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Hydrodynamic interactions between a pair of capsules in simple shear are numerically investigated using afront-tracking finite difference method. The membrane of the capsule is modeled using different hyperelasticconstitutive relations. We also compare the pair interactions between drops to those between capsules. Anincreased viscosity ratio leads to a reduced net cross-stream separation between capsules as well as drops aftercollision. At low viscosity ratios, for the same capillary number drop-pairs show higher cross-stream separationthan those for capsule-pairs, while substantially large viscosity ratios result in almost the same value for bothcases. We investigate pair-collisions between two heterogeneous capsules C1and C2with two different capillarynumbers. The maximum deformation of C1was seen to increase with increasing stiffness (decreasing capillarynumber) of C2, even though the stiffness of C1was kept fixed. The findings are similar for a drop-pair, however,with a smaller maximum deformation for the same combinations of capillary numbers. The final cross-streamdrift of the trajectory of C1decreases with the increasing stiffness of C2, but the relative trajectory betweenthe capsules remains unchanged. The maximum deformation and the cross-stream drift of the trajectory of C1are shown to approximately vary with power-law functions of the ratio of the capillary numbers of C1andC2. An analytical explanation of the dependence on the two capillary numbers is offered. Different membraneconstitutive laws result in similar deformation and drift in trajectory.

Liposomes are representative lipid nanoparticles widelyused for delivering anticancer drugs, DNA fragments, or siRNA to cancercells. Upon targeting, various internal and external triggers have been usedto increase the rate for contents release from the liposomes. Among theinternal triggers, decreased pH within the cellular lysosomes has beensuccessfully used to enhance the rate for releasing contents. However,imparting pH sensitivity to liposomes requires the synthesis of specializedlipids with structures that are substantially modified at a reduced pH.Herein, we report an alternative strategy to render liposomes pH sensitiveby encapsulating a precursor which generates gas bubblesin situinresponse to acidic pH. The disturbance created by the escaping gasbubbles leads to the rapid release of the encapsulated contents from theliposomes. Atomic force microscopic studies indicate that the liposomalstructure is destroyed at a reduced pH. The gas bubbles also render theliposomes echogenic, allowing ultrasound imaging. To demonstrate the applicability of this strategy, we have successfully targeteddoxorubicin-encapsulated liposomes to the pancreatic ductal carcinoma cells that overexpress the folate receptor on the surface.In response to the decreased pH in the lysosomes, the encapsulated anticancer drug is efficiently released. Contents releasedfrom these liposomes are further enhanced by the application of continuous wave ultrasound (1 MHz), resulting in substantiallyreduced viability for the pancreatic cancer cells (14%).

The migration of a capsule enclosed by an elastic membrane in a wall-bounded linearshear is investigated using a front-tracking method. A detailed comparison with themigration of a viscous drop is presented varying the capillary number (in the caseof a capsule, the elastic capillary number) and the viscosity ratio. In both cases,the deformation breaks the flow reversal symmetry and makes them migrate awayfrom the wall. They quickly go through a transient evolution to eventually reach aquasi-steady state where the dynamics becomes independent of the initial positionand only depends on the wall distance. Previous analytical theories predicted thatfor a viscous drop, in the quasi-steady state, the migration and slip velocities scaleapproximately with the square of the inverse of the drop–wall separation, whereasthe drop deformation scales as the inverse cube of the separation. These power lawrelations are shown to hold for a capsule as well. The deformation and inclinationangle of the capsule and the drop at the same wall separation show a crossoverin their variation with the capillary number: the capsule shows a steeper variationthan that of the drop for smaller capillary numbers and slower variation than thedrop for larger capillary numbers. Using the Green’s function of Stokes flow, asemi-analytic theory is presented to show that the far-field stresslet that causes themigration has two distinct contributions from the interfacial stresses and the viscosityratio, with competing effects between the two defining the dynamics. It predicts thescaling of the migration velocity with the capsule–wall separation, however, matchingwith the simulated result very well only away from the wall. A phenomenologicalcorrelation for the migration velocity as a function of elastic capillary number, walldistance and viscosity ratio is developed using the simulation results. The effects ofdifferent membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak,and Skalak – are briefly investigated to show that the behaviour remains similar fordifferent equations.

Micron- to nanometer-sized ultrasound agents,like encapsulated microbubbles and echogenic liposomes,are being developed for diagnostic imaging and ultra-sound mediated drug/gene delivery. This review providesan overview of the current state of the art of the mathe-matical models of the acoustic behavior of ultrasound con-trast microbubbles. We also present a review of the in vitroexperimental characterization of the acoustic properties ofmicrobubble based contrast agents undertaken in our lab-oratory. The hierarchical two-pronged approach of model-ing contrast agents we developed is demonstrated for a lipidcoated (SonazoidTM)and a polymer shelled (polyD-L-lactic acid) contrast microbubbles. The acoustic and drugrelease properties of the newly developed echogenic lipo-somes are discussed for their use as simultaneous imagingand drug/gene delivery agents. Although echogenicity is con-clusively demonstrated in experiments, its physical mecha-nisms remain uncertain. Addressing questions raised herewill accelerate further development and eventual clinicalapproval of these novel technologies.

Background: Mechanical stimulation of bone increases bone mass and fracture healing, at least in part, throughincreases in proliferation of osteoblasts and osteoprogenitor cells. Researchers have previously performedin vitrostudies of ultrasound-induced osteoblast proliferation but mostly used fixed ultrasound settings and have reportedwidely varying and inconclusive results. Here we critically investigated the effects of the excitation parameters oflow-intensity pulsed ultrasound (LIPUS) stimulation on proliferation of MC3T3-E1 preosteoblastic cells in monolayercultures.

Methods:We used a custom-designed ultrasound exposure system to vary the key ultrasound parameters—intensity,frequency and excitation duration. MC3T3-E1 cells were seeded in 12-well cell culture plates. Unless otherwise specified,treated cells, in groups of three, were excited twice for 10 min with an interval of 24 h in between after cell seeding.Proliferation rates of these cells were determined using BrdU and MTS assays 24 h after the last LIPUS excitation.All data are presented as the mean ± standard error. The statistical significance was determined using Student'stwo-sample two-tailedttests.

Results:Using discrete LIPUS intensities ranging from 1 to 500 mW/cm2(SATA, spatial average-temporal average), wefound that approximately 75 mW/cm2produced the greatest increase in osteoblast proliferation. Ultrasound exposuresat higher intensity (approximately 465 mW/cm2) significantly reduced proliferation in MC3T3-E1 cells, suggesting thathigh-intensity pulsed ultrasound may increase apoptosis or loss of adhesion in these cells.Variation in LIPUS frequency from 0.5 MHz to 5 MHz indicated that osteoblast proliferation rate was not frequencydependent. We found no difference in the increase in proliferation rate if LIPUS was applied for 30 min/day or 10 min/day, indicating a habituation response.

Conclusion:This study concludes that a short-term stimulation with optimum intensity can enhance proliferation ofpreosteoblast-like bone cells that plays an important role in bone formation and accelerated fracture healing, alsosuggesting a possible therapeutic treatment for reduced bone mass.

The dynamics of a drop deforming, orienting and moving in a shear flow of aviscoelastic liquid near a wall is numerically investigated using a front-tracking finite-difference method and a semi-analytic theory. The viscoelasticity is modelled usingthe modified FENE-CR constitutive equation. In a Newtonian system, deformation in adrop breaks the reversal symmetry of the system resulting in a migration away fromthe wall. This study shows that the matrix elasticity reduces the migration velocity, thereduction scaling approximately linearly with viscoelasticity (product of the Deborah number De and the ratio of polymer viscosity to total viscosity β). Similar to a Newtonian system, for small Deborah numbers, the dynamics quickly reaches a quasi-steady state where deformation, inclination, as well as migration and slip velocitiesbecome independent of the initial drop–wall separation. They all approximately scaleinversely with the square of the instantaneous separation except for deformation whichscales inversely with the cube of separation. The deformation shows a non-monotonicvariation with increasing viscoelasticity similar to the case of a drop in an unboundedshear and is found to influence little the change in migration. Two competing effectsdue to matrix viscoelasticity on drop migration are identified. The first stems fromthe reduced inclination angle of the drop with increasing viscoelasticity that tries toenhance migration velocity. However, it is overcome by the second effect inhibitingmigration that results from the normal stress differences from the curved streamlinesaround the drop; they are more curved on the side away from the wall comparedwith those in the gap between the wall and the drop, an effect that is also presentfor a rigid particle. A perturbative theory of migration is developed for small ratioof the drop size to its separation from the wall that clearly shows the migrationto be caused by the image stresslet field due to the drop in presence of the wall. The theory delineates the two competing viscoelastic effects, their relative magnitudes, and predicts migration that matches well with the simulation. Using the simulationresults and the stresslet theory, we develop an algebraic expression for the quasi-steadymigration velocity as a function of Ca, De and β. The transient dynamics of themigrating drop is seen to be governed by the finite time needed for development of theviscoelastic stresses. For larger capillary numbers, in both Newtonian and viscoelasticmatrices, a viscous drop fails to reach a quasi-steady state independent of initialdrop–wall separation. Matrix viscoelasticity tends to prevent drop breakup. Drops that† Email address for correspondence: sarkar@gwu.edu

The stabilizing encapsulation of a microbubble-based ultrasound contrast agent (UCA) critically affectsits acoustic properties. Polymers, which behave differently from materials commonly used (i.e.,lipids or proteins)for monolayer encapsulation, have the potential for better stability and improved control of encapsulation prop-erties. Air-filled microbubbles coated with poly(DL-lactic acid) (PLA) are characterized here usingin vitroacousticexperiments and several models of encapsulation. The interfacial rheological properties of the encapsulation aredetermined according to each model using attenuation of ultrasound through a suspension of microbubbles.Then the model predictions are compared with scattered non-linear (sub- and second harmonic) responses. Forthis microbubble population (average diameter, 1.9mm), the peak in attenuation measurement indicatesa weighted-average resonance frequency of 2.5–3 MHz, which, in contrast to other encapsulated microbubbles,is lower than the resonance frequency of a free bubble of similar size (diameter, 1.9mm). This apparently contra-dictory result stems from the extremely low surface dilational elasticity (around 0.01–0.07 N/m) and the reducedsurface tension of the poly(DL-lactic acid) encapsulation, as well as the polydispersity of the bubble population. Allmodels considered here are shown to behave similarly even in the non-linear regime because of the low surface dila-tional elasticity value. Pressure-dependent scattering measurements at two different excitation frequencies (2.25and 3 MHz) revealed strongly non-linear behavior with 25–30 dB and 5–20 dB enhancements in fundamentaland second-harmonic responses, respectively, for a contrast agent concentration of 1.33mg/mL in the suspension.Sub-harmonic responses are registered above a relatively low generation threshold of 100–150 kPa, with up to 20dB enhancement beyond that pressure. Numerical predictions from all models show good agreement with theexperimentally measured fundamental response, but not with the experimental second-harmonic response. Thecharacteristic features of sub-harmonic responses and the steady response beyond the threshold are matchedwell by model predictions. However, prediction of the threshold value depends on estimated properties and sizedistribution. The variation in size distribution from sample to sample leads to variation in estimates of encapsula-tion properties: the lowest estimated value for surface dilational viscosity better predicts the sub-harmonicthreshold.

A recent study [Katiyar and Sarkar (2011). J. Acoust. Soc. Am.130, 3137–3147] showed that incontrast to the analytical result for free bubbles, the minimum threshold for subharmonic generationfor contrast microbubbles does not necessarily occur at twice the resonance frequency. Hereincreased damping—either due to the small radius or the encapsulation—is shown to shift the mini-mum threshold away from twice the resonance frequency. Free bubbles as well as four modelsof the contrast agent encapsulation are investigated varying the surface dilatational viscosity.Encapsulation properties are determined using measured attenuation data for a commercial contrastagent. For sufficiently small damping, models predict two minima for the threshold curve—one attwice the resonance frequency being lower than the other at resonance frequency—in accord withthe classical analytical result. However, increased damping damps the bubble response more attwice the resonance than at resonance, leading to a flattening of the threshold curve and a gradualshift of the absolute minimum from twice the resonance frequency toward the resonance frequency.The deviation from the classical result stems from the fact that the perturbation analysis employedto obtain it assumes small damping, not always applicable for contrast microbubbles.

Echogenic liposomes (ELIP) are an excellent candidate for concurrent imaging and drug delivery applica-tions. They combine the advantages of liposomes-biocompatibility and ability to encapsulate both hydro-phobic and hydrophilic drugs-with strong reflections of ultrasound. The objective of this study is toperform a detailedin vitroacoustic characterization – including nonlinear scattering that has not beenstudied before – along with an investigation of the primary mechanism of echogenicity. Both componentsare critical for developing viable clinical applications of ELIP. Mannitol, a cryoprotectant, added duringthe preparation of ELIP is commonly believed to be critical in making them echogenic. Accordingly, hereELIP prepared with varying amount of mannitol concentration are investigated for their pressure depen-dent linear and non-linear scattered responses. The average diameter of these liposomes is measured tobe 125–185 nm. But they have a broad size distribution including liposomes with diameters over a micro-meter as observed by TEM and AFM. These larger liposomes are critical for the overall echogenicity.Attenuation through liposomal solution is measured with four different transducers (central frequencies2.25, 3.5, 5, 10 MHz). Measured attenuation increases linearly with liposome concentration indicatingabsence of acoustic interactions between liposomes. Due to the broad size distribution, the attenuationshows a flat response without a distinct peak in the range of frequencies (1–12 MHz) investigated. A15–20 dB enhancement with 1.67lg/ml of lipids is observed both for the scattered fundamental andthe second harmonic responses at 3.5 MHz excitation frequency and 50–800 kPa amplitude. It demon-strates the efficacy of ELIP for fundamental as well as harmonic ultrasound imaging. The scatteredresponse however does not show any distinct subharmonic peak for the acoustic excitation parametersstudied. Small amount of mannitol proves critical for echogenicity. However, mannitol concentrationabove 100 mM shows no effect.

The extracellular enzyme matrix metalloproteinase-9(MMP-9) is overexpressed in atherosclerotic plaques and in metastaticcancers. The enzyme is responsible for rupture of the plaques and forthe invasion and metastasis of a large number of cancers. The ability ofultrasonic excitation to induce thermal and mechanical effects has beenused to release drugs from different carriers. However, the majority ofthese studies were performed with low frequency ultrasound (LFUS) atkilohertz frequencies. Clinical usage of LFUS excitations will be limiteddue to harmful biological effects. Herein, we report our results on therelease of encapsulated contents from substrate lipopeptide incorpo-rated echogenic liposomes triggered by recombinant human MMP-9.The contents release was further enhanced by the application ofdiagnostic frequency (3 MHz) ultrasound. The echogenic liposomeswere successfully imaged employing a medical ultrasound transducer(4−15 MHz). The conditioned cell culture media from cancer cells (secreting MMP-9) released the encapsulated dye from theliposomes (30−50%), and this release is also increased (50−80%) by applying diagnostic frequency ultrasound (3 MHz) for 3min. With further developments, these liposomes have the potential to serve as multimodal carriers for triggered release andsimultaneous ultrasound imaging.

Deformation and sedimentation velocities of a viscoelastic drop falling through a Newtonianmedium are numerically investigated using a front-tracking finite difference method. In contrast toa viscous drop, viscoelasticity deforms an initially spherical drop into an oblate shape and decreasesits sedimentation velocity. Further increase of elasticity results in a dimple at the rear end, as theviscoelastic stress at the trailing end of the drop pulls the drop interface inward. The dimplebecomes more prominent with increasing Deborah number, amount of polymeric viscosity, andcapillary number. An approximate analysis is performed to model the stress development along theaxis of symmetry, specifically its increase at the rear end that governs the dimple formation. Foreven higher values of Deborah number, the interfacial tension cannot balance the viscoelasticstresses leading to an unstable situation toward a toroidal shape. We numerically find the criticalDeborah number for the transition. It shows an approximate inverse scaling with capillary number.For unstable cases, downward progressing dimple develops a globular end. Development of theglobular end results in a sudden increase in the cross-sectional area of the drop and a sharp decreaseof the settling velocity.

Variation of subharmonic response from contrast microbubbles with ambient pressure is numericallyinvestigated for non-invasive monitoring of organ-level blood pressure. Previously, several contrastmicrobubbles bothin vitroandin vivoregistered approximately linear (5–15 dB) subharmonicresponse reduction with 188 mm Hg change in ambient pressure. In contrast, simulated subharmonicresponse from a single microbubble is seen here to either increase or decrease with ambient pressure.This is shown using the code BUBBLESIM for encapsulated microbubbles, and then the underlyingdynamics is investigated using a free bubble model. The ratio of the excitation frequency to the natu-ral frequency of the bubble is the determining parameter—increasing ambient pressure increases nat-ural frequency thereby changing this ratio. For frequency ratio below a lower critical value,increasing ambient pressure monotonically decreases subharmonic response. Above an upper criticalvalue of the same ratio, increasing ambient pressure increases subharmonic response; in between, thesubharmonic variation is non-monotonic. The precise values of frequency ratio for these three differ-ent trends depend on bubble radius and excitation amplitude. The modeled increase or decrease ofsubharmonic with ambient pressure, when one happens, is approximately linear only for certain rangeof excitation levels. Possible reasons for discrepancies between model and previous experiments arediscussed.

Deformation of a viscous drop in shear at finite inertia and the streamlines around itare numerically investigated. Inertia destroys the closed streamlines found in Stokesflow. It creates reversed streamlines and streamlines spiralling around the vorticityaxis. Spiralling streamlines spiral either towards the central shear plane or away fromit depending on the viscosity ratio and the inertia. The zones of open or reversedstreamlines as well as streamlines spiralling towards or away from the central shearplane are delineated for varying viscosity ratio and Reynolds number. In contrast tothe infinite extent of the closed Stokes streamlines around a rigid sphere in shear, theregion of the spiralling streamlines in the vorticity direction both for a rigid sphere anda drop shrinks with inertia. Inertia increases deformation, and introduces oscillations indrop shape. An approximate analysis explains the scaling of oscillation frequency anddamping with Reynolds and capillary numbers. The steady-state drop inclination anglewith the flow axis increases with increasing Reynolds number for small Reynoldsnumber. But it decreases at higher Reynolds number, especially for larger capillarynumbers. For smaller capillary numbers, drop inclination reaches higher than 45?(thedirection of maximum extension), critically affecting the interfacial stresses due tothe drop. It changes the sign of first and second normal interfacial stress differences(and thereby these components of the effective stresses of an emulsion of such drops).Increasing viscosity ratio orients the drop towards the flow axis, which increases thecritical Reynolds number above which the drop inclination reaches more than 45?.

Six models of contrast microbubbles are investigated to determine the excitation threshold for sub-harmonic generation. The models are applied to a commercial contrast agent; its characteristic pa-rameters according to each model are determined using experimentally measured ultrasoundattenuation. In contrast to the classical perturbative result, the minimum threshold for subharmonicgeneration is not always predicted at excitation with twice the resonance frequency; instead itoccurs over a range of frequencies from resonance to twice the resonance frequency. The quantita-tive variation of the threshold with frequency depends on the model and the bubble radius. All mod-els are transformed into a common interfacial rheological form, where the encapsulation isrepresented by two radius dependent surface properties—effective surface tension and surface dila-tational viscosity. Variation of the effective surface tension with radius, specifically having anupper limit (resulting from strain softening or rupture of the encapsulation during expansion), playsa critical role. Without the upper limit, the predicted threshold is extremely large, especially nearthe resonance frequency. Having a lower limit on surface tension (e.g., zero surface tension in thebuckled state) increases the threshold value at twice the resonance frequency, in some cases shiftingthe minimum threshold toward resonance.

Linear stability analysis is performed for a mathematical model of diffusion of gases from an encapsulatedmicrobubble. It is an Epstein–Plesset model modified to account for encapsulation elasticity and finite gaspermeability. Although bubbles, containing gases other than air, are considered, the final stable bubble, ifany, contains only air, and stability is achieved only when the surrounding medium is saturated or over-saturated with air. In absence of encapsulation elasticity, only a neutral stability is achieved for zero sur-face tension, the other solution being unstable. For an elastic encapsulation, different equilibriumsolutions are obtained depending on the saturation level and whether the surface tension is smaller orhigher than the elasticity. For an elastic encapsulation, elasticity can stabilize the bubble. However,imposing a non-negativity condition on the effective surface tension (consisting of reference surface ten-sion and the elastic stress) leads to an equilibrium radius which is only neutrally stable. If the encapsu-lation can support a net compressive stress, it achieves actual stability. The linear stability results areconsistent with our recent numerical findings. Physical mechanisms for the stability or instability of var-ious equilibriums are provided.

Effects of drop and matrix viscoelasticity on the retraction of a sheared drop are numerically investigated.Retraction of an Oldroyd-B drop in a Newtonian matrix is initially faster and later slower with increasingdrop Deborah number. The observed behavior is explained using an ordinary differential equation modelrepresenting the dominant balance between various forces during retraction. The initial faster relaxationof viscoelastic drops is due to viscoelastic stresses pulling the drop interface at the tips inward. The laterslower retraction is due to the slowly-relaxing viscoelastic forces at the equator, where they act againstthe capillary force. The drop inclination decreases substantially during retraction unlike in a Newtoniancase. Matrix viscoelasticity slows the relaxation of a Newtonian drop because of the increasingly slowrelaxation of highly stretched polymers near the drop tip with increasing Deborah number. Increasingthe ratio of polymeric to total viscosity further accentuates the viscoelastic effects in both cases. For anOldroyd-B drop in an Oldroyd-B matrix, a competition between the dispersed and the continuous phaseelasticities, represented by their ratio, determines the dynamics; larger values of the ratio leads again toinitial faster and later slower retraction.

Interactions between a pair of equal-size viscous drops in shear are numerically investigated at finiteReynolds number
Re=0.1–10. At low Reynolds number the simulation compares well with aprevious experimental observation. Apart from the usual pairwise motion where drops driven byshear pass over each other
type I trajectory, finite inertia introduces a new type
type IIofreversed trajectory where drops approaching each other reverse their initial trajectories. The newtrajectory is explained by a reversed streamline pattern observed around a single drop in an imposedshear, and is similar to what is also observed for rigid spheres at finite inertia. However, dropdeformability introduces a nonuniform transition from one to the other type of trajectory—dropsdisplay type I trajectory for high and low capillary numbers and type II for intermediate capillarynumbers. The phenomenon is explained by noting that increasing capillary number gives rise tocompeting effects—while it increases drop deformation and therefore increases resistance to slidingmotion, it also increases drop flexibility, decreases inclination angle, and overall effect of the drop’spresence is reduced, all helping them to slide by. The nonuniform behavior—type II trajectory foran intermediate range of capillary numbers—occurs only at Reynolds number above a critical value.Further increase in Reynolds number increases the range of capillary numbers for type II trajectory.For type I trajectory, terminal cross-stream separation increases linearly with increasing inertiaindicating an enhanced shear induced diffusion. Increasing initial streamwise separation aids inreversed
type IItrajectory due to increased overlap with the reversed streamline zone. Increasingcross-stream distance expectedly facilitates
type Isliding motion. For passing drops
type Itrajectory, terminal cross-stream separation is not appreciably affected by capillary number andinitial drop separation.

Deformation of an Oldroyd B drop in a Newtonian matrix under steady shear is simulated using a fronttracking finite difference method for varying viscosity ratio. For drop viscosity lower than that of thematrix, the long-time steady deformation behavior is similar to that of the viscosity matched system—thedrop shows reduced deformation with increasing Deborah number due to the increased inhibiting vis-coelastic normal stress inside the drop. However for higher viscosity ratio systems, the drop response isnon-monotonic—the steady drop deformation first decreases with increasing Deborah number but abovea critical Deborah number, it increases with further increase in Deborah number, reaching higher thanthe viscous case value for some viscosity ratios. We explain the increase in deformation with Deborahnumber by noting that at higher viscosity ratios, strain rate inside the drop is reduced, thereby reducingthe inhibiting viscoelastic stress. Furthermore, similar to the viscosity matched system, the drop inclina-tion angle increases with increasing Deborah number. A drop aligned more with the maximum stretchingaxis at 45 degree of the imposed shear, experiences increased viscous stretching. With increased ratioof polymeric viscosity to total drop viscosity, the drop deformation decreases and the inclination angleincreases. Our simulation results compare favorably with a number of experimental and computationalresults from other researchers.

Gas diffusion from an encapsulated microbubble is modeled using an explicit linear relation for gaspermeation through the encapsulation. Both the cases of single gas (air) and multiple gases (perfluorocarbon insidethe bubble and air dissolved in surrounding liquid) are considered. An analytical expression for the dissolutiontime for an encapsulated air bubble is obtained; it showed that for small permeability the dissolution time increaseslinearly with decreasing permeability. A perfluorocarbon-filled contrast microbubble such as DefinityÒwas pre-dicted to experience a transient growth because of air infusion before it dissolves in conformity with previousexperimental findings. The growth phase occurs only for bubbles with a critical value of initial mole fraction ofperfluorocarbon relative to air. With empirically obtained property values, the dissolution time of a 2.5-microndiameter (same as that of Definity), lipid-coated octafluoropropane bubble, with surface tension 25 mN/m, is pre-dicted to be 42 min in an air-saturated medium. The properties such as shell permeability, surface tension andrelative mole fraction of octafluoropropane are varied to investigate their effects on the time scales of bubblegrowth and dissolution, including their asymptotic scalings where appropriate. The dissolution dynamics scaleswith permeability, in that when the time is nondimensioanlized with permeability, curves for different permeabil-ities collapse on a single curve. Investigation of bubbles filled with other gases (nonoctafluoropropane perfluoro-carbon and sulfur hexafluoride) indicates longer dissolution time because of lower solubility and lowerdiffusivity for larger gas molecules. For such micron-size encapsulated bubbles, lifetime of hours is possibleonly at extremely low surface tension (,1 mN/m) or at extreme oversaturation.(E-mail:sarkar@udel.edu)Ó2009 World Federation for Ultrasound in Medicine & Biology.

A model for gas transport from an encapsulated microbubble into the surrounding medium is developedand investigated incorporating the effects of encapsulation elasticity. Encapsulation elasticity stabilizesmicrobubbles against dissolution and explains the long shelf life of microbubble contrast agent. We con-sider air bubbles as well as bubbles containing perfluorocarbon gas. Analytical conditions between satu-ration level, surface tension and interfacial dilatational elasticity are determined for attaining non-zeroequilibrium radius for these microbubbles. Numerical solution of the equation verifies the stability ofthe equilibrium radii. In an undersaturated medium all encapsulated bubbles dissolve. In a saturatedmedium, an encapsulated bubble is found to achieve a long-time stable radius when interfacial dilata-tional elasticity is larger than equilibrium surface tension. For bubbles with interfacial dilatational elas-ticity smaller than the equilibrium surface tension, stable bubble of non-zero radius can be achieved onlywhen the saturation level is greater than a critical value. Even if they initially contain a gas other than air,bubbles that reach a stable radius finally become air bubbles. The model is applied to an octafluoropro-pane filled lipid-coated 2.5lm bubble, which displayed a transient swelling due to air intake beforereaching an equilibrium size. Effects of elasticity, shell permeability, initial mole fraction, initial radiusand saturation level are investigated and discussed. Shell permeability and mole fraction do not affectthe final equilibrium radius of the microbubble but affect the time scale and the transient dynamics. Sim-ilarly, the ratio of equilibrium radius to initial radius remains unaffected by the variation in initial radius.

Drops driven toward each other by shear at finite inertia follow two distinct types of trajectories.Type I trajectory is similar to the one in Stokes flow where drops slide past each other. However, atfinite inertia, drops display a new type II trajectory, where they reverse their paths. Increasingviscosity ratio results in a transition from type II to type I trajectory. The transition is caused bydecreased drop deformation and increased alignment with the flow at higher drop viscosity; bothdecrease the zone of reversed streamlines that accompanies a drop at finite inertia. The transition isdelineated in a phase diagram of Reynolds number and viscosity ratio for different capillarynumbers. The critical viscosity ratio, where a type II transitions into type I, increases with Reynoldsnumber except at higher capillary numbers, where the critical viscosity ratio shows a slightnonmonotonic variation with Reynolds number. Also, it is nonmonotonic with capillary numbers inthat for a fixed Reynolds number, the critical viscosity ratio first increases with increasing capillarynumber and then decreases. Similar to the Stokes regime, increased viscosity ratio leads to adecreased postcollision cross-stream separation effectively decreasing the shear induced diffusion.Higher viscosity ratio results in an increased separation between drops during encounter, whichresults in a smaller interaction time. With drops placed initially at different shear planes, drops comeunder the influence of the reversed flow zone around a single drop that broadens off the central shearplane. Consequently, the trajectory changes from type I to type II as the offset in the vorticitydirection increases. The change depends on the initial offset in the shear direction as well. The finaldisplacement in the shear direction varies linearly with the initial offset. The net relativedisplacement in the shear direction shows a gradual decrease with increasing offset. The net relativedisplacement in the vorticity direction with increasing offset first increases from a zero value whendrops are placed at the same shear plane to a maximum and then decreases. For certain cases, itreaches a negative value.

The deformation of a Newtonian/viscoelastic drop suspended in a viscoelastic fluidis investigated using a three-dimensional front-tracking finite-difference method.The viscoelasticity is modelled using the Oldroyd-B constitutive equation. Matrixviscoelasticity affects the drop deformation and the inclination angle with the flowdirection. Numerical predictions of these quantities are compared with previousexperimental measurements using Boger fluids. The elastic and viscous stresses atthe interface, polymer orientation, and the elastic and viscous forces in the domainare carefully investigated as they affect the drop response. Significant change in thedrop inclination with increasing viscoelasticity is observed; this is explained in termsof the first normal stress difference. A non-monotonic change – a decrease followedby an increase – in the steady-state drop deformation is observed with increasingDeborah number (De) and explained in terms of the competition between increasedlocalized polymer stretching at the drop tips and the decreased viscous stretchingdue to change in drop orientation angle. The transient drop orientation angle isfound to evolve on the polymer relaxation time scale for highDe. The breakup ofa viscous drop in a viscoelastic matrix is inhibited for smallDe, and promoted athigherDe. Polymeric to total viscosity ratioβwas seen to affect the result throughthe combined parameterβDeindicating a dominant role of the first normal stressdifference. A viscoelastic drop in a viscoelastic matrix with matched relaxation timeexperiences less deformation compared to the case when one of the phases is viscous;but the inclination angle assumes an intermediate value between two extreme cases.Increased drop phase viscoelasticity compared to matrix phase leads to decreaseddeformation.

The dynamics of a liquid capsule enclosed by an elastic membrane in a shear flow is investigated using a front trackingfinite difference method. We compute deformation, orientation and tank-treading of the capsule, as functions of the forcing(capillary number) and the viscosity ratio for two different membrane constitutive equations – Neo-Hookean and Skalak.The computed results compare very well with those obtained by high-order boundary element methods as well as the smalldeformation perturbation analysis. The simulation shows that a drop and a capsule, even under those circumstances thatresult in the same Taylor deformation criterion for both, attain very different shapes. The tank-treading period even fordifferent capillary numbers as well as capsules with different constitutive laws, is primarily determined by the deformationand the viscosity ratio. At low capillary numbers the simulation predicts buckling due to large compressive stresses on themembrane. However, we show that in shear, unlike in extension, the tank-treading motion can inhibit the buckling insta-bility and gives rise to a stable evolution even in presence of membrane compressive stresses. At large capillary numbers thecapsule experiences large bounded shape followed by tip buckling indicating possible membrane breakup.Ó2008 Elsevier Inc. All rights reserved.

Steady shear rheology of a dilute emulsion with viscoelastic inclusions is numerically investigated using direct numerical simulations. Batchelor’sformulation for rheology of a viscous emulsion is extended for a viscoelastic system. Viscoelasticity is modeled using the Oldroyd-B constitutiveequation. A front-tracking finite difference code is used to numerically determine the drop shape, and solve for the velocity and stress fields. Theeffective stress of the viscoelastic emulsion has three different components due to interfacial tension, viscosity difference (not considered here) andthe drop phase viscoelasticity. The interfacial contributions – first and second normal stress differences and shear stresses – vary with Capillarynumber in a manner similar to those of a Newtonian system. However the shear viscosity decreases with viscoelasticity at low Capillary numbers,and increases at high Capillary numbers. The first normal stress difference due to interfacial contribution decreases with increasing drop phaseviscoelasticity. The first normal stress difference due to the drop phase viscoelasticity is found to have a complex dependence on Capillary andDeborah numbers, in contrast with the linear mixing rule. Drop phase viscoelasticity does not contribute significantly to effective shear viscosityof the emulsion. The total first normal stress difference shows an increase with drop phase viscoelasticity at high Capillary numbers. However atlow Capillary numbers, a non-monotonic behavior is observed. The results are explained by examining the stress field and the drop shape.© 2007 Elsevier B.V. All rights reserved.

The deformation of a viscoelastic drop suspended in a Newtonian fluid subjectedto a steady shear is investigated using a front-tracking finite-difference method.The viscoelasticity is modelled using the Oldroyd-B constitutive equation. The dropresponse with increasing relaxation timeλand varying polymeric to the total dropviscosity ratioβis studied and explained by examining the elastic and viscousstresses at the interface. Steady-state drop deformation was seen to decrease from itsNewtonian value with increasing viscoelasticity. A slight non-monotonicity in steady-state deformation with increasing Deborah number is observed at high Capillarynumbers. Transient drop deformation displays an overshoot before settling downto a lower value of deformation. The overshoot increases with increasingβ.Thedrop shows slightly decreased alignment with the flow with increasing viscoelasticity.A simple ordinary differential equation model is developed to explain the variousbehaviours and the scalings observed numerically. The critical Capillary number fordrop breakup is observed to increase with Deborah number owing to the inhibitiveeffects of viscoelasticity, the increase being linear for small Deborah number.

Two-dimensional simulations of fl ow instability at the interface of a three-layer, density-matched, viscosity-stratifi ed Poiseuille fl ow are performed using a front-tracking/fi nite difference method. This is an extension of the study for the stability of two-layer viscosity-stratifi ed fl ow of Cao et al., Int. J. Multiphase Flow, 30, 1485-1508 (2004). We present results for large-amplitude non-linear evolution of the interface for varying viscos-ity ratio m, Weber number We, and phase difference between the perturbations of the two interfaces. Strong non-linear behaviour is observed for relatively large initial perturbation amplitude. The higher viscosity fl uid is drawn out as a fi nger that penetrates into the lower viscosity layer. The fi nger originates at the crest of the perturbation at the interface. The simulated interface shape compares well with previously reported experiments. Increasing interfacial tension retards the growth rate of the interface as expected, whereas increasing the viscosity ratio enhances it. The sinuous instability appears to evolve faster than the varicose one. For certain fl ow parameters the high-viscosity fi nger displays a bulbous tip, which is also seen in our previously conducted experiments and two-layer results, although it is less pronounced. The low-viscosity intruding fi nger does not display this curious bulbous tip. Drop formation is precluded by the two-dimensional nature of the calculations

Deformation and breakup of a viscous drop in a potential vortex are numericallysimulated. Capillary number, Reynolds number, and viscosity and density ratios arevaried to investigate their effects on the drop dynamics. The vortex locally gives riseto an extensional flow near the drop with the axis of extension rotating at a constantrate, as the drop revolves around the vortex centre. The rotation of the axis plays acritical role in the competing dynamics between the flow-induced stretching and theinterfacial tension. The relation between the rotating extensional flow and a shearflow is explored. For low capillary numbers, a periodic state is reached, where thedrop deforms into an ellipsoidal shape and undergoes steady rotation with a distinctphase lag behind the imposed flow. For density-and-viscosity-matched drops, increasedinterfacial tension results in decreased deformation and reduced phase lag. Increasedinertia promotes deformation. In the presence of inertia, decreasing capillary numberleads to a negative phase lag. The rotation of the extension axis inhibits deformation at low values of the Reynolds number. But at high Reynolds numbers, rotation-induced centrifugal forces promote deformation. At low and high viscosity ratios, an increasein viscosity ratio leads to enhancement and reduction in deformation, respectively. At density ratios larger than unity, the drop deformation displays resonance inthat it varies non-monotonically with a distinct peak with variation of interfacialtension and density ratio. The peak corresponds to the natural frequency of thedrop deformation matching with the frequency of rotation due to the vortex. Asimple physical model is used to explain various observations including asymptoticscalings. We also explore different mechanisms for drop breakup at different Reynoldsnumber, and provide critical capillary numbers as functions of other parameters.In particular, vortex-induced resonance offers an alternative mechanism for size-selective drop breakup. Details of flow fields and transients are also presented and discussed.

A viscous drop deforming in a planar oscillating extensional flow is numerically simulated using afront-tracking finite-difference method. The effects of periodic forcing and interfacial tension arestudied at low but finite inertia. The oscillation leads to decreased deformation and bounded dropshapes for conditions for which steady extension results in drop breakup. The drop displays aresonance phenomenon where the deformation reaches a maximum when the forcing frequencymatches the natural frequency of the drop. The large deformation at resonance indicates a possiblemechanism for size selective breakup by flows with appropriate fluctuation frequency. The detailstructure of the flow at different time instants within a period for various values of interfacial tensionand frequency is investigated. The drop dynamics shows a complex phase relation with the forcingflow. Competition between the inertia-induced dynamic pressure and the viscous stresses leads toboth positive and negative values of the phase and a complex variation with interfacial tension andforcing frequency. A second-order ordinary differential equation model with appropriaterepresentation of the pressure and viscous forces is developed that qualitatively explains the phasebehaviors. For the highest inertia case considered in this papersRe = 10.0d, the drop dynamicsbecomes aperiodic at resonance marked by a strong subharmonic component in the frequencyspectrum.

The relation between the normal stress and the imposed strain for a Newtonian emulsion in anoscillating extensional flow is computed at finite Reynolds numbers using numerically simulated dropgeometry. The interfacial stress was determined using Batchelor’s formalism. In the presence of inertia, the phase between the stress and the strain deviates from Stokes’s flow, and leads to a negative elastic modulus at small frequencies. The results are explained by a mass-spring-dashpot model.

Effects of inertia on the rheology of dilute Newtonian emulsion of drops in shear flow areinvestigated using direct numerical simulation. The drop shape and flow are computed by solvingthe Navier-Stokes equation in two phases using Front-tracking method. Effective stress iscomputed using Batchelor’s formulation, where the interfacial stress is obtained from the simulateddrop shape and the perturbation stress from the velocity field. At low Reynolds number, thesimulation shows good agreement with various analytical results and experimental measurements.At higher inertia deformation is enhanced and the tilt angle of the drop becomes larger thanforty-five degree. The inertial morphology directly affects interfacial stresses. The first and thesecond interfacial normal stress differences are found to change sign due to the change in droporientation. The interfacial shear stress is enhanced by inertia and decreases with capillary numberat lower inertia but increases at higher inertia. The total excess stresses including perturbationstress contribution shows similar patterns

Broadband attenuation measurement has been widely used for characterizing ultrasound contrastagents. Chen et al. (2002) recently suggested that broadband attenuation data depend on the center frequency ofthe broadband excitation pulse and, therefore, that they are not a reliable measure of the bubble behavior. Weinvestigated the suitability of measurement of broadband attenuation as a characterizing tool using the contrastagent Definity®as a test case. Analyzing the attenuation data obtained with three broadband unfocusedtransducers with different center frequencies (2.25, 3.5 and 5 MHz), we found that attenuation is independent ofthe transducer used and matches in the overlap regions of any two transducers. Attenuation does not depend onexcitation pressure amplitude as long as the excitation amplitude remains below a critical value (
0.26 MPa),indicating that the measurement of broadband attenuation below critical excitation can, indeed, be used forcharacterization. Furthermore, the linear relationship of attenuation with concentrations of Definity®is alsoinvestigated.

Zero-thickness interface models are developed to describe the encapsulation of microbubblecontrast agents. Two different rheological models of the interface, Newtonian
viscousandviscoelastic, with rheological parameters such as surface tension, surface dilatational viscosity, andsurface dilatational elasticity are presented to characterize the encapsulation. The models are appliedto characterize a widely used microbubble based ultrasound contrast agent. Attenuation ofultrasound passing through a solution of contrast agent is measured. The model parameters for thecontrast agent are determined by matching the linearized model dynamics with measuredattenuation data. The models are investigated for its ability to match with other experiments.Specifically, model predictions are compared with scattered fundamental and subharmonicresponses. Experiments and model prediction results are discussed along with thoseobtained using an existing modelChurch, J. Acoust. Soc. Am.97, 1510
1995and Hoffet al.,J. Acoust. Soc. Am.107, 2272
2000of contrast agents.

The rheology of a dilute emulsion of viscous drops in an oscillating extensional flow is investigated. Deforming drop shape is computedusing a front tracking finite difference method. Excess stresses due to drops are determined using Bachelor’s formula neglecting drop–dropinteractions. We present and discuss the relations between the excess stress and the applied strain rate. We explore the linear extensionalrheology by computing extensional storage and loss moduli. The effects of frequency and surface tension variations are discussed andcompared with analytical models of Oldroyd and Yu and Bousmina. We find that the nature of the excess interfacial stress depends on therelative magnitudes of the time period of oscillation and the relaxation time of the droplet. The excess stress is predominantly elastic (viscous)if the period is much smaller (larger) than the relaxation time. These phenomena are explained using the detail drop dynamics.

Micron-size bubbles encapsulated by a stabilizing layer of surface-active materials are used inmedical ultrasound imaging and drug delivery. Their destruction stimulated by ultrasoundin vivoplays a critical role in both applications. We investigate the destruction process of microbubbles ina commercially available contrast agent by measuring the attenuation of ultrasound through it. Themeasurement is performed with single-cycle bursts from an unfocused transducer
with a centerfrequency of 5 MHzfor varying pressure amplitudes at 50-, 100-, and 200-Hz pulse repetitionfrequencies
PRFwith duty cycles 0.001%, 0.002%, and 0.004%, respectively. At low excitation,the attenuation is found to increase with time. With increased excitation level, the attenuation leveldecreases with time, indicating destruction of microbubbles. There is a critical pressure amplitude
1.2 MPafor all three PRFs, below which there is no significant bubble destruction. Above thecritical pressure amplitudes the rate of destruction depends on excitation levels. But at high-pressureamplitudes the destruction becomes independent of excitation pressure amplitude. The results areinterpreted to identify two different mechanisms of bubble destruction by its signature inattenuation, namely, slow dissolution by diffusion and catastrophic shell rupture. The differentmodes are discussed in detail with their implications in medical applications

Two-dimensional simulations of flow instability at the interface of a two-layer, density-matched, viscos-ity-stratified Poiseuille flow are performed using a front-tracking/finite difference method. We presentresults for the small-amplitude (linear) growth rate of the instability at small to medium Reynolds numberfor varying thickness ration, viscosity ratiom, and wavenumber. We also present results for large-ampli-tude non-linear evolution of the interface for varying viscosity ratio and interfacial tension. For the linearcase, the interfacial mode is neutrally stable forn¼ffiffiffiffimpas predicted by analysis. The growth rate is pro-portional to Reynolds number for smallRe, and increases with viscosity ratio. The growth rate alsoincreases when the thickness of the more viscous layer is reduced. Strong non-linear behavior is observedfor relatively large initial perturbation amplitude. The higher viscosity fluid is drawn out as a finger thatpenetrates into the lower viscosity layer. The simulated interface shape compares well with previouslyreported experiments. Increasing interfacial tension retards the growth rate of the interface as expected,whereas increasing the viscosity ratio enhances it. Drop formation at the small Reynolds number consid-ered in this study is precluded by the two-dimensional nature of the calculations.

A quantitative model of the dynamics of an encapsulated microbubble contrast agent will be avaluable tool in contrast ultrasound (US). Such a model must have predictive ability for widely varyingfrequencies and pressure amplitudes. We have developed a new model for contrast agents, and successfullyinvestigated its applicability for a wide range of operating parameters. The encapsulation is modeled as acomplex interface of an infinitesimal thickness. A Newtonian rheology with surface viscosities and interfacialtension is assumed for the interface, and a modified Rayleigh–Plesset equation is derived. The rheologicalparameters (surface tension and surface dilatational viscosity) for a number of contrast agents (Albunex,Optisonand Quantison) are determined by matching the linearized model dynamics with experimentallyobtained attenuation data. The model behavior for Optison(surface tension 0.9 N/m and surface dilatationalviscosity 0.08 msP) was investigated in detail. Specifically, we have carried out a detailed interrogation of themodel, fitted in the linear regime, for its nonlinear prediction. In contrast to existing models, the new model isfound to capture the characteristic subharmonic emission of Optisonobserved by Shi et al. (1999). A detailedparametric study of the bubble behavior was executed using the ratio of scattering to attenuation (STAR). Itshows that the encapsulation drastically reduces the influence of resonance frequency on scattering cross-section,suggesting possible means of improvement in imaging at off-resonant frequencies. The predictive capability of thepresent model indicates that it can be used for characterizing different agents and designing new ones.

An inertialess jet containing Newtonian drops in a gravitation-free field has been modeled by incorporating thedetailed micro-structural dynamics. Interactions of the drops with the continuous phase, with the wall, and with thejet surface are accounted for, thereby eliminating the need for additional constitutive assumptions. The deformationof the jet and drops, and the dispersion of drops are predicted. The present paper serves two purposes. First, acomplete description of the modeling process and its limitation is presented. Second, preliminary results for singleand for small clusters of drops indicate the potential inherent in this computational approach. The model flow issolved in two dimensions using the boundary element method (BEM). Effects of different drop parameters, such astheir number, relative position, and interfacial tension are investigated.

In Sarkar & Schowalter (2001), we reported results from numerical simulations of dropdeformation in various classes of time-periodic straining flows at non-zero Reynoldsnumber. As often occurs, analytical solutions provide more e ective understanding ofthe structure and signi cance of a phenomenon. Here we describe drop deformationpredicted from analytical solutions to linear time-periodic straining flows. Threedi erent limiting cases are considered: an unsteady Stokes flow that retains all butthe nonlinear advection terms, a Stokes flow that neglects inertia altogether, and aninviscid potential flow. The rst limit is in clear contrast to the common approachin emulsion literature that resorts almost always to the Stokes flow assumption.The analysis clearly shows the forced{damped mass{spring system underlying thephysical phenomena, which distinguishes it from the inertialess Stokes flow. Thepotential flow also depicts resonance, albeit of an undamped system, and provides animportant limit of the problem. The drop deformation is assumed to be small, and aperturbative approach has been employed. The rst-order problem has been solvedto arrive at either an evolution equation (in Stokes and potential flow limits) or thelong-time periodic drop response (for unsteady Stokes analysis). The analytical resultscompare satisfactorily with those obtained from the numerical simulation in Sarkar &Schowalter (2001), and the resonance characteristics are quantitatively explained. Thethree di erent solutions are compared with each other, and the results are presentedfor di erent parameters such as frequency, interfacial tension, viscosity ratio, densityratio and Reynolds number. Furthermore, the simple ODE model presented in theAppendix of Sarkar & Schowalter (2001) is shown to explain the asymptotic limitsof the present solution.

The kinematics of a potential vortex offers an interesting flow history for a rheologically complex material, andearlier work on that subject led us to consider the behavior of a Newtonian drop in three related time dependentflow fields [K. Sarkar, W.R. Schowalter, Deformation of a two-dimensional drop at non-zero Reynolds number intime-periodic extensional flows: numerical simulation, J. Fluid Mech., 2000, submitted for publication; K. Sarkar,W.R. Schowalter, Deformation of a two-dimensional viscous drop in time-periodic extensional flows: analyticaltreatment, J. Fluid Mech., 2000, submitted for publication]. In the work reported here the drop, characterized by anupper-convected Maxwell model (UCM), is suspended in an incompressible Newtonian fluid. Again, three relatedflows are considered. The first is that of a potential vortex, modeled by an extensional flow field near the drop withrotating axes of stretching. The second is a generalization of the first and is calledrotating extensional(RE) flow,in which the frequency of revolution of the flow is varied independently of the shear rate. Finally, we consideroscillating extensional(OE) flow.Calculations were performed at small but non-zero Reynolds numbers using an ADI front-tracking/finite differ-ence method. We have developed an analytic elastic-viscous stress splitting scheme obtained by an integration byparts of the constitutive equation. The scheme explicitly separates the diffusive part of the momentum equation fora wide range of differential constitutive relations. An ADI implementation is executed for the diffusive part. Weinvestigate the effects of periodicity, Reynolds number and relaxation time on the drop dynamics. For a vortex andan RE flow, the long-time deformation reaches a steady value, and the drop attains a revolving, steady elliptic shape.The long-time values of deformation show complex non-monotonic behavior with variation in Weissenberg number,an effect of the decreased damping and increased elasticity, as well as the presence of a shear wave triggered by theUCM constitutive relation. The first two effects are modeled successfully by a simple ODE presented in AppendixA. The wave effects are briefly discussed in Appendix B.

This paper presents applications of boundary element methods to electrical impe- dance tomography. An algorithm for imaging the interior of a domain that consists of regions of constant conductivity is developed, that makes use of a simpler parametrization of the shapes of the regions to achieve efficiency. Numerical results from tests of this algorithm on synthetic data are presented, and show that the method is quite promising.

Ensemble averaging combined with multiple scattering ideas is applied to the Stokes flow over a stochastic rough surface. The surface roughness is modelled by compact protrusions on an underlying smooth surface. It is established that the effect of the roughness on the flow far from the boundary may be represented by replacing the no- slip condition on the exact boundary by a partial slip condition on the smooth surface. An approximate analysis is presented for a sparse distribution of arbitrarily shaped protrusions and explicit numerical results are given for hemispheres. Analogous conclusions for the two-dimensional case are obtained. It is shown that in certain cases a traction force develops on the surface at an angle with the direction of the flow.

Sarkar K, Prosperetti A 1995 “Effective boundary conditions for Laplace Equation with a Rough Boundary,” Proceedings of the Royal Society of London A, 451, 425-453.

A substantial amount of research on acoustic scattering by underwater bubbles is based on the theory of incoherent scattering. More recent work, devoted to much denser bubble assemblies, has instead used effective-media formulations that presuppose coherent effects. Here the mutual relationship between the two approaches is elucidated. It is shown that, underlying the incoherent results, is a WKB approximate solution of the effective equations. As an application, the scattering by tenuous subsurface bubble layers and acoustical bubble counting techniques are examined. Significant differences with previous results are found.

Sarkar K, Meneveau C 1993 “Gradients of potential fields on rough surface: perturbative calculation of the singularity distribution function f(a) for small surface,” Physical Review E, 47, 957-966.

This paper is a continuation of an earlier one [Prosperetti et al., J. Acoust. Soc. Am. 93, 3117-3127 (1993) ] in which the low-frequency backscattering of sound by hemispherical bubble clouds at the ocean's surface was studied. Here, clouds of various geometrical shapes (spheroids, spherical segments, cones, cylinders, ellipsoids) are considered and results in substantial agreement with the earlier ones and with the experiments of Chapman and Harris [J. Acoust. Soc. Am. 34, 1592-1597 ( 1962)] are found. The implication is that the backscattering levels are not strongly dependent on the shape of the clouds, which strengthens the earlier conclusion that bubble clouds produced by breaking waves can very well be responsible for the unexpectedly high backscattering levels observed experimentally. The accuracy of the Born approximation used by others for similar problems is also examined in the light of the exact results. Significant differences are found for gas concentrations by volume of the order of 0.01% or higher. Finally, shallow nonaxisymmetric plumes are briefly considered