On the Perturbation of Hamel Flow Due to Unevenness of the Channel Walls

A method of analyzing the receptivity of longitudinally inhomogeneous flows is proposed. The process of excitation of natural oscillations is studied with reference to the simplest inhomogeneous flow: the two-dimensional flow of a viscous incompressible fluid in a channel with plane nonparallel walls. As physical factors generating perturbations, the cases of a stationary irregularity and localized vibration of the channel walls are considered. By changing the independent variables and unknown functions of the perturbed flow, the problem of the generation of stationary perturbations above an irregularity is reduced to a longitudinally homogeneous boundary-value problem which is solved using a Fourier transform in the longitudinal variable. The same problem is investigated using another method based on representing the required solution in the form of a superposition of solutions of the homogeneous problem and a forced solution calculated in the locally homogeneous approximation. As a result, the problem of calculating the longitudinal distributions of the amplitudes of the normal modes is reduced to the solution of an infinite-dimensional inhomogeneous system of ordinary differential equations. The numerical solution obtained using this method is tested by comparison with an exact calculation based on the Fourier method. Using the method proposed, the problem of flow receptivity to harmonic oscillations of parts of the channel walls is analyzed. The calculations performed show that the method is promising for investigating the receptivity of longitudinally inhomogeneous flow in a laminar boundary layer.     

Investigation of Mechanisms of Excitation and Growth of Unstable Perturbations in Pulsating Flows

General laws of the processes of generation and amplification of secondary perturbations in oscillating viscous fluid flows are studied theoretically. The stability and receptivity are analyzed with reference to perturbations generated by fluctuations of the flow rate of Poiseuille flow induced by small two-dimensional roughnesses of the channel walls. It is shown that the presence of roughness leads to excitation in the flow of perturbations at all multiples of the main flow oscillation frequency. Using the Fourier transform along the streamwise coordinate, the problem of calculating the frequency harmonics is reduced to a system of equations of the Orr-Sommerfeld type interrelated via the oscillatory component of the main flow. On the basis of an investigation of the analytic properties of the Fourier-image it is shown that upstream and downstream of the roughness the perturbation can be represented in the form of a superposition of modes of the time-dependent Poiseuille flow. The modes are classified and their spectrum is calculated. The structure of the mean-square fluctuations generated by free perturbations is investigated. Examples of calculating the evolution of forced perturbations are given for cases in which the scattering of the oscillations of the main flow on the roughness leads to the generation of one or two modes growing downstream.     

Propagation of Perturbations in Plane Poiseuille Flow between Walls of Nonuniform Compliance

The evolution of small perturbations in longitudinally nonuniform flows is studied with reference to the problem of the propagation of flow perturbations in a plane channel with walls of variable elasticity. Using the solution of the problem of the receptivity of the flow to local vibrations of the walls, the problem considered can be reduced to the solution of an integral equation for a single function, namely, the complex vibration amplitude of the walls. A numerical method for solving this equation on the basis of a piecewise-linear approximation of the unknown function is proposed. It is shown that the instability wave amplitude changes discontinuously at the junction of the rigid and elastic channel sections. A second method of investigating the process of propagation of perturbations in the flow considered is proposed. This method is based on laws of evolution of perturbations in nonuniform flows and an analytic solution of the problem of perturbation scattering on the junction of walls with different compliance. On the basis of this method the classical stability theory is generalized to include the case of nonuniform flows.     

Estimates of the evolution of small perturbations in the radial spread (drain) of a viscous ring

This paper studies the evolution of small perturbations in the kinematic and dynamic characteristics of the radial flow of a flat ring filled with a homogeneous Newtonian fluid or an ideal incompressible fluid. When the flow rate is specified as a function of time, the main motion is completely determined by the incompressibility condition regardless of the properties of the medium. A biparabolic equation for the stream function with four homogeneous boundary conditions which simulate adhesion to the expanding (contracting) walls of the ring is derived. Upper bounds for the perturbation are obtained using the method of integral relations for quadratic functionals. The case of an exponential decay of initial perturbations is considered in a finite or infinite time interval. The admissibility of the inviscid limit in this problem is proved, and upper and lower bounds for this limit are estimated.     

Propagation of a Tollmien-Schlichting wave over the junction between rigid and compliant surfaces

The propagation of an instability wave over the junction region between rigid and compliant panels is studied theoretically. The problem is investigated using three different methods with reference to flow in a plane channel containing sections with elastic walls. Within the framework of the first approach, using the solution of the problem of flow receptivity to local wall vibration, the problem considered is reduced to the solution of an integro-differential equation for the complex wall oscillation amplitude. It is shown that at the junction of rigid and elastic channel walls the instability-wave amplitude changes stepwise. For calculating the step value, another, analytical, method of investigating the perturbation propagation process, based on representing the solution as a superposition of modes of the locally homogeneous problem, is proposed. This method is also applied to calculating the flow stability characteristics in channels containing one or more elastic sections or consisting of periodically alternating rigid and compliant sections. The third method represents the unknown solution as the sum of a local forced solution and a superposition of orthogonal modes of flow in a channel with rigid walls. The latter method can be used for calculating the eigenvalues and eigenfunctions of the stability problem for flow in a channel with uniformly compliant walls.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 31–48. Original Russian Text Copyright © 2004 by Manuilovich.     

Numerical Analysis of Laminar-Turbulent Transition in a Circular Pipe with Periodic Inflow Perturbations

The problem of the spatio-temporal evolution of perturbations introduced into the inlet cross-section of a circular pipe is solved numerically. The case of time-periodic inflow perturbations is considered for Re = 4000. It is shown that for relatively small inflow perturbations periodic flow regimes and for greater perturbations chaotic regimes are established.Periodic regimes the flow is a superposition of steady flow and a damped wave propagating downstream. The velocity profile of the steady component differs essentially from both the parabolic Poiseuille and developed turbulent flows and is strongly inhomogeneous in the angular direction. The angular distortion of the velocity profile is caused by longitudinal vortices developing as a result of the nonlinear interaction of inflow perturbations.Chaotic flow regimes develop when the amplitude of the inflow perturbations exceeds a certain threshold level. Stochastic high-frequency pulsations appear after the formation of longitudinal vortices in the regions of maximum angular gradient of the axial velocity. In the downstream part of the flow, remote from the transition region, the developed turbulent regime is formed. The distributions of all the statistical moments along the pipe level off and approach the values measured experimentally and calculated numerically for developed turbulent flows.     

Stability of viscous flow between rotating and stationary disks

The linear problem of the stability of viscous flow between rotating and stationary parallel disks is solved in the locally homogeneous formulation using the method of normal modes. The main flow is assumed to be selfsimilar with respect to the radial coordinate. The system of sixth-order equations, derived for the amplitude functions of the disturbances, is integrated by a finite difference method. The stability characteristics with respect to disturbances of four types are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 79–87, November–December, 1991.     

Numerical simulation of rarefied suspension sedimentation in a container

The evolution of the large-scale velocity perturbations in a homogeneous suspension sedimenting in a rectangular container with rigid horizontal walls and periodic conditions on the vertical boundaries is considered. Numerical simulation of the point-particle motion showed that the density and velocity fluctuations decrease with time. The perturbations are damped due to reshaping of the sedimentation front and the nonlinear interaction of the different modes.     

Stability of the Flow in a Streaky Structure and the Development of Perturbations Generated by a Point Source Inside It

The flow stability in a boundary layer with an inhomogeneous spanwise-periodic velocity profile modeling the streaky structure that develops at a high level of turbulence of the incident flow is analyzed in the three-dimensional formulation for perturbations with an arbitrary transverse period. It is shown that in the presence of inhomogeneity the dispersion relation for the Tollmien-Schlichting waves is split into two branches periodic in the transverse wave number, which correspond to symmetric and antisymmetric modes. The solution for the packet of inhomogeneous-flow modes generated by localized time-periodic fluid injection/ejection is found. The shape of this packet corresponds qualitatively to the shape of the Tollmien-Schlichting wave packet, but the fine perturbation structure inside it is sharply different.     

Reflection and transmission of elastic waves by a periodic array of cracks: Oblique incidence

The interaction of elastic waves with a planar array of periodically spaced cracks of equal length is investigated. The angle of incidence is arbitrary. By the use of Fourier series techniques, the mixed-boundary value problem for a typical strip is reduced to a singular integral equation, which is solved numerically. It is shown that the reflected and transmitted displacement fields are the superposition of an infinite number of homogeneous and inhomogeneous plane body-wave modes. The reflection and transmission coefficients, which correspond to the modes of order zero, are plotted versus the frequency for three angles of incidence. Sharp resonance and antiresonance effects are observed. A check of the accuracy of the computations is provided by the balance of rates of energies.     

Qualitative study of cavitated bifurcation for a class of incompressible generalized neo-Hookean spheres

IntroductionIn 1 958,GentandLindleyobservedthephenomenonofsuddenvoidnucleationinsolidsexperimentallyintensioningahomogenousclose_grainedvulcanizedrubbercylinderforthefirsttime.ButthemathematicalmodelonvoidnucleationandgrowthhasnotbeendescribedasabifurcationproblembasedonthetheoryofnonlinearelasticmechanicsbyBall[1]until1 982 .Inrecentyears,manyinvestigationshavebeenmadeonthisaspect.Theproblemofcavitatedbifurcationforincompressibleisotropichyperelasticmaterialswithpower_lawtypehasbeeninvestig…     

非均匀介质散射问题的体积分方程数值解法

将非均匀介质视为某一均匀背景介质中的扰动，可建立用均匀背景介质格林函数作基本解的体积分方程．给出了配置法求解体积分方程的数值方法，首先解得扰动域内各点以速度扰动为权的波场函数，然后回代计算得到观测面上各接收点的散射波场．与边界元法和Ｂｏｒｎ近似法计算结果比较表明该方法具有很高的精度，可得到穿过非     

Effect of electrohydrodynamic interaction on the stability of a flat plate laminar boundary layer

The stability of a unipolarly charged electrohydrodynamic boundary layer on a flat dielectric plate along which an electric current flows between electrodes located on the plate is investigated within the framework of the linear theory. The solution of the steady-state problem is obtained on the basis of methods developed earlier for conditions typical of aerodynamical experiments and various electric currents and electrode voltages. The effect of the interaction between perturbations of the electric and hydrodynamic flow parameters on the flow stability is estimated within the framework of the locally homogeneous approximation. This effect turns out to be insignificant under the conditions considered. It is shown that steady-state electrohydrodynamic action on the main flow makes it possible to obtain “accelerating” velocity profiles with increased absolute values of the second derivative in the transverse direction. This ensures a significant increase in the critical Reynolds numbers of loss of stability and a narrowing of the growing perturbation wavenumber range.     

Computational study of the axial instability of rimming flow using Arnoldi method

Axial instability of rimming flow has been investigated by solving a linear generalized eigenvalue problem that governs the evolution of perturbations of two‐dimensional base flow. Using the Galerkin finite element method, full Navier–Stokes equations were solved to calculate base flow and this base flow was perturbed with three‐dimensional disturbances. The generalized eigenproblem formulated from these equations was solved by the implicitly restarted Arnoldi method using shift‐invert technique. This study presents instability curves to identify the critical wavelength of the neutral mode and the critical β, which measures the importance of gravity relative to viscosity. The axial instability of rimming flow is examined and three‐dimensional flow was reconstructed by using eigenvector and growth rate at a critical wave number. The critical β value in the axial instability analysis was observed to be comparable to the onset β value of the transition between the bump and the homogeneous film state in 2‐D base flow calculations. Copyright © 2006 John Wiley & Sons, Ltd.     

Homogenization of the equations of the Cosserat theory of elasticity of inhomogeneous bodies

The paper deals with the homogenization of a boundary value problem for an inhomogeneous body with Cosserat properties, which is referred to as the original problem. The homogenization process is understood as a method for representing the solution of the original problem in terms of the solution of precisely the same problem for a body with homogeneous properties. The problem for a body with homogeneous properties is called the accompanying problem, and the body itself, the accompanying homogeneous body. As a rule, a constructive homogenization procedure includes the following three stages: at the first stage, the properties of the inhomogeneous body are used to find the properties of the accompanying homogeneous body (efficient properties); at the second stage, the boundary value problem is solved for the accompanying body; at the third stage, the solution of the accompanying problem is used to find the solution of the original problem. This approach was implemented in mechanics of composite materials constructed of numerous representative elements. A significant contribution to the development of mechanics of composites is due to Rabotnov [1–3] and his students. Recently, the homogenization method has been widely used to solve problems for composites of regular structure by expanding the solution of the original problem in a power series in a small geometric parameter equal to the ratio of the characteristic dimension of the periodicity cell to the characteristic dimension of the entire body. The papers by Bakhvalov [4–6] and Pobedrya [7] were the first in the field. At present, there are numerous monographs partially or completely dealing with the method of a small geometric parameter [8–14]. Isolated problems for inhomogeneous bodies with nonperiodic dependence of their properties on the coordinates were considered by many authors. Most of such papers published before 1973 are collected in two vast bibliographic indices [15, 16]. General methods were considered, and many specific problems of the theory of elasticity of continuously inhomogeneous bodies were solved in Lomakin’s papers and his monograph [17]. The theory of torsion of inhomogeneous anisotropic rods was considered in [18]. In 1991, in his Doctoral dissertation, one of the authors of this paper proposed a version of the homogenization method based on an integral formula representing the solution of the original static problem of inhomogeneous elasticity via the solution of the accompanying problem [19, 20]. An integral formula for the dynamic problem of elasticity was published somewhat later [21]. This integral formula was used to develop a constructive method for the homogenization of the dynamic problem of inhomogeneous elasticity, which can be used in the case of both periodic and nonperiodic inhomogeneity of the properties [22]. The integral formula in the case of the Cosserat theory of elasticity was published in [23]. The present paper briefly presents constructive methods for homogenizing the problems of the Cosserat theory of elasticity based on the integral formula.     

The parameter perturbation method on elastic wave equation in inhomogeneous medium

I.IntroductionTheelasticwaveininhomogeneousmediumiscomplicatedbecauseofthediffracting,scattering,andtransmutingofthewavetapes.Exceptforsomesimpleandregularmediummode1s,thesolutionofelasticwavehasnotbeengotyet.Nowadays,theresearchoftheelasticwavescattering…     

Energy analysis of development of kinematic perturbations in weakly inhomogeneous viscous fluids

The deformation stability relative to small perturbations is analyzed for weakly inhomogeneous viscous media on the assumption that both the main flow and perturbation field are three-dimensional. To test the damping or growth of initial perturbations, sufficient estimates based on the use of variational inequalities in different function spaces (energy estimates) are obtained. The choice of function space determines the measures of the parameter deviations, which may be different for the initial and current parameters. The unperturbed process chosen is a fairly arbitrary unsteady flow of homogeneous incompressible viscous fluid in a three-dimensional region of Eulerian space. At the initial instant, not only the kinematics of the motion but also the density and viscosity of the fluid are disturbed and the medium is therefore called weakly inhomogeneous. On the basis of the integral relation methods developed in recent years, sufficient integral estimates are obtained for lack of perturbation growth in the mean-square sense (in theL ₂ space measure). The rate of growth or damping of the kinematic perturbations depends linearly on the initial variations of the kinematics, density and viscosity. Illustrations of the general result are given. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 56–67, March–April, 2000. The work was supported by the Russian Foundation for Basic Research (projects No. 99-01-00125 and No. 99-01-00250) and by the Federal Special “Integration” Program (project No. 426).     

Wave Regimes of Viscous Film Flow down a Vertical Cylinder

The flow of a viscous liquid film down a vertical cylinder in the gravity field is considered. In the case of small Reynolds numbers for long-wave perturbations on a cylinder of radius much greater than the film thickness, the problem can be reduced to a single nonlinear equation for the evolution of the film thickness perturbation. For axially symmetric solutions, this equation coincides with the well-known Sivashinsky-Kuramoto equation. The results of a numerical analysis of this equation for three-dimensional stationary traveling solutions of the problem are reported. The effect of the problem parameters on the solution behavior is demonstrated. Soliton type solutions are presented.     

Evolution of the interface in a stratified anisotropic porous material

It is shown that the boundary-value problem describing the evolution of the interface during impregnation of a stratified inhomogeneous anisotropic porous material with a viscous fluid can be reduced to a similar problem for a stratified inhomogeneous isotropic material by nonorthogonal transformation of the coordinates. As a result, the well-known estimates of the problem parameters determining the interface configuration for impregnation of an isotropic material can be extended to the anisotropic case.     

Existence of Standing Pulse Solutions to an Inhomogeneous Reaction–Diffusion System

We prove the existence of locally unique, symmetric standing pulse solutions to homogeneous and inhomogeneous versions of a certain reaction–diffusion system. This system models the evolution of photoexcited carrier density and temperature inside the cavity of a semiconductor Fabry–Pérot interferometer. Such pulses represent the fundamental nontrivial mode of pattern formation in this device. Our results follow from a geometric singular perturbation approach, based largely on Fenichel's theorems and the Exchange Lemma.