Investigation of the interaction between a viscous incompressible fluid layer and walls of a channel formed by coaxial vibrating discs

The interaction between a viscous incompressible fluid layer and walls of a channel formed by two concentric discs moving perpendicularly to their planes due to vibration of the base on which the channel is mounted is investigated. The case of two absolutely rigid discs with elastic suspension and the case in which one of the discs is an elastic plate with the rigid restrain on the edges are considered. The velocity and pressure distributions over the fluid and the laws of motion of the walls and their frequency characteristics which make it possible to determine the resonance vibration frequencies of the mechanical system considered are found.     

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.     

The Stability of a Compressible Rayleigh Layer

The aim of this work is to determine the linear stability of a compressible Rayleigh layer and to ascertain what role unsteady effects play. A Rayleigh layer is formed when an infinite flat plate is impulsively set in motion in its own plane with constant velocity beneath an initially quiescent fluid. When the fluid is compressible there is a motion both parallel and normal to the plate. The classical boundary-layer scaling is employed to determine solutions which are expressed in terms of a similarity variable and are valid for a large range of Mach, Prandtl and Reynolds numbers. Solutions are presented for both an adiabatic and iso-thermal temperature boundary condition at the plate. The temporal stability of the flow is considered by solving an Orr–Sommerfeld system: here the underlying flow is calculated at a certain time and the instantaneous stability to viscous travelling waves is determined. The stability is seen to be altered by changing the Mach number (an increase of which decreases the stability of the flow), and also by cooling and heating the wall. These results are limited by the fact that the growth of the layer in time is not taken into account. To include this we consider the large Reynolds number limit and use a triple-deck structure to determine the modes characteristics. The triple-deck approach is used to determine an asymptote to the lower branch of the neutral curve and unsteady effects can be included in a consistent manner. For the upper branch, however, a five-deck structure is required due to the fact that the critical layer is now distinct from the viscous sublayer. The upper-branch stability is only calculated to the first order which is sufficient to give an insight into the stability characteristics.     

Laminated piezoelectric plate for active underwater acoustic control

The pressure reflected from a bi-laminated piezoelectric plate has been determined using the Thomson-Haskell matrix method. The plate is composed of a piezoelectric layer with grounded vacuum and an elastic layer in contact with the fluid. An incident plane wave in the fluid medium strikes the plate at different angles. The required electric potential across the piezoelectric layer to cancel the reflection from the fluid/elastic boundary has been determined for the piezoelectric material PZT-5 at various thicknes parameters and incident frequencies. Project supported by the National Natural Science Foundation of China (No. 10172039).     

Oscillatory Motion of a Viscous Fluid in Contact with a Flat Layer of a Porous Medium

Analytical solutions are obtained for two problems of transverse internal waves in a viscous fluid contacting with a flat layer of a fixed porous medium. In the first problem, the waves are considered which are caused by the motion of an infinite flat plate located on the fluid surface and performing harmonic oscillations in its plane. In the second problem, the waves are caused by periodic shear stresses applied to the free surface of the fluid. To describe the fluid motion in the porous medium, the unsteady Brinkman equation is used, and the motion of the fluid outside the porous medium is described by the Navier–Stokes equation. Examples of numerical calculations of the fluid velocity and filtration velocity profiles are presented. The existence of fluid layers with counter-directed velocities is revealed.     

Finite Difference Method for the Acoustic Radiation of an Elastic Plate Excited by a Turbulent Boundary Layer: A Spectral Domain Solution

A finite difference method is developed to study, on a two-dimensional model, the acoustic pressure radiated when a thin elastic plate, clamped at its boundaries, is excited by a turbulent boundary layer. Consider a homogeneous thin elastic plate clamped at its boundaries and extended to infinity by a plane, perfectly rigid, baffle. This plate closes a rectangular cavity. Both the cavity and the outside domain contain a perfect fluid. The fluid in the cavity is at rest. The fluid in the outside domain moves in the direction parallel to the system plate/baffle with a constant speed. A turbulent boundary layer develops at the interface baffle/plate. The wall pressure fluctuations in this boundary layer generates a vibration of the plate and an acoustic radiation in the two fluid domains. Modeling the wall pressure fluctuations spectrum in a turbulent boundary layer developed over a vibrating surface is a very complex and unresolved task. Ducan and Sirkis [1] proposed a model for the two-way interactions between a membrane and a turbulent flow of fluid. The excitation of the membrane is modeled by a potential flow randomly perturbed. This potential flow is modified by the displacement of the membrane. Howe [2] proposed a model for the turbulent wall pressure fluctuations power spectrum over an elastomeric material. The model presented in this article is based on a hypothesis of one-way interaction between the flow and the structure: the flow generates wall pressure fluctuations which are at the origin of the vibration of the plate, but the vibration of the plate does not modify the characteristics of the flow. A finite difference scheme that incorporates the vibration of the plate and the acoustic pressure inside the fluid cavity has been developed and coupled with a boundary element method that ensures the outside domain coupling. In this paper, we focus on the resolution of the coupled vibration/interior acoustic problem. We compare the results obtained with three numerical methods: (a) a finite difference representation for both the plate displacement and the acoustic pressure inside the cavity; (b) a coupled method involving a finite difference representation for the displacement of the plate and a boundary element method for the interior acoustic pressure; (c) a boundary element method for both the vibration of the plate and the interior acoustic pressure. A comparison of the numerical results obtained with two models of turbulent wall pressure fluctuations spectrums - the Corcos model [3] and the Chase model [4] - is proposed. A difference of 20 dB is found in the vibro-acoustic response of the structure. In [3], this difference is explained by calculating a wavenumber transfer function of the plate. In [6], coupled beam-cavity modes for similar geometry are calculated by the finite difference method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Hydroelastic interaction between water waves and thin elastic plate floating on three-layer fluid

The wave-induced hydroelastic responses of a thin elastic plate floating on a three-layer fluid, under the assumption of linear potential flow, are investigated for two-dimensional cases. The effect of the lateral stretching or compressive stress is taken into account for plates of either semi-infinite or finite length. An explicit expression for the dispersion relation of the flexural-gravity wave in a three-layer fluid is analytically deduced. The equations for the velocity potential and the wave elevations are solved with the method of matched eigenfunction expansions. To simplify the calculation on the unknown expansion coefficients, a new inner product with orthogonality is proposed for the three-layer fluid, in which the vertical eigenfunctions in the open-water region are involved. The accuracy of the numerical results is checked with an energy conservation equation, representing the energy flux relation among three incident wave modes and the elastic plate. The effects of the lateral stresses on the hydroelastic responses are discussed in detail.     

On the stability of a plate under longitudinal compression on a two-layer half-space with lower layer prestressed by gravity forces

In the plane (plane strain) and axially symmetric statements, we study the problem of stability, under the action of longitudinal compressing forces, of an infinite elastic plate in two-sided contact with an elastic half-space. The upper layer of finite depth is described by the usual equations of linear theory of elasticity; the lower layer, which is geometrically nonlinear, incompressible, and infinite in depth, is prestressed by gravity forces. The total adhesion between the layer of finite depth and the lower half-space is realized. It is also assumed that the same adhesion takes place between the upper layer of the half-space and the plate with the contact tangential stresses taken into account.The results can be used to calculate the working capacity of coated bodies and layered composites and in problems of geophysics.The problem of stability of an infinite elastic plate under longitudinal compression under conditions of two-sided contact with an elastic base was studied earlier in the monograph [1] (Fuss-Winkler base) and in [2–4].     

Experimental studies of interfacial instabilities in multilayer pressure-driven flow of polymeric melts

The interfacial deformation and stability of two-(A-B) as well as three-layer symmetric (A-B-A) and asymmetric (A-B-C) pressure-driven flow of viscoelastic fluids has been investigated. Flow visualization in conjunction with digital image processing has been used to observe and measure the rate of encapsulation and interfacial stability/instability of the flow. Specifically, the encapsulation behavior as well as stability/instability of the interface and the corresponding growth or decay rate of disturbances as a function of various important parameters, namely, number of layers and their arrangement, layer depth ratio, viscosity and elasticity ratio as well as disturbance frequency, have been investigated. Based on these experiments, we have shown that the encapsulation phenomena occurs irrespective of the stability/instability of the interface and in cases when both encapsulation and instability occur simultaneously their coupling leads to highly complex and three-dimensional interfacial wave patterns. Moreover, it has been shown that the simple notion that less viscous fluids encapsulate more viscous fluids is incorrect and depending on the wetting properties of the fluid as well as their first and second normal stresses the reverse could occur. Additionally, in two- and three-layer flows it has been shown that by placing a thin, less viscous layer adjacent to the wall longwave disturbances can be stabilized while short and intermediate wavelength disturbances are stabilized when the more elastic fluid is the majority component. Furthermore, in three-layer flows it has been demonstrated that in the linear instability regime no dynamic interaction between the two interfaces is possible for short and intermediate wavenumber disturbances. However, in the nonlinear stability regime dynamic interactions between interfaces have been observed in this range of disturbance wavenumbers leading to highly chaotic flows. Finally, in the parameter space of this study no subcritical bifurcations were observed while supercritical bifurcations resulting in waves with a pointed front and a gradual tail were observed.     

Numerical solution of the problem of the mixing of the boundary layers shed from the trailing edge of a wing

During the mixing of viscous incompressible flows with different velocities, in the vicinity of a trailing edge an interaction region with a three-layer structure is formed, similar to that in the case of symmetric shedding with equal velocities. The boundary layers developing on the upper and lower sides of the airfoil form a viscous mixing layer, or vortex sheet, which separates the flows downstream of the trailing edge. The boundary value problem corresponding to the flow in the viscous sublayer in the vicinity of the trailing edge of a flat plate is solved for high Reynolds numbers using an efficient numerical method for solving the equations of asymptotic interaction theory.     

Oscillations of a Circular Cylinder in a Linearly Stratified Fluid

The plane problem of the motions of a three-layer fluid initiated by the oscillations of a circular cylinder is solved in the linear formulation in the Boussinesq approximation. The cylinder is completely immersed in the linearly stratified middle layer, and the upper and lower layers are homogeneous and bounded by rigid horizontal walls. The fluid is assumed to be ideal and incompressible. The added mass and damping coefficients are calculated as functions of the oscillation frequency of the cylinder and the layer thicknesses.     

On a Fluid-Structure Model in Which the Dynamics of the Structure Involves the Shear Stress Due to the Fluid

In this paper we consider a model for fluid-structure interaction. The hybrid system describes the interaction between an incompressible fluid in a three-dimensional container with interior a fixed domain and a thin elastic plate, the interface, which coincides with a flexible flat part of the surface of the vessel containing the fluid. The motion of the fluid is described by the linearized Navier–Stokes equations and the deformation of the plate by the classical plate equations for in-plane motions, modified to include the viscous shear stress which the fluid exerts on the plate as well as damping of Kelvin–Voigt type. We establish the existence of a unique weak solution of the interactive system of partial differential equations by considering an appropriate variational formulation. Uniform stability of the energy associated with the model is shown under the assumption that the potential plate energy is dominated by the dissipation induced by the viscosity of the fluid. The retention of the physical parameters in the problem is an a priori requirement in this physical condition.      

Nonorthogonal stagnation-point flow over a lubricated surface

An attempt is made to study a steady two-dimensional flow of a viscous incompressible fluid incident at some angle onto a plate lubricated with a thin layer of a power-law fluid. Similar and nonsimilar solutions of the governing partial differential equations are obtained numerically by imposing the continuity of velocity and shear stress at the interface layer between the fluid and the lubricant. The Keller box method is applied to obtain the solutions. The limiting cases for full and no-slip conditions are compared.     

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.     

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.     

Viscoelastic shear horizontal wave in graded and layered plates

In this paper, viscoelastic shear horizontal (SH) wave propagation in functionally graded material (FGM) plates and laminated plates are investigated. The controlling differential equation in terms of displacements is deduced based on the Kelvin–Voigt viscoelastic theory. The SH wave characteristics is controlled by two elastic constants and their corresponding viscous coefficients. By the Legendre polynomial series method, the asymptotic solutions are obtained. In order to verify the validity of the method, a homogeneous plate is calculated to make a comparison with available data. Through three different graded plates, the influences of gradient shapes on dispersion and attenuation are discussed. The viscous effects on the displacement and stress shapes are illustrated. The different boundary conditions are analyzed. The influential factors of the viscous effect are analyzed. Finally, two multilayered (two layer and five layer) viscoelastic plates that are composed of the same material volume fraction are calculated to show their differences from the graded plate.     

Streamwise vibrations of a plate in a viscous fluid flow in a channel,induced by forced transverse vibrations of the plate

Aeroelastic vibrations of a plate aligned at a zero angle of attack in a viscous incompressible fluid flow in a channel with parallel walls are considered within the framework of a plane model. Forced vibrations of the plate in the transverse direction give rise an unsteady component of the flow friction force, induced by the perturbation of the fluid flow velocity by the vibrating plate. Under the assumption of the laminar character of the fluid flow, it is demonstrated that this force can excite streamwise vibrations of the plate if the channel width is small as compared with the plate length; these streamwise vibrations have the same order as the transverse vibrations of the plate excited by external forces.     

Propagation of waves in a suspension of solid particles

A theory is developed for the propagation of waves in a suspension of elastic or rigid solid particles in a viscous or inviscid compressible fluid, using a homogenization process. We study the case where the characteristic length of the particles is small compared with the wave length. In the case of a viscous fluid, a law similar to Darcy's law for the average velocity of the suspension is established, and in the case of macroscopic homogeneity and isotropy, the propagation of a plane wave displays one dilatational, damped and dispersive wave. In the case of a barotropic inviscid fluid, the average acceleration of the suspension depends, in a linear way, on the mean pressure gradient and in the case of macroscopic homogeneity and isotropy, the propagation of a plane wave displays one dilatational, undamped and non dispersive wave.     

Influence of fluid viscosity on waves in an initially deformed,compressible, elastic layer interacting with a fluid medium

The propagation of harmonic waves along the interface of an initially stressed, compressible layer and a viscous, compressible fluid half-space is investigated. A dispersion relation that does not depend on the form of the elastic potential is derived on the basis of the three-dimensional linearized elasticity equations for elastic bodies with uniform initial deformations and on the linearized Navier-Stokes equations for a viscous Newtonian fluid at rest. The phase velocities and attenuation coefficients of the elastic modes are determined numerically as functions of the thickness of the elastic layer using a Murnaghan-type three-invariant elastic potential.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 9, pp. 3–9, September, 1994.     

Stability of a fluid interface under tangential vibrations

The stability of the interface between two immiscible fluids of different density which occupy a plane horizontal layer performing harmonic horizontal oscillations is considered. Within the framework of the ideal fluid model a transformation reducing the problem of small plane perturbations to the Mathieu equation is found. Resonance instability domains associated with the formation of capillary-gravitational waves are investigated. A model which takes into account dissipation processes due to the presence of viscous friction is constructed. The role of the viscous dissipation in suppressing resonance instability is discussed. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–31, May–June, 1998. The work was carried out with partial support from the Russian Foundation for Basic Research (project No. 95-01-00386).