Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The locked pendulum mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.     

Dynamic adsorption in a radial flow of a solution around a spherical cavity

The paper studies the local variation of the concentration of a neutral dilute solution during its radial flow around a spherical cavity in the approximations of an adsorption layer and the Langmuir adsorption kinetics. The authors used the boundarylayer method and the method of asymptotic series expansion of the solution in a small parameter, which is the ratio of the time of establishing an adsorption equilibrium to the time of establishing a steady diffusion layer around the cavity. The equations obtained for a zeroth approximation were studied analytically and numerically. In the case of highfrequency oscillations of the cavity in the solution, a solution of the problem was found that corresponds to the process of straightened adsorption or pumping an admixture into the adsorption layer.     

LDA measurements in low Reynolds number turbulent boundary layer

LDA measurements of the mean velocity in a low Reynolds number turbulent boundary layer allow a direct estimate of the friction velocity U from the value of /y at the wall. The trend of the Reynolds number dependence of / is similar to the direct numerical simulations of Spalart (1988).     

Three-mode interactions in harmonically excited systems with quadratic nonlinearities

An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ₃2₂ and ₂2₁ to a harmonic excitation of the third mode, where the _m are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitudeF of the excitation as a control parameter. As the excitation amplitudeF is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. AsF is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. AsF is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.     

Concentration-dependent behaviour of the shear viscosity of coal-fuel oil suspensions

A function correlating the relative viscosity of a suspension of solid particles in liquids to their concentration is derived here theoretically using only general thermodynamic ideas, with out any consideration of microscopic hydrodynamic models. This function ( _r = exp (1/2B ^* C ²)) has a great advantage over the many different functions proposed in literature, for it depends on a single parameter,B ^*, and is therefore concise. To test the validity of this function, a least-squares regression analysis was undertaken of available data on the viscosity and concentration of suspensions of coal particles in fuel oil, which promise to be a useful alternative to fuel oil in the near future. The proposed function was found to accurately describe the concentration-dependent behaviour of the relative viscosity of these suspensions. Furthermore, an attempt was made to obtain information about the factors affecting the value ofB ^*, however the results were only qualitative because of, among other things, the inaccuracy of the viscosity measurements in such highly viscous fluids. shear viscosity of the suspension - ₀ shear viscosity of the Newtonian suspending medium - _r = /₀ relative viscosity - solid volume concentration - c solid weight concentration - _m maximum attainable volume concentration of solids - solid volume concentration at which the relative viscosity of the suspension becomes infinite - c _m maximum attainable solid weight concentration - _s density of the solid phase - _l density of the liquid phase - _m density of the suspension - k _n coefficients of theø-power series expansion of _r - { _j } sets of parameters specifying the thermodynamic state of the solid phase of a suspension - T absolute temperature (K) - f (c, T, _j) formal expression for the relative variation of the viscosity with concentration = [1 / (/c)] _T,j - d median size of the granulometric distribution - _B plastic or Bingham viscosity - K consistency factor - n flow index - g ([c _m –c],T, _j ) function including an asymptotic divergence asc tends toc _m , formally describing the concentration dependent behaviour of the shear viscosity of a suspension - A (T, _j) regression analysis parameters - B (T, _j) regression analysis parameters - B ^* (T, _j ) regression analysis parameters     

On the linear instability of the Hall-Stewartson vortex

It is demonstrated that the Hall-Stewartson leading-edge vortex is linearly unstable to viscous perturbations of the center-mode type. Center modes are found to occur in two reigons of Reynolds-number-wave-number space, in limits in which the axial wave number is large. The appropriate center-mode equations in these neighborhoods are established, and it emerges that the two sets are identical. The single system of equations, which depends on the azimuthal wave number m and a distance parameter only, is solved numerically for various values of m and . Highly unstable modes are found for large positive , and the results are shown to be in good agreement with proposed asymptotic expansions when >1. To lowest order, unstable modes have phase surfaces that rotate with the fluid: in addition constant phase surfaces propagate upstream but the group velocity is directed downstream. The growth rate of the instability increases faster than Reynolds number to the quarter power. This, together with the finding that the length scale of the unstable modes found goes to zero as the Reynolds number tends to infinity, makes this instability an unusual one.This work was supported by the Air Force Office of Scientific Research under contract AFOSR-89-0346 monitored by Dr. L. Sakell, and by the U.S. Army Research Office at the Mathematical Sciences Institute of Cornell University.     

Rotating convection in a finite cylinder

This paper is devoted to analyzing numerically experimental observations of azimuthally travelling waves that appear in rotating convection in a circular container at intermediate Prandtl numbers. The instability is a Hopf bifurcation that gives rise to a pattern precessing generally counter to the rotation direction. Two types of modes can be differentiated, the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the center. Results are presented for Prandtl number 6.8 and aspect ratio d/h equal to 2.5 as a function of the rotation rate. For rigid insulating sidewalls, and rigid thermally conducting top and bottom lids, the results agree well with those mesured experimentally.     

Real-Space Renormalization Estimates for Two-Phase Flow in Porous Media

We present a spatial renormalization group algorithm to handle immiscibletwo-phase flow in heterogeneous porous media. We call this algorithmFRACTAM-R, where FRACTAM is an acronym for Fast Renormalization Algorithmfor Correlated Transport in Anisotropic Media, and the R stands for relativepermeability. Originally, FRACTAM was an approximate iterative process thatreplaces the L × L lattice of grid blocks, representing the reservoir,by a (L/2) × (L/2) one. In fact, FRACTAM replaces the original L× L lattice by a hierarchical (fractal) lattice, in such a way thatfinding the solution of the two-phase flow equations becomes trivial. Thistriviality translates in practice into computer efficiency. For N=L ×L grid blocks we find that the computer time necessary to calculatefractional flow F(t) and pressure P(t) as a function of time scales as N^1.7 for FRACTAM-R. This should be contrasted with thecomputational time of a conventional grid simulator N^2.3. The solution we find in this way is an accurateapproximation to the direct solution of the original problem.     

Bifurcations of two-dimensional into three-dimensional wave regimes for a vertically flowing liquid film

The three-dimensional steady traveling wave regimes of a viscous liquid film flowing down a vertical wall which branch off from two-dimensional nonlinear waves are investigated. The numerical calculations are based on a model system of equations valid for intermediate Reynolds numbers. It is shown that there exist two fundamentally different types of three-dimensional steady traveling waves branching off from plane waves. One of these possesses checkerboard symmetry in the distribution of the maxima of the wave profile thickness and is the more interesting. An important difference in the breakdown of plane waves of the first and second families is also demonstrated. The wave characteristics of certain three-dimensional regimes are calculated as functions of the bifurcation parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 109–114, September–October, 1990.     

Solitary wave trains in a cold plasma

The existence of traveling solitary waves, the products of modulation instability in a cold quasi-neutral plasma, is considered. Solitary waves of this type (solitary wave trains) are formed as a result of bifurcation from a nonzero wave number of the linear wave spectrum. It is shown that the complete system of equations describing the wave process in a cold plasma has solutions of the solitary wave train type, at least when the undisturbed magnetic field is perpendicular to the wave front. Sufficient conditions of existence of solitary wave trains in weakly dispersive media are also formulated.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 154–161, September–October, 1996.     

Visualization of the structure of supersonic turbulent boundary layers

A series of flow visualizations has been performed on two flat-plate zero-pressure-gradient supersonic boundary layers. The two different boundary layers had moderate Mach numbers of 2.8 and 2.5 and Re 's of 82, 000 and 25, 000 respectively. A number of new visualization techniques were applied. One was a variation of conventional schlieren employing selective cut-off at the knife edge plane. Motion pictures of the flow were generated with this technique. Droplet seeding was also used to mark the flow, and high speed movies were made to show structure evolution. Still pictures were also taken to show details within the large-scale motions. Finally, Rayleigh scattering was used to construct planar images of the flow. Together, these techniques provide detailed information regarding the character and kinematics of the large-scale motions appearing in boundary layers in supersonic flow. Using these data, in concert with existing hot-wire data, some suggestions are made regarding the characteristics of the average large-scale motion.This work was supported by the Air Force Office of Scientific Research under Grant 89-0120, monitored by Dr. James M. McMichael. Also, the authors wish to thank Prof. R. B. Miles for his contributions to the Rayleigh scattering portion of this project.     

A free boundary problem modeling thermal instabilities: Stability and bifurcation

In this paper, we analyze a simple free boundary model associated with solid combustion and some phase transition processes. There is strong evidence that this one-phase model captures all major features of dynamical behavior of more realistic (and complicated) combustion and phase transition models. The principal results concern the dynamical behavior of the model as a bifurcation parameter (which is related to the activation energy in the case of combustion) varies. We prove that the basic uniform front propagation is asymptotically stable against perturbations for the bifurcation parameter above the instability threshold and that a Hopf bifurcation takes place at the threshold value. Results of numerical simulations are presented which confirm that both supercritical and subcritical Hofp bifurcation may occur for physically reasonable nonlinear kinetic functions.     

Hopf bifurcation in a three-dimensional system

In this paper,Liapunor-Schmidl reduction and singularity theory are employed to discuss Hopf and degenerate Hopf bifureations in global parametric region in a three-dimensional system x=-βx+y, y=-x-βy(1-kz), z=β[α(1-z)-ky²], The conditions on existence and stability are given.     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

Inertial phase separation in rotating self-gravitating media

The centrifugal separation of foreign inclusions (particles) in a rotating spherical volume of a self-gravitating medium is considered in the hydrodynamic approximation. Using the full Lagrangian approach, the particle trajectories and radial concentration profiles are studied for a rigid-body velocity distribution in the carrier phase. The regimes of continuum and free-molecular flow around the particles are considered. The cases of a heavy (with density greater than that of the carrier phase and traveling toward the center) and a light-weight (traveling toward the periphery) admixture are investigated. Analytical and numerical solutions corresponding to steady-state spherically symmetric boundary conditions for the dispersed phase are found. It is shown that the presence of rotation may result in a significant angular anisotropy of the radial particle concentration distributions and, in particular, in the formation of ring-shaped accumulation zones of heavy inclusions in the equatorial plane. The solutions obtained can be used to explain the mechanisms of onset of density nonuniformities in planet cores, the formation of planetary systems from gas-particle clouds, and the behavior of aerosol particles in atmospheric vortices.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 86–100.Original Russian Text Copyright © 2004 by Ahuja, Belonoshko, Johansson, and Osiptsov.     

Numerical Analysis of Axisymmetric Buckling of Plates under Radial Compression

Nonlinear boundaryvalue problems of axisymmetric buckling of simply supported and clamped plates under radial compression are formulated for a system of six firstorder ordinary differential equations with independent fields of finite displacements and rotations. Multivalued solutions are obtained by the shooting method with specified accuracy. Bifurcation of the solutions of the problem is studied, and a parametric bifurcation diagram is constructed for various values of the loading parameter. Curves of buckling modes are given for three branches of the solution. The numerical results agree with available theoretical data.     

Symmetry-Breaking Bifurcations of Charged Drops

It has been observed experimentally that an electrically charged spherical drop of a conducting fluid becomes nonspherical (in fact, a spheroid) when a dimensionless number X inversely proportional to the surface tension coefficient is larger than some critical value (i.e., when <_c). In this paper we prove that bifurcation branches of nonspherical shapes originate from each of a sequence of surface-tension coefficients ), where ₂=_c. We further prove that the spherical drop is stable for any >₂, that is, the solution to the system of fluid equations coupled with the equation for the electrostatic potential created by the charged drop converges to the spherical solution as t provided the initial drop is nearly spherical. We finally show that the part of the bifurcation branch at =₂ which gives rise to oblate spheroids is linearly stable, whereas the part of the branch corresponding to prolate spheroids is linearly unstable.     

Asymptotic Modeling of Nonlinear Wave Processes in Shock‐Loaded Elastoplastic Materials

Nonlinear wave processes in shockloaded elastoplastic materials are modeled. A comparison of the results obtained with experimental data and numerical solutions of exact systems of dynamic equations shows that the model equations proposed qualitatively describe the stressdistribution evolution in both the elasticflow and plasticflow regions and can be used to solve one and twodimensional problems of pulsed deformation and fracture of elastoplastic media.     

Nonlinear Modes of Motion of Thin Circular Cylindrical Shells

Free flexural vibrations of a simply supported shell are studied within the framework of the nonlinear theory of flexible shallow shells. It is assumed that largeamplitude flexural vibrations are coupled with radial vibrations of the shell. Modal equations are derived by the Bubnov–Galerkin method. Periodic solutions are obtained by the Krylov–Bogolyubov method. The skeleton curve of the soft type obtained using a nonlinear finitedimensional shell model agrees with available experimental data.     

Doubly periodic and quasiperiodic wave regimes in a liquid film flowing down an inclined plane,their stability and bifurcations

A nonlinear evolution equation frequently encountered in modeling the behavior of disturbances in various nonconservative media, for example, in problems of the hydrodynamics of liquid film flow, is considered. Wave solutions of this equation, regular in space and both periodic and quasiperiodic in time, branching off from steady and steady-state traveling waves are found numerically. The stability and bifurcations are analyzed for some of the solutions obtained. As a result, a bifurcation chain is found for solutions stable with respect to disturbances of the same spatial period. It is shown that the bifurcations are related to the loss of certain symmetries of the initial solution. It is demonstrated that as the bifurcation parameter increases it is possible to distinguish in the structure of the solutions intervals of quiet behavior and intervals of intense outbursts.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 98–107, July–August, 1992.