Dynamics of gas bubbles in viscoelastic fluids. II. Nonlinear viscoelasticity

The nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear viscoelastic media are examined. The constitutive equation [Upper-Convective Maxwell (UCM)] used for the fluid is suitable for study of large-amplitude excursions of the bubble, in contrast to the previous work of the authors which focused on the smaller amplitude oscillations within a linear viscoelastic fluid [J. S. Allen and R. A. Roy, J. Acoust. Soc. Am. 107, 3167-3178 (2000)]. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of viscoelastic constitutive equations into bubble dynamics equations. The numerical method used to solve the governing system of equations (one integrodifferential equation and two partial differential equations) is outlined. An energy balance relation is used to monitor the accuracy of the calculations and the formulation is compared with the previously developed linear viscoelastic model. Results are found to agree in the limit of small deformations; however, significant divergence for larger radial oscillations is noted. Furthermore, the inherent limitations of the linear viscoelastic approach are explored in light of the more complete nonlinear formulation. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that tissue viscoelasticity may be an important consideration for the risk assessment of potential cavitation bioeffects.     

Nonlinear oscillations of gas bubbles in viscoelastic fluids

Nonlinear oscillations of gas bubbles in viscoelastic fluids of a three-constant Oldroyd model are theoretically investigated. The equation of motion for a bubble in a viscoelastic fluid subject to a pulsating pressure, and the pressure equation at the bubble wall are obtained. By numerical calculations on these equations, the effects of relaxation time and retardation time on frequency response curves, and the relation between the maximum pressure at the bubble wall and the initial radius of the bubble are clarified.     

Acoustically coupled gas bubbles in fluids: time-domain phenomena.

In previous work [C. Feuillade, J. Acoust. Soc. Am. 98, 1178-1190 (1995)] a coupled oscillator formalism was introduced for describing collective resonances, scattering, and superresonances, of multiple gas bubbles in a fluid. Subsequently, time-domain investigations of the impulse response of coupled systems have disclosed the exact conditions which determine whether the ensemble scattering behavior should be described using: either (a), a multiple scattering; or (b), a self-consistent methodology. The determining factor is the Q of the individual scatterers, and their typical spatial separations in the medium. For highly damped or sparse systems, e.g., scattering from loose schools of swimbladder fish, or from a gassy seabed containing entrained bubbles, the multiple scatter counting approach should be applicable. For more strongly coupled systems, e.g., a dense cloud of resonating bubbles in the water column, energy exchange may be due primarily to radiative cycling rather than scattering, in which case a self-consistent approach is indicated. The result has implications for both volume and bottom scattering applications.     

Linear quantum Enskog equation. I. Homogeneous quantum fluids

The quantum-statistical generalization of the well-known classical, linear revised Enskog equation is derived for spatially uniform systems. This new quantum kinetic equation allows the study of equilibrium time correlation functions and their associated transport coefficients of normal quantum fluids where static correlations and degeneracy effects due to particle statistics (both are treated exactly) are important. Furthermore, we derive the quantum-statistical analog of the classical ring operator. These microscopic and systematic derivations are based on a recently developed superoperator formalism (including cluster expansion techniques) that, as a main feature, allows a clear distinction between static and dynamic correlations, which is crucial in the discussion of the Enskog approximation.     

Undulatory swimming in viscoelastic fluids

The effects of fluid elasticity on the swimming behavior of the nematode Caenorhabditis elegans are experimentally investigated by tracking the nematode's motion and measuring the corresponding velocity fields. We find that fluid elasticity hinders self-propulsion. Compared to Newtonian solutions, fluid elasticity leads to up to 35% slower propulsion. Furthermore, self-propulsion decreases as elastic stresses grow in magnitude in the fluid. This decrease in self-propulsion in viscoelastic fluids is related to the stretching of flexible molecules near hyperbolic points in the flow.     

粘弹包膜气泡的声散射特性

在线性条件下研究了单个粘弹包膜气泡的声散射特性,并用WT散射法求解了造影剂的多气泡散射特性。首先建立了超声造影剂中单个粘弹包膜气泡的声散射物理模型,结合包膜内外声波方程和边界条件解得散射矩阵,并得到散射系数。通过数值计算用简化散射截面积曲线来表征单个气泡的声散射特性,用声色散曲线和衰减系数曲线来表征气泡的集总散射特性。研究结果表明:增加包膜的弹性参数μ_e的值,不仅使得气泡共振频率增高,还使气泡散射共振峰值相应加大;增加包膜的粘性参数μ_v的值,将使得散射的共振峰值降低;气泡的另两个参数λ_e和λ_v对散射特性几乎没有影响。     

Bernoulli excitation and detection of gas bubbles.

A simple method is proposed for detecting and sizing bubbles in pipeline fluid flow. This is based on changing the pressure of the fluid, which in turn excites volume oscillations in the bubble. If the change in pressure is of sufficient brevity and magnitude, the transient distortion results in excitation of the bubble into radiative oscillation at its natural frequency. In a moving fluid, the Bernoulli equation predicts that such a pressure change can be achieved through a suitable gradient in the flow velocity. In the experiments described here, this is achieved by altering the cross-sectional area of the pipe in which the fluid is flowing. We demonstrate the efficacy of this excitation method and, by detecting the radiated sound using a nearby hydrophone, determine the size of individual bubbles from their characteristic oscillation frequency.     

Ultrasonic scattering cross sections of shell-encapsulated gas bubbles immersed in a viscoelastic liquid: first and second harmonics

The oscillations of gas bubbles, without shell, immersed in viscoelastic liquids and driven by an acoustic wave have been the subject of several investigations. They demonstrate that the viscosity coefficient and the spring constant of the liquid have significant influence on the scattering cross section of the gas bubble. For shell-encapsulated gas bubbles, the investigations have been concentrated to bubbles immersed in a pure viscous liquid. This present work computes the ultrasonic scattering cross section, first and second harmonics, of shell-encapsulated gas bubbles immersed in a viscoelastic liquid. The theoretical model of the bubble oscillation is based on the generalized Rayleigh-Plesset equation of motion of a spherical cavity immersed in a viscoelastic liquid represented by a three-parameter linear Oldroyd model. The scattering cross section is computed for Albunex type of bubble (shell thickness=15 nm, shell shear viscosity=1.77 Pas, shell modulus of rigidity=88.8 MPa) irradiated by a 3.5 MHz ultrasonic pressure wave with an amplitude of 30 kPa. The results demonstrate that encapsulated bubbles respond independently of the surrounding liquid being pure viscous or viscoelastic as long as the surrounding liquid shear viscosity is as low as 10(-3) Pas. Nevertheless, for higher shear viscosities, the bubble responds differently if the surrounding liquid is pure viscous or viscoelastic. In general, the scattering cross sections of first and second harmonics are larger for the viscoelastic liquid.     

Linear hydrodynamics and viscoelasticity of nematic elastomers

We develop a continuum theory of linear viscoelastic response in oriented monodomain nematic elastomers. The expression for the dissipation function is analogous to the Leslie-Ericksen version of anisotropic nematic viscosity; we propose the relations between the anisotropic rubber moduli and new viscous coefficients. A new dimensionless number is introduced, which describes the relative magnitude of viscous and rubber-elastic torques. In an elastic medium with an independently mobile internal degree of freedom, the nematic director with its own relaxation dynamics, the model shows a dramatic decrease in the dynamic modulus in certain deformation geometries. The degree to which the storage modulus does not altogether drop to zero is shown to be both dependent on frequency and to be proportional to the semi-softness, the non-ideality of a nematic network. We consider the most interesting geometry for the implementation of the theory, calculating the dynamic response to an imposed simple shear and making predictions for effective moduli and (exceptionally high) loss factors. Received 16 October 2000 and Received in final form 10 December 2000     

Shear waves in viscoelastic wormlike micellar fluids

Low frequency (61 Hz) shear wave speeds have been measured in viscoelastic wormlike micellar (WM) fluids for a concentration range of 20/12-160/96 mM CTAB/NaSAL. The strain induced birefringence of the WM fluids was exploited to optically track the shear pulse using crossed polarizing filters and high speed video. It was found that shear speed increases roughly linearly with concentration at a rate of 3.5 mm s(-1) mM(-1) CTAB. Further, comparison with elastic and loss moduli obtained from rheology data show that shear wave propagation is dominated by the elastic modulus for this frequency range.     

Non-hydrodynamic modes in viscoelastic behaviour of simple fluids

ABSTRACT

We discuss the role of non-hydrodynamic processes in viscoelastic transition in pure liquids. In particular, using both analytical results and molecular dynamics simulations, we clarify the effect of the shear stress relaxation on the transverse dynamics. We use as an example the Lennard-Jones fluids. We analyse the frequency dependence of the shear modulus and its connection to the non-hydrodynamic shear relaxation mode. The analysis of the relaxation times of the non-hydrodynamic modes in longitudinal and transverse dynamics makes evidence that the emergence of the non-hydrodynamic transverse excitations outside the propagation gap is not directly connected to the onset of the positive sound dispersion.     

Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles

To facilitate practical medical applications such as cancer treatment utilizing focused ultrasound and bubbles, a mathematical model that can describe the soft viscoelasticity of human body, the nonlinear propagation of focused ultrasound, and the nonlinear oscillations of multiple bubbles is theoretically derived and numerically solved. The Zener viscoelastic model and Keller–Miksis bubble equation, which have been used for analyses of single or few bubbles in viscoelastic liquid, are used to model the liquid containing multiple bubbles. From the theoretical analysis based on the perturbation expansion with the multiple-scales method, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, which has been used as a mathematical model of weakly nonlinear propagation in single phase liquid, is extended to viscoelastic liquid containing multiple bubbles. The results show that liquid elasticity decreases the magnitudes of the nonlinearity, dissipation, and dispersion of ultrasound and increases the phase velocity of the ultrasound and linear natural frequency of the bubble oscillation. From the numerical calculation of resultant KZK equation, the spatial distribution of the liquid pressure fluctuation for the focused ultrasound is obtained for cases in which the liquid is water or liver tissue. In addition, frequency analysis is carried out using the fast Fourier transform, and the generation of higher harmonic components is compared for water and liver tissue. The elasticity suppresses the generation of higher harmonic components and promotes the remnant of the fundamental frequency components. This indicates that the elasticity of liquid suppresses shock wave formation in practical applications.     

Effects of medium viscoelasticity on bubble collapse strength of interacting polydisperse bubbles

Due to its physical and/or chemical effects, acoustic cavitation plays a crucial role in various emerging applications ranging from advanced materials to biomedicine. The cavitation bubbles usually undergo oscillatory dynamics and violent collapse within a viscoelastic medium, which are closely related to the cavitation-associated effects. However, the role of medium viscoelasticity on the cavitation dynamics has received little attention, especially for the bubble collapse strength during multi-bubble cavitation with the complex interactions between size polydisperse bubbles. In this study, modified Gilmore equations accounting for inter-bubble interactions were coupled with the Zener viscoelastic model to simulate the dynamics of multi-bubble cavitation in viscoelastic media. Results showed that the cavitation dynamics (e.g., acoustic resonant response, nonlinear oscillation behavior and bubble collapse strength) of differently-sized bubbles depend differently on the medium viscoelasticity and each bubble is affected by its neighboring bubbles to a different degree. More specifically, increasing medium viscosity drastically dampens the bubble dynamics and weakens the bubble collapse strength, while medium elasticity mainly affects the bubble resonance at which the bubble collapse strength is maximum. Differently-sized bubbles can achieve resonances and even subharmonic resonances at high driving acoustic pressures as the elasticity changes to certain values, and the resonance frequency of each bubble increases with the elasticity increasing. For the interactions between the size polydisperse bubbles, it indicated that the largest bubble generally has a dominant effect on the dynamics of smaller ones while in turn it is almost unaffected, exhibiting a pattern of destructive and constructive interactions. This study provides a valuable insight into the acoustic cavitation dynamics of multiple interacting polydisperse bubbles in viscoelastic media, which may offer a potential of controlling the medium viscoelasticity to appropriately manipulate the dynamics of multi-bubble cavitation for achieving proper cavitation effects according to the desired application.     

Dynamics of supercooled fluids

The dynamics of supercooled fluids is investigated. This approach is based on a static mean field theory of freezing developed by Grewe and Klein and the dynamical theory of Cahn and Hilliard. A nonlocal term is introduced to the standard Landau-Ginsburg-Wilson Hamiltonian and the resulting dynamics are explored. It is shown that for short times the supercooled liquid is unstable to density fluctuations of a nonzero wavevectork. Thek at which this instability first appears is on the order of the inverse range of the interaction potential.After Oct. 1, 1982: Institute for Basic Studies National Bureau of Standards Washington DC 20234, USASupported in part by grants from the ARO, NSF     

In-water gas combustion in linear and annular gas bubbles

A new pulsed-cyclic method of in-water gas combustion was developed with separate feed of fuel gas and oxygen with the focus on development of new technologies for heat generators and submerged propellers. The results of calorimetric and hydrodynamic measurements are presented. In-water combustion of acetylene, hydrogen, and propane was tested with the operation frequency of 2–2.5 Hz and with a linear injector. The combustion dynamics of combustion of stoichiometric mixture with propane (C₃H₈+5O₂) was studied for a bubble near a solid wall; the produced gas bubble continues expansion and oscillations (for the case of linear and annular bubbles). It was demonstrated that gas combustion in annular bubbles produces two same-magnitude pulses of force acting on the wall. The first pulse is produced due to expansion of combustion products, and the second pulse is produced due to axial cumulative processes after bubble collapse. This process shapes an annular vortex which facilitates high-speed convective processes between combustion products and liquid; and this convection produces small-size bubbles.     

Propagation of acoustic wave in viscoelastic medium permeated with air bubbles

Based on the modification of the radial pulsation equation of an individual bubble, an effective medium method (EMM) is presented for studying propagation of linear and nonlinear longitudinal acoustic waves in viscoelastic medium permeated with air bubbles. A classical theory developed previously by Gaunaurd (Gaunaurd GC and \"{U}berall H, {\em J. Acoust. Soc. Am}., 1978; 63: 1699--1711) is employed to verify the EMM under linear approximation by comparing the dynamic (i.e. frequency-dependent) effective parameters, and an excellent agreement is obtained. The propagation of longitudinal waves is hereby studied in detail. The results illustrate that the nonlinear pulsation of bubbles serves as the source of second harmonic wave and the sound energy has the tendency to be transferred to second harmonic wave. Therefore the sound attenuation and acoustic nonlinearity of the viscoelastic matrix are remarkably enhanced due to the system's resonance induced by the existence of bubbles.     

Bubble oscillation and inertial cavitation in viscoelastic fluids

Non-linear acoustic oscillations of gas bubbles immersed in viscoelastic fluids are theoretically studied. The problem is formulated by considering a constitutive equation of differential type with an interpolated time derivative. With the aid of this rheological model, fluid elasticity, shear thinning viscosity and extensional viscosity effects may be taken into account. Bubble radius evolution in time is analyzed and it is found that the amplitude of the bubble oscillations grows drastically as the Deborah number (the ratio between the relaxation time of the fluid and the characteristic time of the flow) increases, so that, even for moderate values of the external pressure amplitude, the behavior may become chaotic. The quantitative influence of the rheological fluid properties on the pressure thresholds for inertial cavitation is investigated. Pressure thresholds values in terms of the Deborah number for systems of interest in ultrasonic biomedical applications, are provided. It is found that these critical pressure amplitudes are clearly reduced as the Deborah number is increased.