Viscous Fluid Medium Research Articles

Axisymmetric vibrations of a viscous-fluid-filled piezoelectric spherical shell and the associated radiation of sound

Axisymmetric vibrations of a viscous-fluid-filled piezoelectric sphere, with radial polarization, submerged in a compressible viscous fluid medium are investigated. The oscillations are harmonically driven by an axisymmetrically applied electric potential difference across the surface of the shell. A theoretical formulation cast the piezoelectric shell problem into a corresponding problem of an elastic shell with the contribution of piezoelectricity confined to slightly modified in vacuum natural frequencies and their associated mode shapes. It is noted that the fluid inside the shell will have a dominating influence on the vibrational characteristics of the submerged shell. The circular components of the natural frequency spectra closely follow those of the fluid-filled shell in vacuo. Furthermore, the corresponding damping components of those natural frequencies are rather small, making acoustic radiation and under-damped oscillation possible for an infinite number of natural frequencies. The characteristics of natural frequencies are elucidated using a fluid-filled polyvinglindene fluoride (PVDF) shell submerged in both air and water as an example. It is found that the piezoelectric parameters that contribute to the shell's natural frequencies is of a small order for thin PVDF shells, and is thereby negligible. It is noted that, with the mechanical constant typically associated with piezoelectric materials, fluid viscosity could have a significant effect on some vibrations. In certain cases, a natural frequency associated with a minimum viscous damping and a maximum of total damping (indicating highly efficient acoustic radiation) is possible with such a frequency. The vibrational characteristics, fluid loading, and energy flow are evaluated for a fluid-filled PVDF shell submerged in air and water. The inclusion of fluid inside the shell is shown to produce various narrow band peaks responses, vibrational absorbing frequencies, and non-dissipating frequencies. Those vibrational characteristics could have many potential applications. For example, the interior fluid could offer the option of generating a desired narrow band near resonant sound radiation while keeping power dissipation due to fluid viscosity to a minimum. Those well-defined narrow band characteristics also open up possibilities of using a vibrating, fluid-filled shell as a micro scale sensor for sensing and detection applications.

Viscosity and Diffusion Effects at the Boundary Surface of Viscous Fluid and Thermoelastic Diffusive Solid Medium

AbstractThis paper concentrates on the wave motion at the interface of viscous compressible fluid half-space and homogeneous isotropic, generalized thermoelastic diffusive half-space. The wave solutions in both the fluid and thermoelastic diffusive half-spaces have been investigated; and the complex dispersion equation of leaky Rayleigh wave motion have been derived. The phase velocity and attenuation coefficient of leaky Rayleigh waves have been computed from the complex dispersion equation by using the Muller’s method. The amplitudes of displacements, temperature change and concentration have been obtained. The effects of viscosity and diffusion on phase velocity and attenuation coefficient of leaky Rayleigh waves motion for different theories of thermoelastic diffusion have been depicted graphically. The magnitude of heat and mass diffusion flux vectors for different theories of thermoelastic diffusion have also been computed and represented graphically.

Axisymmetric Vibrations of a Piezoelectric Spherical Shell Submerged in a Compressible Viscous Fluid Medium

Abstract Axisymmetric vibrations of a hollow piezoelectric sphere submerged in a compressible viscous fluid medium are investigated. The piezoelectric sphere is radially polarized. The differential equations governing the shell motion are obtained by the use of Hamilton’s principle. Based on the classical bending theory of shells, it is shown that all the piezoelectric contributions can be included in the in vacuo natural frequencies and their corresponding mode shapes. As such, the previous theory on elastic shell vibration becomes readily extendable. The flow field, determined by the boundary layer theory, is coupled to the shell motion through no-slip and no-penetrating conditions. It is found that the contribution of the piezoelectric parameters in the thin shell’s free vibration is of small order and is negligible. Natural frequencies and their associated vibration characteristics are numerically obtained and presented for a Polyvinglindene fluoride (PVDF) shell submerged in water. Dynamic responses of a submerged piezoelectric sherical shell, and the associated radiation of sound are investigated. The oscillations are harmonically driven by an axisymmetrically applied electric potential difference across the surface of the shell. The vibrational, fluid loading, and energy flow characteristics are derived and evaluated for a PVDF shell submerged in water. The essential feature of the modal response is determined by various critical frequencies, such as resonant frequencies and vibration-absorbing frequencies. Viscous effect is found noticeable in several cases.

Three-dimensional viscous finite element formulation for acoustic fluid–structure interaction

A three-dimensional viscous finite element model is presented in this paper for the analysis of the acoustic fluid–structure interaction systems including, but not limited to, the cochlear-based transducers. The model consists of a three-dimensional viscous acoustic fluid medium interacting with a two-dimensional flat structure domain. The fluid field is governed by the linearized Navier–Stokes equation with the fluid displacements and the pressure chosen as independent variables. The mixed displacement/pressure based formulation is used in the fluid field in order to alleviate the locking in the nearly incompressible fluid. The structure is modeled as a Mindlin plate with or without residual stress. The Hinton–Huang’s 9-noded Lagrangian plate element is chosen in order to be compatible with 27/4 u/p fluid elements. The results from the full 3D FEM model are in good agreement with experimental results and other FEM results including Beltman’s thin film viscoacoustic element [W.M. Beltman, P.J.M. Van der Hoogt, R.M.E.J. Spiering, H. Tijdeman, Implementation and experimental validation of a new viscothermal acoustic finite element for acousto-elastic problems, J. Sound Vib. 216 (1) (1998) 159–185] and two and half dimensional inviscid elements [A.A. Parthasarathi, K. Grosh, A.L. Nuttall, Three-dimensional numerical modeling for global cochlear dynamics, J. Acoust. Soc. Am. 107 (2000) 474–485]. Although it is computationally expensive, it provides a benchmark solution for other numerical models or approximations to compare to besides experiments and it is capable of modeling any irregular geometries and material properties while other numerical models may not be applicable.

Ultrasonic scattering by a fluid cylinder of elliptic cross section, including viscous effects

This paper analyzes acoustic scattering by a viscous compressible fluid cylinder of elliptic cross section submerged in an unbounded viscous nonheat-conducting compressible fluid medium. The classical method of eigenfunction expansion along with the appropriate wave field expansions and the pertinent boundary conditions are used to develop a solution in the form of infinite series involving Mathieu and modified Mathieu functions of complex arguments. The complications arising due to the nonorthogonality of angular Mathieu functions corresponding to distinct wave numbers in addition to the problems associated with the appearance of additional angular-dependent terms in the boundary conditions are all avoided in an elegant manner by expansion of the angular Mathieu functions in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of the unknown scattering coefficients. A multiprecision code was developed for computing the Mathieu functions of complex argument in terms of complex Fourier coefficients that are themselves calculated by numerically solving appropriate sets of eigen-systems. The numerical results point to the imperative influence of fluid viscosity in notable reduction of pressure amplitudes at intermediate and high frequencies. They also reveal the central role of the cross sectional ellipticity in conjunction with the angle of incidence in altering the pressure directivities. Limiting cases are considered, and fair agreements with well-known solutions are obtained.

Ultrasonic Scattering by a Spheroidal Suspension Including Dissipative Effects

This study treats the axisymmetric interaction of a plane compressional sound wave with a viscoelastic (viscous) prolate spheroidal particle (droplet) suspended in a boundless viscous fluid medium. The method of eigen‐function expansion along with the novel features of Havriliak‐Negami model for viscoelastic material behavior and the linearized equations of Navier‐Stokes for wave motion in a viscous non‐heat‐conducting compressible fluid are employed to develop a closed form series solution involving Spheroidal harmonics of complex argument. The complications arising due to the non‐orthogonality of angular Spheroidal functions corresponding to different wave numbers as well as problems associated with the appearance of additional angular dependent terms in the boundary conditions are all avoided by expansion of the angular Spheroidal functions in terms of Lengedre functions which are themselves expanded in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of unknown scattering and transmission coefficients. The presented solution demonstrates that the acoustic characteristics of particulate suspensions are strongly influenced by spheroidicity of particle cross‐section. Limiting case involving an elastic (ideal) spheroidal particle (droplet) in an ideal fluid is considered and fair agreement with available solution is established.

Ultrasonic Scattering by a Viscoelastic Fiber of Elliptic Cross‐Section Suspended in a Viscous Fluid Medium

This study treats the two‐dimensional interaction of a plane compressional sound wave with a viscoelastic fiber of elliptic cross‐section submerged in a boundless viscous nonheat‐conducting compressible fluid medium. The classical method of eigen‐function expansion along with the novel features of Havriliak‐Negami model for viscoelastic material behavior and the pertinent boundary conditions are used to develop a closed form series solution involving Mathieu and modified Mathieu functions of complex argument. The complications that arise because of the nonorthogonality of angular Mathieu functions corresponding to different wave numbers, as well as problems associated with the appearance of additional angular dependent terms in the boundary conditions, are all avoided by expansion of the angular Mathieu functions in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of the unknown scattering coefficients. The presented solution demonstrates that acoustic characteristics of fiber suspensions are strongly influenced by cross‐sectional ellipticity of the fibers in addition to the dynamic viscoelastic properties of the fiber material. It also shows that the common rigid circular fiber approximations used to model fiber suspensions can not capture the important resonance and damping effects associated with the highly viscoelastic noncircular fibers. The proposed model is valid for a wide range of cross‐sectional geometries (aspect ratios) and incident wave frequencies where most numerical methods fail. Limiting case involving an elastic elliptic cylinder in an ideal fluid is considered and fair agreement with a recent solution is established.

Dynamic Viscoelastic and Multiple Scattering Effects in Fiber Suspensions

This study considers the most fundamental problem of 2‐D acoustic scattering in fiber suspensions. It treats the interaction of a plane compressional sound wave with a cluster of two flexible fibers submerged in a boundless viscous fluid medium. The dynamic viscoelastic properties of the fibers and the viscosity of the surrounding fluid are rigorously taken into account in the solution of the problem. The translational addition theorem for cylindrical wave functions, the Havriliak‐Negami model for viscoelastic material behaviors and the appropriate wave field expansions and the pertinent boundary conditions are employed to develop a closed‐form solution in the form of an infinite series. The prime objectives are to investigate the influence of dynamic viscoelastic properties of fiber material as well as multiple scattering interaction effects on acoustic scattering and its associated quantities. The analytical results are illustrated with a numerical example in which two identical viscoelastic fibers are insonified by a plane sound wave at broadside/end‐on incidence. The backscattering form function amplitude and the spatial distribution of the total acoustic pressure are numerically evaluated and discussed for representative values of the parameters characterizing the system. The effects of incident wave frequency, angle of incidence, material properties, and proximity of the two fibers are examined. A limiting case involving a pair of rigid cylinders in an ideal fluid is considered, and fair agreement with a well‐known solution is established.

Theoretical analysis of damping effects of guided elastic waves at solid∕fluid interfaces

A theoretical description of ideal and viscous fluid media is proposed to address the problem of modeling damping effects of surface acoustic waves and more generally of any guided elastic waves at the interface between viscous fluids and solids. It is based on the Fahmy-Adler eigenvalue representation of the elastic propagation problem, adapted to provide Green’s function of the considered media. It takes advantage of previous efforts developed to numerically stabilize Green’s-function computation process. This function is used to compute a harmonic admittance according to the Blötekjaër approach. The influence of acoustic radiation and viscosity effects on different kinds of waves excited on various substrates is reported and discussed.

Acoustic diffraction by a disk in a viscous medium

The diffraction of a time-harmonic acoustic plane wave by a circular disk in a viscous fluid medium is considered by using the linearized equations of viscous fluid flow and the no-slip condition on the rigid disk. Sets of dual integral equations for the fluid velocity and pressure are derived for an arbitrary disk radius and an arbitrary angle of incidence. The dual integral equations are solved by an analytic reduction to sets of linear algebraic equations. In the cases of normal or tangential incidence, numerical results are presented for the fluid velocity in the plane of the disk and the scattered acoustic disturbance in the far field.

Acoustic diffraction by a half-plane in a viscous fluid medium.

We consider the diffraction of a time-harmonic acoustic plane wave by a rigid half-plane in a viscous fluid medium. The linearized equations of viscous fluid flow and the no-slip condition on the half-plane are used to derive a pair of disjoint Wiener-Hopf equations for the fluid stresses and velocities. The Wiener-Hopf equations are solved in conjunction with a requirement that the stresses are integrable near the edge of the half-plane. Specific wave components of the scattered velocity field are given analytically. A Padé approximation to the Wiener-Hopf kernel function is used to derive numerical results that show the effect of viscosity on the velocity field in the immediate vicinity of the edge of the half-plane.

Characteristics for generalized hermomechanical fields in viscous fluid media

In the context of a general theory of characteristics, thermomechanical processes are considered in fluid media with the final heat-propagation velocity. The equations of propagation of characteristics for two variants of the theory of a viscous fluid medium are derived, and ways of applying them for determining displacement velocities are indicated.

Thermal processes in semimoment asymmetric hydrodynamics and the method of characteristics

A resolving system of equations of the asymmetric theory of viscous fluid media with account for the propagation of thermal processes in it is derived under the condition of equality of the rotation of the local trihedron to the mean rotation of the field of permutation, and the equation of propagation of characteristics is obtained.

Nonlinear Stability of an Electrified Interface Supporting Surface Charges between Two Viscous Fluids

The charged boundary between two viscous fluid media is taken to be under the influence of a vertical constant electric field producing surface charges. We illustrate the solution of the nonlinear perturbation using the method of multiple scales. In the neutral stability the relation between surface charge density and surface viscous force is obtained. Because the viscosity ratio has the same value as the dielectric constant ratio a singularity in the nonlinear coefficient of Schrödinger equation is observed and gradually creates a growth in the instability. Previous works have failed to recognize this singularity by not including the effects of viscosity. The absence of surface viscous force leads to decay in the surface charge density and a growth in the stability occurs. Global stability and local instability are observed in the stability diagrams. The duality mechanism is observed for the electric field, the surface charge density, and the stratified kinematic viscosity. Further, the behavior away from neutral stability is explained by examination of the dynamical system. It is observed that the stabilizing or destabilizing role for the surface charge density has been related to the destabilizing or stabilizing role of the density influence.

Sound-pressure level calculations using the RRA algorithm for depth-dependent speeds of sound valid at turning points and focal points

Sound-pressure level (SPL) calculations are made along individual ray paths for arbitrary, one-dimensional, depth-dependent speeds of sound using an enhanced version of the RRA (recursive ray acoustics) algorithm. The SPL calculations are valid (i.e., finite) at turning points and focal points and do not require the use of Airy functions. The SPL calculations include the effects of frequency-dependent volume attenuation and frequency-dependent attenuation due to surface and bottom reflections. The ocean surface and bottom are treated as boundaries between viscous fluid media. Although the ocean surface is modeled as a planar boundary, the bathymetry is an arbitrary function of horizontal range. Sound speed versus depth and bathymetric data are represented by orthogonal function expansions. Computer simulation results from preliminary test cases are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Sound-pressure level calculations using the RRA algorithm for depth-dependent speeds of sound valid at turning points and focal points

Sound-pressure level (SPL) calculations are made along individual ray paths for one-dimensional, depth-dependent, speeds of sound using an enhanced version of the recursive ray acoustics (RRA) algorithm. The SPL calculations are valid (i.e., finite) at turning points and focal points and do not require the use of Airy functions. The SPL calculations include the effects of frequency-dependent volume attenuation and frequency-dependent attenuation due to surface and bottom reflections. The ocean surface and bottom are treated as boundaries between viscous fluid media. Although the ocean surface is modeled as a planar boundary, the bathymetry is an arbitrary function of horizontal range. Sound speed versus depth and bathymetric data are represented by orthogonal function expansions. Computer simulation results from preliminary test cases are presented. [Work supported by NAVSEA.]

Bubble Dynamics in Polymer Melts

AbstractA system has been developed for studying the dynamic behaviour of gas bubbles in liquid media and is currently being used to investigate the behaviour of a single nitrogen bubble in molten LDPE. It involves the formation of an initial gas-liquid interface from a capillary by direct injection of the gas into the melt, subsequent growth until an equilibrium state is reached and the application of ambient pressure ramps which result in bubble expansion or contraction. The response depends on the rate of change of the ambient pressure. Experiments have been conducted at various temperatures above the crystalline melting temperature, Tm, of LDPE in the range 120°C-150°C. The numerical data acquired are in the form of pressure and temperature values, and these data are superimposed on stored video images of the bubbles for further processing on an image analyser whereby bubble sizes and shapes are determined.The results obtained reveal the effect of melt viscoelasticity on the dynamics of bubbles in highly viscous fluid media. Furthermore, the changes in bubble shapes with time due to buoyancy effects are also shown. Clearly, the bubble response in the melt is governed by a combination of viscous and elastic effects. However, in the absence of suitable viscous solutions to the problem, we have attributed the behaviour to a time-dependent elastic response, but only as a first order approximation.

Analysis of shear horizontal surface waves at the boundary between a piezoelectric crystal and a viscous fluid medium

Surface shear horizontal wave propagation at the boundary of a piezoelectric substrate with a viscous fluid is theoretically analyzed. The analytical solution clearly shows the dependence of the velocity change, the wave propagation loss, and the wave amplitude profile (thus the energy confinement near the surface), in terms of the liquid viscosity and density, the layer thickness, and the wave frequency. It is shown that the viscous fluid loading produces some guidance near the surface but also some damping of the wave. The propagation loss is due only to viscous coupling and not to a mode conversion and viscous coupling as is the case with Rayleigh surface acoustic waves (SAWs). Closed-form expressions are derived for the attenuation coefficient and the fractional velocity change (thus the frequency change) in terms of the piezoelectric crystal and viscous liquid parameters. The theory, applicable to both a Bleustein–Gulyaev (BG) wave and a surface skimming bulk wave (SSBW), indicates that surface shear waves could be used in the implementation of sensitive acoustic wave liquid-phase-based detectors, viscosity sensors, and/or biosensors. These sensors will not experience the high propagation loss associated with Rayleigh SAWs.

Natural frequencies of an elastic spherical shell submerged in a compressible viscous fluid medium

The frequency equations previously derived for free, axisymmetric vibrations of an elastic spherical shell submerged in a compressible viscous fluid is reduced to a simple polynomial expression. Its complete solutions are numerically obtained and presented for a steel shell submerged in water. Although free oscillatory motion is damped due to acoustic radiation and viscous dissipation, viscosity slightly reduces the total damping in some cases. Because damping due to viscous dissipation is always positive, the reduction of damping in these shell modes implies that viscosity reduces acoustic radiation.

The effect of viscosity on free oscillations of fluid-filled spherical shells

In order to clarify the effect of fluid viscosity on the vibrations of elastic shells, the axisymmetric free oscillations of a fluid-filled spherical shell are studied. The dynamic response of the shell is determined by the classical normal mode method, while a boundary layer approximation is employed for the compressible viscous fluid medium. The study shows that the fluid viscosity in general will produce a damping effect on the shell motion. It is also shown that the change on the real and imaginary components of each natural frequency due to viscosity are equal in magnitude but opposite in sign. Therefore, the circular frequency of a vibrating spherical shell will decrease with the increase of viscosity of the contained fluid. The study further reveals that viscous damping reaches a peak value in the vicinity of the upper branch in vacuo frequencies and then drops rapidly to a minimum. It is noted that the minimum damping is so small that the shell motion is almost steady state (non-dissipating).