Nonlinear Viscous Flows with a Free Surface

An asymptotic solution of the problem of time evolution of a periodic wave on the surface of a viscous, infinitely deep fluid in the approximation quadratic in the wave amplitude is proposed.     

Nonlinear Capillary-Gravity Waves on the Charged Surface of an Ideal Fluid

A second-order asymptotic expression for the profile of a capillary-gravity wave traveling over the charged surface of an ideal incompressible fluid is calculated analytically. Two types of steady-state profiles of nonlinear periodic capillary-gravity waves are found. For a certain fixed dimensionless surface charge the shape of the tops of the nonlinear waves changes: from blunt to pointed for short waves and from pointed to blunt for long waves.     

Evolution of long nonlinear waves on the interface of a stratified viscous fluid flow in a channel

The dynamics of disturbances of the interface between two layers of incompressible immiscible fluids of different densities in the presence of a steady flow between the horizontal bottom and lid is studied analytically and numerically. A model integrodifferential equation is derived, which takes into account long-wave contributions of inertial layers and surface tension of the fluids, small but finite amplitude of disturbances, and unsteady shear stresses on all boundaries. Numerical solutions of this equation are given for the most typical nonlinear problems of transformation of both plane waves of different lengths and solitary waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 49–61, July–August, 2007.     

Spatial Motion of a Rod in a Viscous Fluid Flow

Equations of spatial motion of a curved finitelength rod in a viscous fluid flow are derived. Analytical solutions of problems on the motion of a straight rod under conditions of pure shear, simple shear, and uniaxial extension of the fluid are obtained. Longitudinal stability of the straight rod during its spatial motion is considered. Effective viscosity of a suspension filled by rigid straight rods is evaluated.     

Generation of Periodic Motions by a Disk Performing Torsional Oscillations in a Viscous, Continuously Stratified Fluid

The solution of the problem of nonlinear generation of periodic internal waves by a boundary flow on a vertical cylinder or a horizontal disk performing torsional oscillations in an exponentially stratified fluid is constructed. The calculations are in satisfactory agreement with the results of experiments in which both horizontal and inclined disks of various diameters and a model propeller performing periodic torsional oscillations, including oscillations against a background of uniform rotation, are used as perturbation sources. The experiments were carried out over a wide range of parameters including the laminar, transition, and turbulent flow regimes. The limits of applicability of the proposed analytic theory of wave radiation are determined.     

Control Problems for the Steady-State Equations of Magnetohydrodynamics of a Viscous Incompressible Fluid

Control problems for a steady-state model of the magnetohydrodynamics of a viscous incompressible fluid in a bounded domain with an impermeable, perfectly conducting boundary are formulated. The resolvability of the problems is studied, the use of the Lagrange principle is justified, and optimality systems are analyzed.     

Flow Asymptotics of a Viscous Compressible Fluid with Discontinuous Initial Data

Asymptotics of a continuous solution to a plane problem on the motion of a viscous incompressible fluid with discontinuous initial velocity and pressure fields is studied by the boundarylayer method with simultaneous stretching of space and time coordinates.     

Nonlinear Waves and the Origin of Bubbles in Fluidized Beds

Early experiments in the mid-1940s established two different regimes of behavior of fluidized systems. These are broadly classified into systems that exhibit massive phase segregation, leading to particle-free regions called bubbles, and those that do not. Explaining the origin of bubbles and of these two regimes has represented both a technological and scientific challenge since then. The late 1960s through the 1970s saw a series of illuminating experiments that established many features of the flow regimes and their characteristics through both flow visualizations and quantitative measurements. Recent numerical and theoretical work has come close the resolving the problem. This paper represents the written version of the talk given at the Symposium in honor of Leen van Wijngaarden's retirement. In it, I review the history of progress on the problem in two giant 25-year steps.     

Flow of a Nonlinearly Viscous Fluid over the Surface of a Rotating Flat Disk

The flow of a nonlinearly viscous (power-law) fluid over the surface of a rotating flat disk is investigated. A solution form which makes it possible to reduce the complete system of partial differential equations to a system of ordinary differential equations is found. This system is integrated using the Runge-Kutta method and reduction to a Cauchy problem on the basis of Newton's method. The velocity and pressure fields in a power-law fluid film flowing over the surface of a rotating flat disk are found numerically.     

Steady Film Flow of a Highly Viscous Heavy Fluid with Mass Supply

Asymptotic models of a thin layer of highly viscous heavy incompressible Newtonian fluid are constructed for steady axisymmetric (plane) flow on a curved rigid surface with distributed or point mass supply on a surface section near the axis (plane) of symmetry. Examples of analytical and numerical investigations of the free-surface shape and hydrodynamic-parameter fields are given. The models constructed are generalized for the case of a viscoplastic fluid and solutions which can be used for describing extrusive volcanic eruptions are obtained.     

Bifurcation of a Main Steady-State Viscous Fluid Flow in a Plane Divergent Channel

The evolution of steady-state viscous incompressible fluid flows in a plane divergent channel is investigated. For the classical formulation of the Jeffery-Hamel problem a complete solution is given as a function of the determining parameters. For a fixed angle of divergence the behavior of the main unimodal flow is determined as a function of the Reynolds number. Critical values at which the flow pattern bifurcates and the steady-state unimodal flow ceases to exist are found. The mechanism of bifurcation is established and its diagram is constructed. This mechanism and the diagram were not previously known in the scientific literature in connection with the investigation of the Jeffery-Hamel problem. The critical Reynolds number at which bifurcation occurs is given as a function of the channel divergence angle. The results may be of interest for hydromechanical, technological, and geophysical applications.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 25–36.Original Russian Text Copyright © 2005 by Akulenko and Kumakshev.     

Nonlinear traveling waves in a compressible Mooney-Rivlin rod I. Long finite-amplitude waves

In literature, nonlinear traveling waves in elastic circular rods have only been studied based on single partial differential equation (pde) models, and here we consider such a problem by using a more accurate coupled-pde model. We derive the Hamiltonian from the model equations for the long finite-amplitude wave approximation, analyze how the number of singular points of the system changes with the parameters, and study the features of these singular points qualitatively. Various physically acceptable nonlinear traveling waves are also discussed, and corresponding examples are given. In particular, we find that certain waves, which cannot be counted by the single-equation model, can arise. The project supported by the Research Grants Council of the HKSAR, China (City U 1107/99P) and the National Natural Science Foundation of China (10372054 and 10171061)     

Investigation of the Relative Motion of a Viscous Fluid and a Porous Medium Using the Brinkman Equations

Exact solutions are obtained for the following three problems in which the Brinkman filtration equations are used: laminar fluid flow between parallel plane walls, one of which is rigid while the other is a plane layer of saturated porous medium, motion of a plane porous layer between parallel layers of viscous fluid, and laminar fluid flow in a cylindrical channel bounded by an annular porous layer.     

黏滞阻尼器在超高层结构体系中的效率研究

250m以上的超高层结构主要有框架-核心筒、框筒-核心筒和巨型框架-核心筒等三种结构体系．由于超高层结构的高宽比通常较大（5以上）,结构的侧向变形往往成为控制因素．随着消能减震技术的不断成熟,黏滞阻尼器越来越多的应用于建筑结构领域．本文分别将黏滞阻尼器均匀布置在上述三种超高层结构体系中,对比其消能减震效果并分析原因,最后建议了黏滞阻尼器在这些结构中的较优布置．     

Effect of the Hydrodynamic Interaction of a Large Number of Particles on Their Sedimentation Rate in a Viscous Fluid

The gravity-induced sedimentation of a large number of identical spherical particles in a viscous fluid is considered. The hydrodynamic interaction of all the particles is taken into account. The problem is modeled numerically for random locations of 1–100 particles forming different configurations. A dependence of the mean sedimentation rate on the particle number and concentration in the cloud is obtained and compared with previous results.     

Resolution of a Paradox Involving Viscous Dissipation and Nonlinear Drag in a Porous Medium

The modelling of viscous dissipation in a porous medium saturated by an incompressible fluid is discussed, for the case of Darcy, Forchheimer and Brinkman models. An apparent paradox relating to the effect of inertial effects on viscous dissipation is resolved, and some wider aspects of resistance to flow (concerning quadratic drag and cubic drag) in a porous medium are discussed. Criteria are given for the importance or otherwise of viscous dissipation in various situations.     

Motion of a Rod in a Viscous Flow

The equations of the dynamics of a finitelength curved rod in a viscous flow are derived. The longitudinal stability of the rod against small deflections from a rectilinear form is studied for two types of flow (pure and simple shear). The minimum flexural rigidity of the rod that ensures rod stability for any orientation in the flow is found. The effective viscosity of a suspension filled with rectilinear discrete fibers is estimated.     

Solitary Waves in Nonlinear Beam Equations: Stability,Fission and Fusion

We continue work by the second author and co-workers onsolitary wave solutions of nonlinear beam equations and their stabilityand interaction properties. The equations are partial differentialequations that are fourth-order in space and second-order in time.First, we highlight similarities between the intricate structure ofsolitary wave solutions for two different nonlinearities; apiecewise-linear term versus an exponential approximation to thisnonlinearity which was shown in earlier work to possess remarkablystable solitary waves. Second, we compare two different numericalmethods for solving the time dependent problem. One uses a fixed griddiscretization and the other a moving mesh method. We use these methodsto shed light on the nonlinear dynamics of the solitary waves. Earlywork has reported how even quite complex solitary waves appear stable,and that stable waves appear to interact like solitons. Here we show twofurther effects. The first effect is that large complex waves can, as aresult of roundoff error, spontaneously decompose into two simplerwaves, a process we call fission. The second is the fusion of twostable waves into another plus a small amount of radiation.     

Exact Analytic Solutions of the Nonlinear Long-Wave Equations in the Case of Axisymmetric Fluid Vibrations in a Parabolic Rotating Vessel

A class of exact analytic solutions of the system of nonlinear long-wave equations is found. This class corresponds to the axisymmetric vibrations of an ideal incompressible homogeneous fluid in a rotating vessel in the shape of a paraboloid of revolution. The radial velocity of these motions is a linear function, and the azimuthal velocity and free surface displacements are polynomials in the radial coordinate with time-dependent coefficients. The nonlinear vibration frequency is equal to the frequency of the lowest mode of linear axisymmetric standing waves in the parabolic vessel.