976.     

Stability of a viscous liquid jet in a high-frequency alternating electric field

The stability of a liquid electrolyte placed in a tangential electric field oscillating harmonically at high frequency is considered assuming that the liquid is viscous and Newtonian. It is shown that, if the Peclet number calculated from the thickness of the Debye layer is small, the problem can be solved separately for the electrodynamic part of the problem in the Debye layer and for the hydrodynamic part of the problem in the jet. The linear stability of the trivial solution of the problem is investigated. A dispersion relation is derived and used to study the effect of the amplitude and frequency of electric field oscillations on the stability of the jet. It is shown that the presence of the external oscillating field has a stabilizing effect on the jet. The basic stability regimes as functions of the control parameters of the problem and bifurcation changes in the regimes are investigated.     

Poynting oscillations of a rigid disk supported by a neo-Hookean shaft

Simultaneous axial and torsional oscillations of a rigid disk attached to an elastomeric shaft are investigated. Five cases are solved exactly. The uncoupled, small amplitude axial and torsional oscillations of the disk are investigated for neo-Hookean and Mooney-Rivlin shafts with static stretch. The finite torsional vibration of the load superimposed on a static stretch of the shaft is studied for the Mooney-Rivlin model. Solutions for both small and finite amplitude, uniaxial vibrations of the body superimposed on a pretwisted neo-Hookean shaft with static stretch are derived. Simple bounds on the period for the finite motion are provided; and various universal frequency relations for neo-Hookean and Mooney-Rivlin materials are identified.Finally, the main problem of finite, uniaxial vibrations accompanied by a small twisting motion is studied for the neo-Hookean model. The exact periodic solution for the axial response is obtained; and the coupled, small torsional motion is then determined by Hill's equation. A stability criterion for the Mathieu-Hill equation is used to obtain stability maps in a physical parameter space. Geometrical conditions sufficient for universal stability of the motion are read from this graph. Instability of the torsional oscillation, the beating phenomenon and exchange of energies, and the relation of the stability diagram to amplitude bounds on the uncoupled, linearized motion sufficient to assure universal stability predicted for small amplitude vibrations, are discussed and described graphically with the aid of a numerical model. It is shown that an unstable configuration may be stabilized by increasing the diameter of the disk.     

Stability analysis of a noise control system in a duct by using delay differential equation

The paper deals with the criteria for the closed- loop stability of a noise control system in a duct. To study the stability of the system, the model of delay differential equation is derived from the propagation of acoustic wave governed by a partial differential equation of hyperbolic type. Then, a simple feedback controller is designed, and its closed- loop stability is analyzed on the basis of the derived model of delay differential equation. The obtained criteria reveal the influence of the controller gain and the positions of a sensor and an actuator on the closed-loop stability. Finally, numerical simulations are presented to support the theoretical results.     

Nonaxisymmetric compressive failure of a circular crack parallel to a surface of halfspace

Considered in this work is the fracture stability of a circular crack parallel to the surface of a halfspace subjected to around applied compression. A state of subcritical initial strain is assumed. The failure criterion is based on the loss of local stability and quantified in terms of a critical eigen value. Analysis involves reducing the problem to a system of Fredholm equations of the second kind where the solutions are identified with harmonic potential functions. Critical loads are determined for nonaxisymmetric forms of loss in stability in the region local to the crack; their values would depend on the material properties and geometric parameters. Numerical results are displayed graphically for a crack in an elastic solid and a composite made of aluminum/boron/silicate glass with epoxy-maleimic resin. They depend on the number of angularly dependent harmonies that introduce the nonaxisymmetry in the problem.     

Convective stability of a medium containing a heavy solid additive

Two problems of convective stability in a medium containing settling heavy solid particles are discussed. A study is made of the stability of steady convective flow of a medium containing an additive between vertical plates heated to different temperatures and also of the stability of a flat layer of a medium containing an additive which is heated from below. It is shown that the presence of settling solid particles has a significant stabilizing effect on convective stability.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 105–115, May–June, 1976.The author thanks E. M. Zhukhovitskii for directing the work, V. E. Nakoryakov and participants in the seminars directed by him, and also A. G. Kirdyashkin for providing valuable discussions of the results.     

Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 101–106, January–February, 1976.     

Stability of a viscoelastic plate in fluid flow

The stability of an infinite viscoelastic plate on an elastic foundation in a viscous incompressible flow is studied. The Navier-Stokes system is linearized for an exponential velocity profile. The problem is reduced by a Fourier-Laplace transform to a system of ordinary differential equations, whose solution is found in the form of convergent series. The roots of the dispersion relation that characterize the stability of the system are found numerically. The effect of the viscosities of the fluid and the plate on the stability of the waves propagating upstream and downstream is studied. The results are compared with available data on the stability of a viscoelastic plate in an ideal fluid flow. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 66–74, July–August, 2006.     

The stability of growth of a through-wall circumferential crack in a pipe subject to combined bending and tensile loadings

The stability of growth of a through-wall circumferential crack in a pipe is analysed for the case where the material has a high crack growth resistance, the analysis being based on the tearing modulus procedure. Rotations and lateral displacements are applied at the ends of the pipe, and this allows the combined effects of bending and tensile loadings on the stability of crack growth to be assessed. The general conclusion is that tensile loadings can have an adverse effect on crack stability, in accord with the conclusion reached in the author's earlier studies of plane strain crack growth in a beam. The stability results are compared with those obtained by Tada, Paris and Gamble, who allowed the tensile loadings to affect the position of the neutral axis, but did not consider instability in terms of the deformations produced by these loadings.     

Effect of a Standing Acoustic Wave on the Development of Large-Scale Convection in a Horizontal Layer

The effect of a standing acoustic wave on the development of long-wave convective perturbations in a horizontal layer with thermally insulated boundaries is investigated. The main two-dimensional flow is determined. A nonlinear amplitude equation with spatially-periodic coefficients is derived for investigating the stability of the main flow and secondary convection flows in the neighborhood of the stability threshold. The intensity of the acoustic field is assumed to be low. It is shown that the acoustic action leads to destabilization of the layer. Plane and three-dimensional perturbations are critical at large and small Prandtl numbers, respectively. Nonlinear one-dimensional steady-state solutions of the amplitude equation are obtained and their stability is investigated.     

Effect of vibration on the stability of a plane displacement front in a porous medium

The effect of linearly polarized vibration on the stability of a plane displacement front in a porous medium is studied. The problem of the stability of the motion of a plane displacement front traveling at a constant velocity U under the action of vibration normal to the front is considered. It is shown that under the action of vibration the dynamics of the plane displacement front can be described by the Mathieu equation with a dissipative term. Using the standard averaging method, in the case of high-frequency vibration it is revealed that vibration can only increase the stability of the system. It is found that the vibration stabilizes the plane displacement front with respect to part of the perturbation spectrum.     

Stability in a case of motion of a paraboloid over a plane

We solve a nonlinear orbital stability problem for a periodic motion of a homogeneous paraboloid of revolution over an immovable horizontal plane in a homogeneous gravity field. The plane is assumed to be absolutely smooth, and the body–plane collisions are assumed to be absolutely elastic. In the unperturbed motion, the symmetry axis of the body is vertical, and the body itself is in translational motion with periodic collisions with the plane.The Poincare´ section surfacemethod is used to reduce the problemto studying the stability of a fixed point of an area-preserving mapping of the plane into itself. The stability and instability conditions are obtained for all admissible values of the problem parameters.     

Effects of a crack on the stability of a non-linear rotor system

The stability of a rotor system presenting a transverse breathing crack is studied by considering the effects of crack depth, crack location and the shaft's rotational speed. The harmonic balance method, in combination with a path-following continuation procedure, is used to calculate the periodic response of a non-linear model of a cracked rotor system. The stability of the rotor's periodic movements is studied in the frequency domain by introducing the effects of a perturbation on the periodic solution for the cracked rotor system.It is shown that the areas of instability increase considerably when the crack deepens, and that the crack's position and depth are the main factors affecting not only the non-linear behaviour of the rotor system but also the different zones of dynamic instability in the periodic solution for the cracked rotor. The effects of some other system parameters (including the disk position and the stiffness of the supports) on the dynamic stability of the non-linear periodic response of the cracked rotor system are also investigated.     

Dynamics of a Fluid in a Rotating Horizontal Cylinder

The behavior of a low-viscosity fluid in a rotating horizontal circular cylinder is investigated experimentally. The stability of the centrifuged layer, the motion of the fluid with respect to the cavity, the excitation of inertial waves on the fluid surface, and the effect of the waves on the stability and flow structure are studied over a wide region of relative occupancy of the cavity. The results are analyzed from the viewpoint of vibrational mechanics in which the motion is generated by the oscillations of the fluid with respect to the cavity and the gravity force plays the role of the force oscillating in the cavity reference system.     

Automatic control of a modified tractor to work on steep side slopes

The main purpose of this paper is to investigate the control and stability of a tractor moving over obstacles on steep side slopes. To this end, a modified slope tractor is considered and the lateral stability of the vehicle is studied. The modified tractor is able to roll against the side slope and change the position of its mass centre. The tractor behaviour in a sloped field on a straight path is simulated in ADAMS software environment. A two-layer controller consisting of a Fuzzy upper layer and a PI lower layer is designed for the control of tractor stability. The dynamics of hydraulic actuators adjusting the body roll angle is also included. The stability of controlled and uncontrolled tractors is investigated by several simulations. It is shown that in comparison with the two types of ordinary and uncontrolled slope tractors, the controller is capable of maintaining the stability of modified tractor and preventing it from rollover and instability on various side slopes and obstacles.     

Stability of a plane-parallel convective flow of a liquid in a horizontal layer

We consider the stationary plane-parallel convective flow, studied in [1], which appears in a two-dimensional horizontal layer of a liquid in the presence of a longitudinal temperature gradient. In the present paper we examine the stability of this flow relative to small perturbations. To solve the spectral amplitude problem and to determine the stability boundaries we apply a version of the Galerkin method, which was used earlier for studying the stability of convective flows in vertical and inclined layers in the presence of a transverse temperature difference or of internal heat sources (see [2]). A horizontal plane-parallel flow is found to be unstable relative to two critical modes of perturbations. For small Prandtl numbers the instability has a hydrodynamic character and is associated with the development of vortices on the boundary of counterflows. For moderate and for large Prandtl numbers the instability has a Rayleigh character and is due to a thermal stratification arising in the stationary flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–100, January–February, 1974.