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The influence of heat conduction on acoustic streaming is considered including the effect of variable tube wall temperature. The calculations are carried out in the limiting case when the boundary layer is thin compared to the tube radius.

The general three‐dimensional high‐frequency oscillatory flow near a stagnation point is investigated through the method of matched asymptotic expansions. The nonlinear Reynolds's stresses create a steady streaming motion in the fluid. It is found that this streaming is minimized when the surface near the stagnation point is axisymmetric and is maximized when the surface is a saddle.

Individual air bubbles held in the central interior of a water‐filled chamber by a 20‐kHz pressure‐gradient force, were observed dissolving in undersaturated water and growing by rectified diffusion in saturated water. When bubbles grew or dissolved in the absence of nearby acoustic streaming, present theories for normal diffusion and for rectified diffusion in the absence of streaming gave reasonable agreement with experiment. When crispations appeared on a bubble, acoustic streaming accompanied the surface activity and the rate of dissolution or growth of the bubble was intensified up to twentyfold. Measurements of threshold sonic pressures for onset of crispation revealed that, in general, the larger the bubble the lower the pressure required. Present theories for diffusion in the presence of acoustic streaming imply, in contrast to experiment, that streaming should inhibit rectified diffusion in saturated and undersaturated liquids. For gas bubbles dissolving in undersaturated liquids, the effects of sound on mass transfer can be made negligible, and experimental measurements of the diffusion coefficient of the gas in the liquid can be obtained.

The first-order properties of acoustic waves (i.e., the to and fro particle displacement and velocity) can produce a number of second-order phenomena: cavitation, acoustic streaming, surface instability and radiation pressure. The dependence of cavitation induced phenomena (erosion, luminescense, chemical reactions) on the physical and acoustical parameters of a system are discussed. Some of the past work carried out in this field is analyzed and reinterpreted. In the light of this, it appears that the extent of the solubility of a gas has a pronounced effect on cavitation related phenomena in addition to the effect of other variables such as the ambient liquid temperature, the hydrostatic pressure, the specific heat ratio, the thermal conductivity of dissolved gas and the intensity and frequency of acoustic field. A summary of the application of sonic and ultrasonic energy to industrial processing operations is also provided. This discussion includes how the other second-order effects (e.g., interfacial instability) are related to the enhancement of these operations. The wide variety of processes in which the applications of acoustic energy has a beneficial effect suggests the versatility and broad commercial potential of sonochemical engineering.

Theoretical results are obtained for heat transfer from a circular cylinder oscillating in an unbounded viscous fluid which is otherwise at rest. The amplitude of the oscillation is assumed small compared to the radius of the cylinder, which for most of the examples considered is assumed to be at a constant temperature. The analysis is based upon use of the acoustic streaming flow field and consideration is given to the cases of small and large streaming Reynolds numbers. For large streaming Reynolds numbers, a solution for the previously undetermined steady streaming flow field is computed. The results obtained cover a wide range of Prandtl number. The method of matched asymptotic expansions is exploited in the analysis and the computed results are also supplemented by an approximate method based on an integrated form of the governing equations. The relationship between the present work and other relevant contributions in the literature is discussed. In a final section, attention is devoted to a technique for determining the temperature distribution which results when a line source of heat is embedded at the centre of the oscillating cylinder.

Experiments are described involving a membrane set into oscillatory motion (40 kc) by contact with a vibrating tip of small area. Small scale acoustic streaming is observed in a liquid above the membrane, especially near the area of contact; its character depends on the membrane tension. Loosely adhering films are removed from certain areas of the membrane. The cleared areas form a pattern, indicative of the surface stress distribution set up by the eddying liquid. The main features of the reported results are accounted for, at least, qualitatively, by application of acoustic streaming theory to idealized cases. Referring to the above results, the possibility is discussed of motion induced in biological cells by external vibrating bubbles.

A type of acoustic cavitation has been studied that occurs at the face of a vibrator with a small diameter. A gas bubble grows to near‐resonant size by accumulation of microbubbles. The process then reverses and the parent bubble rapidly throws off a cloud of microbubbles. This phase is accompanied by severe surface distortion of the main bubble, with emission of strong harmonics and a good deal of white noise. Under suitable conditions, a cyclic process is set up at a very regular repetition rate that depends on the acoustic pressure and the physical properties of the liquid. Acoustic streaming has an important role in maintaining the cycle. High‐speed movies have been taken, and these, along with measurements of the acoustic emission, show clearly the progress of the cycle. A technique is discussed for using this process to generate bubbles of defined size.

The present work considers enhancement of mass transport resulting from superimposing an ultrasonically induced convective transport on a diffusional transport. As a result of the nonlinearities in the Navier‐Stokes equation, a time‐dependent secondary flow called acoustic streaming can be produced when an acoustic wave is passed through a medium. Between adjacent vortices or cells, molecular diffusion is the only means of transport; however, within each cell, mass transport is primarily by convection. Increases in the rate of mass transfer of the order of 150% above the normal diffusional flux were found. The theoretical analysis and results presented have application to systems in which ultrasonics may be used to increase mass transfer through membranes (e.g., dialysis), and to increase the efficiency of very active catalytic systems and of solid‐liquid extraction systems.

It is shown that the third‐order differential equations commonly used in connection with acoustic streaming are special approximations to the law of rotational motion for a fluid element. An expression is derived for the torque exerted by stresses on a spherical element of fluid, about its center of gravity. The average torque in a steady‐state sound field consists partly of (1) a sonic torque, exerted by the sound field and partly of (2) a viscous torque due to the induced steady flow. Expressions are also derived for the instantaneous rate of increase of moment of momentum of a spherical fluid element, due to angular acceleration, expansion and shearing. Interpretation is given to individual terms in various streaming equations. In general, any field consisting of superposed elementary waves will give rise to a nonzero distribution of sonic torque. In the special case of a directional source the distribution is similar to that of the gradient of the magnitude of radiation pressure.

Propagation of an ultrasonic wave through an absorbing liquid contained in a tube gives rise to a gradient of pressure along the axis which is determined by observing the velocity of flow of the liquid through a side tube connected in parallel with the main tube. The intensity of the ultrasonic wave is found by recording the voltage applied to the exciting transducer, the parameters of which are measured in the vicinity of resonance. Obstruction of the sound field by the intensity measurement is thus avoided. The observations are used to evaluate the absorption coefficients of liquids over the range 130 kc/s to 1560 kc/s. The limits and accuracy of the technique are discussed and a new theoretical approach enables the pressure gradient to be derived from second-order acoustic variables without recourse to the concept of radiation pressure.

Ultrasound in the absence of cavitation was applied in a special manner to a process in which the sonic effect on surface layers and mass transport of reactants could be studied. The process selected involved cyclic anodic and cathodic polarization of a platinum electrode in dilute acid. The test electrode was a small platinum sphere mounted on the tip of an 85‐kc/sec Mason horn. Sound applied during selected parts of the cycle caused pronounced changes in recordings of the electrode potential versus time. These changes probably result from the transport of dissolved gases, via small‐scale acoustic streaming, to and from the electrode surface. An order‐of‐magnitude approach is given to the theory for mass transport associated with the acoustic streaming.

I - Absorption of Finite-Amplitude Waves.- II - Acoustic Radiation Pressure.- III - Acoustic Streaming.- IV - Pulsations of Cavitation Voids.- V - Experimental Investigations of Ultrasonic Cavitation.- VI - The Cavitation Zone.

We initiate a study of acoustic streaming in a stratified fluid. A set of equations describing nondimensional field quantities is first derived. By introducing successive approximations, we next derive equations for the primary and the secondary motion. These equations are briefly discussed.

This paper is a continuation of an analysis of acoustic streaming begun in a previous paper where streaming at high Reynolds numbers (order of several hundred) was treated. The corresponding problem for low Reynolds numbers (order of ten) is now considered, and it is shown that the flow pattern obtained around a cylinder in a sound field is opposite to the one corresponding to high Reynolds numbers. The Reynolds number is defined as R = U0a/v, where U0 is the particle velocity in the incident sound wave, a is the radius of the cylinder, and v is the kinematic viscosity. The two types of flow associated with different values of Reynolds numbers seem to correspond to the two types of circulations that have been experimentally observed and reported in the literature.

The acoustic streaming of an idealized elastico-viscous fluid near a cylindrical obstacle is considered using the boundary-layer type of approximation. It is shown that this streaming phenomena can be greatly influenced by the presence of elasticity in the fluid.

Research on cell suspensions, single cells, macromolecular suspensions, and model (nonliving) systems reveal a variety of changes brought about by sound in the absence of gross‐heating or transient (“collapse”‐type) cavitation. Many of these phenomena are explainable in terms of acoustic streaming, radiation pressure, and other characteristics of ultrasound that arise from nonlinearity. Commonly, the sonic effect depends on nonuniformity in the sound field. Interestig results are obtained with techniques by which one can set up nonuniform vibration in the walls or membranes of individual cells. In ultrasonic beams acting on tissue, it is possible that variations may arise from gradients in the incident field, or from inhomogeneities in the tissue, which scatter sound. Gaseous pockets or bubbles would be especially effective as such inhomogeneities.

Most aspects of acoustic streaming were considered understood in 1954. The time dependence of its onset, however, was controversial when Liebermann's measurements, valid for slowly developing streaming, were criticized. The criticism cited a report that streaming was present only 20 msec after turning on the acoustic source. Using a thermistor bead as flow meter, we have observed the time dependence of the onset of streaming in two fluids of widely different viscosity, each in two vessels of different geometry. The time dependence of the streaming velocity is described as follows: There is zero time derivative at the start. This is followed in succession by a positive second derivative, inflection point, and negative second derivative which persists during an approach to a steady state flow. From the start to the inflection point, the flow velocity v depends on time t according to v = btp where b and p are empirical constants. The exponent p appears to be a nearly fundamental property of streaming onset w...

An approximate solution is developed for sonically‐induced steady flow near a fluid‐solid interface. The result is valid, subject to stated conditions, for the flow near any portion of surface in the vicinity of which the irrotational oscillatory velocity distribution ua is known. The principal condition on the validity is that the acoustic boundary layer parameter (ν/ω)12 (where ω is the angular frequency and ν is the kinematic viscosity coefficient for the fluid) should be small compared to the scale of ua. Applications of the general result are made to special situations, one case of particular interest being that of a small source near a rigid plane. The conclusion is reached that small compressible bodies, and especially resonant gas bubbles, resting on boundaries, are likely sites of pronounced microstreaming a sound field.

An experimental arrangement is described which permits controllable generation of small‐scale acoustic streaming, i.e., “microstreaming” near a plane boundary S. This arrangement, which consists essentially of a cylindrical bar oscillating longitudinally above the boundary S in question, has been used in studies of sonically‐produced effects at a fluid‐solid interface. The present paper deals with first‐order pressure and velocity fields in the thin layer of fluid between S and the vibrating face of the bar. Theoretical expressions are developed, based on a simple model in which incompressible flow is assumed. Experimental determination of the pressure field was accomplished by means of a special hydrophone arrangement described herein, which measures the pressure at points along the plane boundary. The first order theory agrees well with experiment, and is thus judged satisfactory as basis for a detailed treatment of the microstreaming flow, to be given in a subsequent paper. A byproduct of the work is a method for calibration of small hydrophones.

Theories for calculating steady streaming associated with sound fields are reviewed, comparing the methods and approximations of various authors. Two illustrative problems are worked out, both for rectilinear flow due to irrotational sound fields. The first deals with a single attenuated plane wave traveling down a tube, as in Cady's quartz wind experiments. In the second, a pair of crossed plane waves is treated, giving rise to a quite different kind of streaming. In obtaining solutions, attention is given to boundary conditions; here, gradients of She excess static pressure, another second‐order quantity, come into consideration. Significantly, streaming speeds depend critically upon α, the attenuation constant, where α may be due to any common cause, such as heat conduction, scattering, thermal relaxation, etc. From these results it appears that streaming measurements cannot be used to distinguish between absorption mechanisms. Numerical values are given for a few cases; high flow speeds may be expected in a bubbly medium.

A survey is given of several analytical techniques that can be applied to acoustic streaming phenomena. A method is selected for examining intense acoustic field-viscous fluid flow interaction problems in which the flow oscillation magnitude is much larger than any time-mean flow component. The method itself has a wide range of applicability and provides a clear and explicit statement of the approximations involved. The major goal of the paper is to illustrate this method by applying it to a new problem; namely, the interaction of an intense transverse resonant acoustic field with simple channel flow. The solution, while simple, possesses some very interesting heat transfer aspects.

In a previous article, the author described an experimental arrangement suitable for generating small‐scale near‐boundary acoustic streaming, i.e., “microstreaming,” in a precise and controllable manner. Such microstreaming is of particular interest insofar as it can have considerable influence on processes taking place at a solid‐liquid interface. In the present paper, results of a comprehensive study of the microstreaming flow field associated with the aforementioned experimental system are discussed in some detail. Photographs are shown of the various flow configurations, and data are presented on streaming speeds as a function of significant experimental parameters. Theoretical expressions are developed along lines of conventional acoustic streaming theory to describe various features of the flow. The correlation between theory and experimental results is found to be good when certain criteria, described herein, are fulfilled.

A radially symmetric progressive acoustic beam has been set up in a closed, gas‐filled tube to cause streams driven by the Eckart, volume, source of verticity. A careful determination of the acoustic pressure at all points of a cross section of the beam has been used successfully to predict the stream velocities at this cross section, for argon. The work with argon and the results of further experiments with dry nitrogen and moist air indicate that: (1) the Navier‐Stokes equation, the equation of continuity, and the acoustic “equation of state” p1 = ρ1c02+Rρ1, provide a valid basis for second‐order studies of streaming; (2) thermal conduction effects may be introduced into the Eckart streaming equations in the same manner as they appear in the attenuation formulas; (3) the macroscopic quantity, bulk viscosity, is made up of contributions from the intramolecular processes of vibrational and rotational relaxation.

Vortices are generated by acoustic oscillations as a second-order viscous effect. Traveling tangential modes in an axially perforated solid propellant rocket give a single vortex swirling about the axis. Tangential velocity at the periphery, typically a few hundred feet per second, can be predicted approximately using acoustic streaming theory modified by heat addition and radial flow effects. Serious consequences of the vortex are 1) torque, which gives undesired roll, especially in large vehicles as torque varies as (radius), and 2) pressure rise caused largely by reduction in the effective area of the nozzle throat. Standing tangential modes produce a pair of vortices rotating in opposite directions for each pressure antinode. These give no net torque and are much less destructive. Experimental data include 3200 frames/ sec cine-films of the vortices, torque measurements, and radial pressure gradient determinations in the head end of a motor. The traveling tangential modes can be initiated readily by tangential injection of nitrogen into the motor.

Observations of acoustic cavitation have been in helium I and II using an extensional‐mode vibrator capable of very high intensities at 10 kHz. Remote monitoring of the drive oscillatory velocity was achieved. In helium I clouds of small vapor bubbles were visible, as well as larger bubbles near the expected radial resonance size. For comparison, observations were also made using two other normal liquids—nitrogen and water—at temperatures near their boiling points. In helium II observation of acoustic streaming with emission of white noise suggested the presence of even smaller vapor bubbles. A strong subharmonic signal with low threshold excitation has not been fully explained.

A photographic study employing smoke as the indicating medium has shown the existence of a new type of streaming near a heated horizontal cylinder in the presence of a horizontal transverse sound field. This phenomenon, called “thermoacoustic streaming,” is characterized by the development of two vortices above the cylinder; the fluid pattern resembles vortex shedding behind a cylinder in forced flow normal to its axis. In the presence of sound waves whose half‐wavelength is six or more times greater than the diameter of the heated cylinder, the formation of the vortex flow is a function of the sound intensity only; for such wavelengths the vortices begin to appear at 140 db (re 0.0002 μbar) and become fully developed at 146 db. This type of streaming is a flow phenomenon which is much stronger than isothermal streaming for the same geometry and sound intensity. It appears that thermoacoustic streaming will have important practical applications, particularly in the field of heat transfer.

It has been suggested by Eckart (1948), Liebermann (1949) and others that measurement of acoustic streaming in a fluid provides an independent means of evaluating the ratio of the second coefficient of viscosity to the coefficient of shear viscosity. Arguments are put forward here to support the view (Nyborg 1953) that the streaming velocity is directly dependent on the coefficient of sound absorption. The second coefficient of viscosity influences the velocity of streaming only through its relationship to the coefficient of sound absorption. It is concluded that measurements of streaming velocity do not yield information about the second coefficient of viscosity other than may be obtained from the direct measurement of sound absorption. The theory of streaming based on the gradient of radiation pressure has been established quantitatively by a solution of the Navier-Stokes equations. A description is given of a method of determining the sound-absorption coefficient of a liquid from measurements of the velocity of streaming, and some typical results are stated.

This paper reviews the problem of unstable combustion in solid-propellant rockets, with special emphasis on the status of theory and, where possible, the comparison between theory and experiment. The manifestation of the problem in terms of the appearance of periodic pressure waves, and the consequences which ensue when these reach sufficient amplitude are first outlined. A rocket motor is then viewed as an acoustic cavity. The possible sources and sinks of acoustic energy are enumerated, and the gain-loss balance is discussed with respect to self-excited oscillation, and also with respect to influences of outside perturbations on a stable cavity which has a high Q . The quantitative approach to the representation of, the various gain-loss mechanisms in the domain of linear stability theory is then reviewed. Particular emphasis is put on the modifications to ordinary acoustic stability theory which are required because of the existence of a mean flow field. The contributions to linear instability of the response of the burning propellant, both to the pressure and velocity components of the acoustic field, are then discussed. Attention is then turned to the fundamental theory of the interaction of the burning surface with a pressure oscillation. The linear (small perturbation) theory of the acoustic response of this region is reviewed in terms of the analytic models, and postulates that have been made, and their theoretical results. Some qualitative comparison with experiment is made. The remaining part of the paper is then devoted to nonlinear effects. The theory of nonlinear instability in such systems is first reviewed to the extent that it exists. Then, the consequences of oscillation of finite amplitude arising from nonlinear effects are discussed. Under this heading, amplitude limitation in unstable cavities, wave distortion, and changes in the mean properties of the system are considered. In the alst category, changes in the mean burning rate due to nonlinear pressure response and due to the erosion associated with the velocity component of the acoustic field are noted. The induction of vortex flow by acoustic streaming, as described by Swithenbank and Sotter and by Flandro, is discussed with particular reference to its consequences in producing increased erosion, decreased nozzle flow, and angular torques. Brief mention is also made of the effects of the amplitude oscillations on combustion, efficiency and the composition of the product gases.

By using special acoustical and optical arrangements, vortex motions are discovered in the vicinity of small vibrating sources in liquids. Of particular importance is the vigorous eddying that typically takes place near a vibrating gas bubble when it rests on a boundary. This bubble‐associated streaming has been observed under a variety of circumstances, and apparently occurs quite generally. Especially violent and chaotic motions appear near bubbles of resonant size. The attendant highly localized convection and viscous stresses are probably very important as intermediate physical mechanisms for sonic irradiation effects.

THERE are many natural phenomena in which nonlinear interactions of time-dependent inputs give rise to steady—that is, time-independent outputs. One of these is a steady streaming belonging to a class of secondary flows sometimes called acoustic streaming. It occurs when a circular cylinder oscillates normal to its axis in an unbounded Newtonian fluid1–3. We report here on the steady secondary flow induced when a long thin cylinder oscillates as described in a viscoelastic liquid. We found that the direction of steady streaming is opposite to that found for the bulk of fluid when the experiment is performed with a Newtonian fluid.

Recent studies have led to a satisfactory solution to the problem of acoustic streaming in the vicinity of a sphere for s/a≪1, where s is the oscillation amplitude, and a is the radius of the sphere. The results of these studies have now been applied to the calculation of the inherent energy losses. The energies in the dc vortices and in the primary ac vortices have been computed, and it is shown that the energy in the former is negligible in comparison with the energy in the latter. The results are restricted to cases where s/a≪1, and a(ω/2ν)12≫1, where ν is the kinematic viscosity and ω is the frequency of oscillation.

Normal- and high-speed cinemicrographs of events resulting from highly localized 25-kc vibration of small regions of an individual cell wall in Elodea leaf cells and in plastic cell models are discussed. In plant cells, complex patterns of ordered agitation are set up, similar to parts of the patterns observed in model cells. In models containing Newtonian fluids, steady circulation results with streamline positions and directions being functions of viscosity and the vibration pattern of the wall, the general features being accounted for by the theory of acoustic streaming. In models containing weak agar gels, a combination of plastic and fluid behavior is observed as setting ensues; immediately after vibration begins, suspended particles ``flow'' for a short distance following the streamlines of a viscous fluid, with displacement vectors distributed like the velocity vectors in acoustic streaming, but, as the sound ceases, return slowly almost to their original positions.

The acoustic equations pertaining to an aerosol containing small solid particles in suspension are developed. In order to study the acoustic streaming of this medium, that is, the steady unidirectional motion of particles which is superimposed upon the oscillatory acoustic motion, it it necessary to carry the analysis to so-called second order acoustic variables. It is shown that the inclusion of the dynamic and thermal properties of the suspended particles significantly influences the attenuation of sound and the streaming of the medium. Since the well known but presently unexploited process of sonic agglomeration of particulate aerosols depends to a large extent upon the acoustic streaming phenomenon, it is suggested that this work may be useful in further development of this process. A numerical example describing the streaming of an aerosol in a duct under the influence of acoustic excitation is developed in some detail.

The choice of equipment and detergent media for ultrasonic cleaning depends first and foremost on the type of contaminant. Since mechanical destruction of the contaminant film under the action of cavitation and acceleration of the process of chemical interaction between the washing liquid and contaminant under the action of acoustic streaming occur simultaneously in the sound field, it is convenient to classify all contaminants according to three basic criteria: 1. the ability to withstand the action of microimpact loading; 2. the bond strength with the surface to be cleaned; 3. the nature of the chemical interaction with the detergent.

Roll torque of considerable magnitude is generated in some solid-propellaiit rocket motors during periods of high-intensit y acoustic combustion instability. Forces normal to the motor axis sometimes accompany the roll perturbations. These effects appear to result from nonlinear viscous phenomena (acoustic streaming) that occur in the presence of intense acoustic wave motion in the combustion gases. Direct measurements of the moments and associated vortex flow disturbances originating within the burning cavity verify this theoretical description. Unstable motors using a cylindrical grain perforation geometry are especially subject to roll torque generation. However, similar effects are exhibited in the more complex grain configurations. Star-perforated cavities are capable of producing several levels of roll torque, depending upon the number of star-points. The secondary force effects are sensitive to initial grain temperature and thus can be minimized in some motors by prerun temperature conditioning.

By controlled irradiation of erythrocyte suspensions at 20 kHz it is demonstrated that shear associated with acoustic microstreaming can be an important mechanism for biological effects of sound. Two effective sources of acoustic microstreaming are stable oscillating gas bubbles and transversely oscillating wires. The threshold displacement amplitude for achieving critical shear can be reduced by increasing the solvent viscosity and reducing the radius of the source of acoustic streaming. The threshold stress was found to decrease by 55% or more when the sample was heated to 45°C or higher for 10 min. This suggests that synergism exists between mechanical and thermal mechanisms for sonic effects. Mass transfer associated with small‐scale acoustic streaming controls the rate of cell disruption.

Naturally‐isolated living cells were subjected to ultrasonic vibration by means of the tip of a steel needle applied directly to the cell surface or inserted into a drop of a suspension of cells in sea water. The needle was machined into the tip of a stainless‐steel cone, and the base of the cone was glued to one end of an electroded and polarized barium titanate hollow cylinder. This composite transducer was driven at its resonant frequency of approximately 85 000 cps. Ultrasound applier to the surfaces of egg cells of marine invertebrates produces rotation, translation, deformation, and fragmentation of the nucleoli; rotation and deformation of the nuclei; acoustic streaming of nucleoplasm and cytoplasm; and deformation of the cellular surface and fragmentation of the cell. Cells and intracellular bodies are often attracted to the sound source as a result of acoustic‐radiation pressure, and, especially for whole cells in suspension, this attractive tendency is typically superposed on a tendency of the b...

The acceleration of the liberation of free gas present in a liquid in bubble form in a sound field is attributable to a number of factors. The occlusion of the bubbles to the surface is clearly accelerated by their coalescence under the action of the Bjerknes force, owing to the entrainment of bubbles by acoustic streaming, and under the influence of the acoustic radiation pressure. Due to the complexity of the degassing process as a whole, we shall proceed as before and analyze its individual stages, assessing the influence of the various factors in the example of a single gas bubble or pair of bubbles.