Intermittency in a cantilever plate in a randomly fluctuating fluid flow

Fluid–elastic systems nearing dynamic instabilities are known to be sensitive to fluctuations in fluid flow. A cantilever plate in axial flow with random temporal fluctuations, is examined numerically for its dynamical behaviour. The numerical model comprises of a nonlinear structural model for the flexible plate, coupled with unsteady lumped vortex model for the fluid forces. As the mean flow velocity is increased, the system transitions to limit cycle oscillations from a state of rest, through a regime of intermittent oscillations. The conditions for onset and disappearance of intermittency are discussed and are interpreted using stochastic bifurcation theories. While the onset of intermittency is found to be unaffected by the time scales of the flow fluctuations, they are observed to affect the length of the intermittency regime. The effect of plate flexibility on intermittency is also discussed.     

Thermal stresses in a two-dimensional infinite medium containing a rigid inclusion embedded in a line crack

This paper examines the problem of finding thermal stresses, caused by a symmetric indentation of a line crack by an inclusion in an infinite isotropic elastic heat conducting solid. The thermal and elastic problems are reduced to a system of triple integral equations. In each case the solution of the triple integral equations is obtained in a closed form. The expressions for the stress intensity factor at the edge of the line crack, the strain energy density function and the resultant pressure applied to the inclusion are obtained. The expression for the displacement component is also obtained. Finally the results for the physical quantities are displayed graphically.     

Accelerating motion of a sphere in a maxwell fluid

The translatory accelerating motion of a sphere due to an arbitrarily applied force in an unlimited Maxwell fluid is considered. The exact solutions for the velocity of the sphere for three particular types of accelerating motion are presented. The first is for a falling sphere; the second is for the decelerating motion of a sphere after the force which maintains the sphere with a constant velocity is removed; the third is for the motion of the sphere subjected to an impulsive force. The exact solutions are expressed in terms of real, regular, definite integrals which can be evaluated by numerical technique. Also presented are the asymptotic solutions for the velocity of the sphere in all three cases which are valid for small values of time.     

Displacement field around a circular hole in a viscoelastic plate

Moiré experimental techniques are used to measure displacement fields in viscoelastic plates undergoing large deformations at elevated temperatures. These experimental procedures are applicable to determining displacement fields in nonlinear materials. As preliminary information, the material properties are determined from creep studies. The moiré method is used to determine the strains under constant load and isothermal conditions. Tests are conducted for several combinations of load and temperature for 2.5 decades of time. Assuming thermorheologically simple behavior, the data are shifted to establish the creep extensional compliance over ten decades in time. The constitutive equations are formulated as integral equations, the kernels of which are the functions that were measured in this work. These equations are solved exactly for the infinitesimal case. The finite case is then approximated by an incremental superposition of a series of successiye infinitesimal solutions. The results are applied to a plate initially containing a circular hole, and are shown to agree closely with the experimental measurements.     

Wave dispersion of a gas in a liquid

A novel method of dispersing a gas in a liquid by pressure pulses generated by waves propagating from a hydrodynamic oscillation generator is proposed. Devices that realize this method, wave dispersers, are created and investigated experimentally. The amplitude-frequency characteristics of the wave processes in the dispersers and the size distribution density functions of the gas bubbles are obtained. It is shown that there are optimal pressure values at the disperser inlet at which a minimum bubble size is achieved. The average diameter of the gas bubbles produced by water wave dispersers in the optimal operation regimes varied on the range from 0.3 to 0.6 mm, depending on the gas flow-rate.     

Lifetime of a narrow channel in a liquid

Lifetime determination is considered for a narrow channel in a liquid collapsing in response to gravitational and capillary forces. Working formulas are derived to relate the lifetime to the properties of the liquid and the channel parameters. The calculations are compared with experiments on the effects of a focused high-power electron beam acting on a liquid and solid.Translated from Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 121–129, January 1974.     

Motion of a cylinder in a viscous electroconductive medium with a magnetic field

The method of force sources is used to consider the planar problem of the motion of a circular cylinder in a viscous electroconductive medium with a magnetic field. The conventional and magnetic Reynolds numbers are assumed to be small. Expressions are obtained for the hydrodynamic reaction forces of the medium, acting on the moving cylinder. It is shown that as a result of the flow anisotropy in the medium, caused by the magnetic field, in addition to the resistance forces on bodies moving at an angle to the field, there are deflecting forces perpendicular to the velocity vector. The velocity field disturbances at great distances from the moving cylinder are determined.The problems of viscous electroconductive flow about solid bodies in the presence of a magnetic field constitute one of the divisions of magnetohydrodynamics. Motion of an electroconductive medium in a magnetic field gives rise to inductive electromagnetic fields and currents which interact with the velocity and pressure hydrodynamic fields in the medium [1, 2]. Under conditions of sufficiently strong interaction, the number of independent flow similarity parameters in MHD is considerably greater than in conventional hydrodynamics. This circumstance complicates the theoretical analysis of MHD flow about bodies, and therefore we must limit ourselves to consideration of individual particular flow cases.Here we consider the linear problem of the motion of an infinite circular cylinder in a viscous incompressible medium with finite electroconductivity located in a uniform magnetic field.There are many studies devoted to the flow of a viscous electroconductive medium with a magnetic field about solid bodies (see, for example, [3–5]). Because of this, some of the results obtained here include previously known results, which will be indicated below. In contrast to the cited studies, the examination is made by the method of force sources, suggested in [6]. This method permits obtaining integral equations for the distribution of the forces acting on the surface of the moving body. Their solution is obtained for small Reynolds and Hartmann numbers. Then the nature of the velocity disturbances at great distances from the body are determined. These results are compared with conventional viscous flow about a cylinder in the Oseen approximation.     

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.     

Edge effects in a sandwich plate: a plane problem

Edge effects in a rectangular sandwich plate with isotropic components are studied. The mathematical model is represented by the homogeneous equations of linear elasticity, which is indicative of an approximate approach in edge-effect theory. The initial equations are reduced to inhomogeneous ones and an exact problem is formulated. Approximate solutions are found by the mesh method. Discrete problems are based on the concept of base scheme. The mesh equations are written in an explicit form and then solved using a computation optimization procedure. As an example, edge-effect zones in a real composite are analyzed.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 124–133, December 2004.     

Motion of a flexible finite-length filament in a flow of a viscous fluid

The two-dimensional problem of the configuration of a flexible filament of finite length in a deformable viscous fluid is solved. The flexuural stresses in the filament and the inertial and gravitational forces are not taken into account. The equilibrium equations are obtained. The friction force that acts on the filament surface from the side of the viscous fluid is proportional to the flow rate. The specific features of the evolution of a bent filament under the conditions of pure and simple shear of a fluid are studied numerically. Analytical solutions are obtained for the evolution of a rectilinear filament; in particular, the stretching force in the filament is found. For the indicated types of flow, the stability of a rectilinear filament against small perturbations is investigated. Volzhskii Polytechnical Institute at the Volgograd State Technical University, Volgograd 404121. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 144–153, March–April, 2000.     

Propagation of a stress pulse in a viscoelastic rod

Experimental work is reported on the propagation of a stress pulse in a viscoelastic waveguide. The data obtained are compared with results of analysis using one-dimensional wave-propagation theory. The waveguide used in this work is a low-density polyethylene rod 1/2 in. in diameter and 30-in. long. Stress input to the waveguide and the resulting particle velocity at three stations are measured using a crystal stress transducer, two Faraday-principle velocity transducers and a capacitor transducer. The experiment is described mathematically as a boundary-value problem formulated in terms of the one-dimensional equation of motion, the strain-displacement relationship, a hereditary constitutive equation and the stress-boundary condition. Fourier transform and inversion yield an integral expression for velocity which is evaluated numerically at three stations using measured values for the stress-boundary condition, material attenuation and phase velocity. The analytical results compare favorably with the experimental data. The one-dimensional theory appears adequate to describe pulse propagation of this type. The attenuation and phase velocity used here are found to be a linear function and a logarithmic increasing function of frequency respectively.     

Dynamics of a small bubble in a compressible fluid

An efficient (simplified) method for solving problems of spherically symmetric dynamics of a small gas bubble in a compressible fluid is proposed. The method is based on the joint use of the full problem statement (the gas dynamics equations for the gas and the fluid) and its relevant simplifications. Some approximate statements are discussed. In the proposed method, the rarefaction and compression of the gas during the slow motion of the bubble surface is assumed to be uniform over the bubble volume. At the same time the fluid in the thin zone adjacent to the bubble is considered to be slightly compressible. Otherwise the gas dynamics equations are used for the gas and the fluid. The dynamics of the fluid in the thick external zone are described by the linear acoustics only. The proposed simplified method and two others used in literature are estimated by comparison of their numerical results with those obtained in full statement. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

Secondary flows in a layer with a free surface

The loss of stability of a plane-parallel incompressible viscous heat-conducting fluid flow in a horizontal layer subject to a longitudinal temperature gradient is considered. The lower surface of the layer is assumed to be rigid, while the upper one is free with a surface tension coefficient depending linearly on temperature. Both boundaries are assumed to be thermally-insulated. The critical value of the temperature gradient as a function of other relevant parameters is determined by analyzing the spectrum of the linearized problem. Secondary flows arising after the onset of instability are determined from an analysis of the full nonlinear problem using the expansion of the solution in a power series in terms of a supercritical state parameter in the vicinity of the bifurcation point. Three types of secondary flows are studied: plane two-dimensional waves propagating along the temperature gradient; plane waves travelling at a certain angle to the gradient; and three-dimensional waves propagating along the gradient. A numerical method of problem solution, based on the polynomial approximation, is described.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 85–98, September–October, 1994.     

Acoustic field in a borehole within fluid formations with a horizontal interface

The acoustic field in a cylindrical borehole embedded in a horizontally layered infinite medium is studied. The modes in each layer are found to consist of continuous as well as discrete ones, the orthogonality and completeness of which are proven. The relevant weights associated with the two kinds of modes are determined by solving a set of integral equations deduced from the boundary conditions at the layer interface. This technique is not limited to low frequencies, but applies well to frequencies prevailing in typical logging environment. Reflection, transmission and coupling coefficients of different modes are evaluated. The numerical results are verified by comparison with the method of real axis integration (RAI) and the hybrid mehtod proposed by Tsang [“Transient acoustic waves in a fluid-filled borehole with a horizontal bed boundary separating two solid formations”, J. Acoust. Soc. Am. 81, 844–853 (1987)]. The scaled laboratory experiments are carried out and the corresponding results are presented for comparison with the numerical computation.     

Numerical simulations of a filament in a flowing soap film

Experiments concerning the properties of soap films have recently been carried out and these systems have been proposed as experimental versions of theoretical two‐dimensional liquids. A silk filament introduced into a flowing soap film, was seen to demonstrate various stable modes, and these were, namely, a mode in which the filament oscillates and one in which the filament is stationary and aligns with the flow of the liquid. The system could be forced from the oscillatory mode into the non‐ oscillatory mode by varying the length of the filament. In this article we use numerical and computational techniques in order to simulate the strongly coupled behaviour of the filament and the fluid. Preliminary results are presented for the specific case in which the filament is seen to oscillate continuously for the duration of our simulation. We also find that the filament oscillations are strongly suppressed when we reduce the effective length of the filament. We believe that these results are reminiscent of the different oscillatory and non‐oscillatory modes observed in experiment. The numerical solutions show that, in contrast to experiment, vortices are created at the leading edge of the filament and are preferentially grown in the curvature of the filament and are eventually released from the trailing edge of the filament. In a similar manner to oscillating hydrofoils, it seems that the oscillating filaments are in a minimal energy state, extracting sufficient energy from the fluid to oscillate. In comparing numerical and experimental results it is possible that the soap film does have an effect on the fluid flow especially in the boundary layer where surface tension forces are large. Copyright © 2004 John Wiley & Sons, Ltd.     

The motion of a closely-fitting sphere in a fluid-filled tube

Singular perturbation techniques are used to investigate the slow, asymmetric flow around a sphere positioned eccentrically within a long, circular, cylindrical tube filled with viscous fluid. The results apply to situations in which the sphere occupies virtually the entire cross section of the cylinder, so that the clearance between the particle and tube wall is everywhere small compared with both the sphere and tube radii. The technique is an improvement over conventional “lubrication-theory” analyses.Asymptotic expansions, valid for small dimensionless clearances, are obtained for the hydrodynamic force, torque and pressure drop for flow past a stationary sphere, as well as for the case of a sphere translating or rotating in an otherwise quiescent fluid. These expansions are employed to predict the macroscopic behavior of both a neutrally-buoyant sphere suspended in a Poiseuille flow, and a sedimenting sphere in a vertical tube.The results find application in capillary blood flow, pipeline transport of encapsulated materials, and falling-ball viscometers.     

Interaction of a screw dislocation with a notch in a piezoelectric bi-material

Summary The electro-elastic interaction of a screw dislocation and a notch in a piezoelectric bi-material is analyzed. The electro-elastic fields induced by the dislocation are derived using the conformal mapping and the image-dislocation approach, where the solution for a piezoelectric bi-material without a notch is used as a base. The stress and the electric displacement intensity factors of the notch and the image force on the dislocation are given explicitly. We find that intensity factors are expressed in terms of the effective material constants, while the radial component of the image force is independent of the notch angle and the angular position of the dislocation in the polar coordinate system. Numerical results for the image force are provided for the use when one of the two media is purely elastic. They illustrate the behavior of the dislocation in the neighborhood of the notch.     

Dynamics of a cavitational cavity in a non-newtonian fluid

The dynamics of a spherical cavity in a non-Newtonian fluid, described by the Reiner-Rivlin rheological equation [1], is investigated. The equation of radial cavity motion is obtained, where the gas in the cavity is subject to a polytropic law and surface tension is taken into account. The equation of cavity motion is solved numerically for a number of values of the transverse viscosity coefficient. The influence of the transverse viscosity on the collapse process of vapor and gas-filled cavities is shown. Numerical computations are also carried out for the rate of energy dissipation and the pressure distribution in the fluid.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 170–173, July–August, 1973.The authors are grateful to A. T. Listrove for attention to the research.     

Electroelastic analysis of a piezoelectric cylindrical fiber with a penny-shaped crack embedded in a matrix

The electroelastic response of a penny-shaped crack in a piezoelectric cylindrical fiber embedded in an elastic matrix is investigated in this study. Fourier and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. They are then reduced to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor, energy release rate and energy density factor for piezoelectric composites are obtained to show the influence of applied electric fields.