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.
Zero-thickness interface models are developed to describe the encapsulation of microbubblecontrast agents. Two different rheological models of the interface, Newtonianviscousandviscoelastic, 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, 15101995and Hoffet al.,J. Acoust. Soc. Am.107, 22722000of 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.
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.
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
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 transducerwith a centerfrequency of 5 MHzfor varying pressure amplitudes at 50-, 100-, and 200-Hz pulse repetitionfrequenciesPRFwith 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 amplitude1.2 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