Measurement and modeling on hydrodynamic forces and deformation of an air bubble approaching a solid sphere in liquids

The interaction between bubbles and solid surfaces is central to a broad range of industrial and biological processes. Various experimental techniques have been developed to measure the interactions of bubbles approaching solids in a liquid. A main challenge is to accurately and reliably control the relative motion over a wide range of hydrodynamic conditions and at the same time to determine the interaction forces, bubble–solid separation and bubble deformation. Existing experimental methods are able to focus only on one of the aspects of this problem, mostly for bubbles and particles with characteristic dimensions either below 100 μm or above 1 cm. As a result, either the interfacial deformations are measured directly with the forces being inferred from a model, or the forces are measured directly with the deformations to be deduced from the theory. The recently developed integrated thin film drainage apparatus (ITFDA) filled the gap of intermediate bubble/particle size ranges that are commonly encountered in mineral and oil recovery applications. Equipped with side-view digital cameras along with a bimorph cantilever as force sensor and speaker diaphragm as the driver for bubble to approach a solid sphere, the ITFDA has the capacity to measure simultaneously and independently the forces and interfacial deformations as a bubble approaches a solid sphere in a liquid. Coupled with the thin liquid film drainage modeling, the ITFDA measurement allows the critical role of surface tension, fluid viscosity and bubble approach speed in determining bubble deformation (profile) and hydrodynamic forces to be elucidated. Here we compare the available methods of studying bubble–solid interactions and demonstrate unique features and advantages of the ITFDA for measuring both forces and bubble deformations in systems of Reynolds numbers as high as 10. The consistency and accuracy of such measurement are tested against the well established Stokes–Reynolds–Young–Laplace model. The potential to use the design principles of the ITFDA for fundamental and developmental research is demonstrated.     

Rheology of interfacial layers

The response of interfacial layers to deformations in size and shape depends on their composition. The corresponding main mechanical quantities are elasticity and viscosity of dilation and shear, respectively. Hence, the interfacial rheology represents a kind of two-dimensional equivalent to the traditional bulk rheology. Due to growing interest in the quantitative understanding of foams and emulsions, more works are dedicated to studies on interfacial rheology. This overview presents the theoretical basis for traditional and recently developed experimental tools and discusses their application to different interfacial systems. While dilational rheology provides information on the composition of mixed interfacial layers, the shear rheology gives answers essentially on structures formed at an interface. The most frequently used methods at present are the oscillating drop and bubble tensiometry methods for dilational deformations and oscillating ring/bicone rheometers for shear deformations.     

A bubble breakage model for finite Reynolds number flows

The breakage frequency of bubbles in turbulent liquid flows is modeled as the inverse of the breakage time by Martinez-Bazan et al. [J. Fluid Mech. 401: 157–182; 1999]. In this definition of the breakage frequency, it is assumed that the breakage probability is unity and hence all bubbles will break. This assumption is reasonable in turbulent flows at extremely high Reynolds numbers in which the turbulence energy dissipation is very high. For systems characterized by finite Reynolds numbers the energy dissipation rate decreases rapidly and the breakage probability is reduced significantly. In the present study, the breakage frequency model by Martinez-Bazan et al. has been extended to include the effect that only a fraction of the bubbles breaks at finite Reynolds numbers. For this model extension, an adjusted version of the breakage probability formula proposed by Coulaloglou and Tavlarides [Chem. Eng. Sci. 32: 1289–1297; 1977] was employed. The extended breakage frequency model for finite Reynolds number flows has been evaluated by comparison to recent experimental single bubble breakage data. It can be concluded that extensive experimental analyses are required to gather sufficient experimental data for improved understanding of the physical phenomena and for model validation. In particular, the bubble breakage analysis must be performed simultaneously with the characterization of the local turbulence properties in the flow.     

The thermocapillary migrations of two bubbles in microgravity environment

The thermocapillary motion of two bubbles along their line of centers in a uniform temperature gradient is investigated theoretically. The bubbles are moving in the direction of the temperature gradient. And the interaction between the leading bubble and the trailing one becomes significant as the separation distance between them is decreased greatly so that the bubble interaction is considered in this case. The appropriate equations of momentum and energy are solved using the method of reflections. In order to proceed analytically, sets of transformations between two coordinates are obtained. By using these transformations and the reflection process, accurate migration velocities of these two bubbles in the microgravity environment are derived for the limit of small Marangoni and Reynolds numbers. These results are employed to describe the thermocapillary motion of two bubbles and to estimate the effects of bubble size and the thermal gradient on the interaction between two bubbles. All of our results for the migration of the two bubbles demonstrate that the approach of the second bubble to the first one intensifies the mutual interaction between these two bubbles and yields some interesting thermocapillary motions.     

Gas Holdups of Small and Large Bubbles in a Large-scale Bubble Column with Elevated Pressure

Gas holdups of large bubbles and small bubbles were measured by means of dynamic gas disengagement approach in the pressured bubble column with a diameter of 0. 3 m and a height of 6.6 m. The effects of su-perficial gas velocity, liquid surface tension, liquid viscosity and system pressure on gas holdups of small bub-bles and large bubbles were investigated. The holdup of large bubbles increases and the holdup of small bub-bles decreases with an increase of liquid viscosity. Meanwhile, the holdup of large bubbles decreases with in-creasing the system pressure. A correlation for the holdup of small bubbles was obtained from the experimen-tal data.     

On the Viscosity of Dilute Emulsions

The slow motion of a liquid droplet in a shear flow in the presence of surfactants is studied. The effects of the interfacial viscosity, Gibbs elasticity, surface diffusion and bulk diffusion of surfactants in both phases are taken into account. The analytical solution of the problem for small Reynolds and Peclet numbers gives a simple criterion for estimation of the tangential mobility of the droplet interface. By applying the standard procedure for averaging of the stress tensor flux at an arbitrary surface of the dilute emulsion, an analytical formula for the viscosity of emulsions in the presence of surfactants is derived. The result is a natural generalization of the well-known formula of Einstein for the viscosity of monodisperse dilute suspensions and of the expressions derived by Taylor and Oldroyd for the viscosity of monodisperse dilute emulsions taking into account the Marangoni effect. Copyright 2001 Academic Press.     

Strain-induced yielding in bubble clusters

We study how shearing clusters of two or four bubbles induces bubble separation or topological rearrangement. The critical deformation at which this yielding occurs is measured as a function of shear rate, liquid composition, and liquid content in the cluster. We establish a geometrical yield criterion in the quasistatic case on the basis of these experimental data as well as simulations. In the dynamic regime, the deformation where the cluster yields increases with the strain rate, and we derive a scaling law describing this phenomenon based on the dynamical inertial rupture of the liquid meniscus linking the two bubbles. Our experiments show that the same scaling law applies to two- and four-bubble clusters.     

Asymmetric deformation of bubble shape: cause or effect of vortex-shedding?

Two perpendicular projections of rising bubbles were observed in counter-current downstream diverging flow. Evidently, the bubbles did not enter the boundary layer at the channel wall and a plug liquid flow assumption was acceptable in our experimental equipment. This confirmed that the experiment was appropriate for simulation of bubble rises in a quiescent liquid column. Recent data obtained by a high-speed camera permitted recording over a period of 60 s. Image analysis by a tailor-made program provided a time-series of quantities related to the position, size, and shape of bubbles. In addition to determination of the aspect ratio of the equivalent oblate ellipsoid, deviation from this shape was investigated in respect of the difference between the bubble’s centre of mass and the geometrical centre of bubble projection. Autocorrelation of the data indicated that the bubble inclination oscillated harmonically with a frequency of 5–10 Hz; cross correlation showed that the horizontal shift of the centre of mass, as well as the horizontal velocity, increased with increasing bubble inclination, and the vertical shift of the centre of mass increased with an increases in the absolute value of the bubble inclination. There is no significant phase shift in the oscillation of these quantities. The bulky bottom side of the bubbles is in accordance with the model of bubble oscillation induced by instability of the equilibrium of gravity and surface tension forces. The oscillation frequency dependence on surface forces (Eötvös number) is evident, while viscosity does not play a significant role in low-viscosity liquids. Therefore, vortex-shedding is more likely to be an effect of the oscillation and not its cause.     

On the influence of the surfactant''s polar group on the local and terminal velocities of bubbles

The local and the terminal velocities, the size and the degree of bubbles’ shape deformations were determined as a function of distance from the position of the bubble formation (capillary orifice) in solutions of n-octyltrimethylammonium bromide, n-octyldimethylphosphine oxide, n-octyl-β-D-glucopyranoside and n-octanoic acid.

These surface-active compounds have different polar groups but an identical hydrocarbon chain (C8) in the molecule. The motion of the bubbles was monitored and recorded using a stroboscopic illumination, a CCD camera, and a JVC professional video. The recorded bubble images were analyzed by the image analysis software. The bubbles accelerated rapidly and their shape was deformed immediately after detachment from the capillary. The extent of the bubbles’ shape deformation (ratio of horizontal and vertical diameters) was 1.5 in distilled water and dropped rapidly down to a level of ca. 1.05–1.03 with increasing surfactant concentration. After the acceleration period the bubbles either attained a constant value of the terminal velocity (distilled water and high concentrations of the solutions), or a maximum in the velocity profiles was observed (low concentrations). The values of the terminal velocity diminished drastically with increasing concentration, from the value of 35 cm/s in water down to about 15 cm/s, while the bubble diameter decreased by ca. 10% only. The surfactant adsorption at the surface of the bubbles was evaluated and the minimum adsorption coverages required to immobilize the bubbles’ surface were determined. It was found that this minimum adsorption coverage was ca. 4% for n-octyldimethylphosphine oxide, n-octyl-β-D-glucopyranoside, n-octanoic acid and 25% for n-octyltrimethylammonium bromide. The difference in the adsorption coverage together with the surfactants’ surface activities indicate that it is mainly the adsorption kinetics of the surfactants that governs the fluidity of interfaces of the rising bubbles.     

Theory of non-equilibrium force measurements involving deformable drops and bubbles

Over the past decade, direct force measurements using the Atomic Force Microscope (AFM) have been extended to study non-equilibrium interactions. Perhaps the more scientifically interesting and technically challenging of such studies involved deformable drops and bubbles in relative motion. The scientific interest stems from the rich complexity that arises from the combination of separation dependent surface forces such as Van der Waals, electrical double layer and steric interactions with velocity dependent forces from hydrodynamic interactions. Moreover the effects of these forces also depend on the deformations of the surfaces of the drops and bubbles that alter local conditions on the nanometer scale, with deformations that can extend over micrometers. Because of incompressibility, effects of such deformations are strongly influenced by small changes of the sizes of the drops and bubbles that may be in the millimeter range. Our focus is on interactions between emulsion drops and bubbles at around 100 μm size range. At the typical velocities in dynamic force measurements with the AFM which span the range of Brownian velocities of such emulsions, the ratio of hydrodynamic force to surface tension force, as characterized by the capillary number, is ~ 10^{− 6} or smaller, which poses challenges to modeling using direct numerical simulations. However, the qualitative and quantitative features of the dynamic forces between interacting drops and bubbles are sensitive to the detailed space and time-dependent deformations. It is this dynamic coupling between forces and deformations that requires a detailed quantitative theoretical framework to help interpret experimental measurements. Theories that do not treat forces and deformations in a consistent way simply will not have much predictive power. The technical challenges of undertaking force measurements are substantial. These range from generating drop and bubble of the appropriate size range to controlling the physicochemical environment to finding the optimal and quantifiable way to place and secure the drops and bubbles in the AFM to make reproducible measurements. It is perhaps no surprise that it is only recently that direct measurements of non-equilibrium forces between two drops or two bubbles colliding in a controlled manner have been possible. This review covers the development of a consistent theory to describe non-equilibrium force measurements involving deformable drops and bubbles. Predictions of this model are also tested on dynamic film drainage experiments involving deformable drops and bubbles that use very different techniques to the AFM to demonstrate that it is capable of providing accurate quantitative predictions of both dynamic forces and dynamic deformations. In the low capillary number regime of interest, we observe that the dynamic behavior of all experimental results reviewed here are consistent with the tangentially immobile hydrodynamic boundary condition at liquid–liquid or liquid–gas interfaces. The most likely explanation for this observation is the presence of trace amounts of surface-active species that are responsible for arresting interfacial flow.     

Comparison of simple and pure shear for an incompressible isotropic hyperelastic material under large deformation

The concepts of simple and pure shear are well known in continuum mechanics. For small deformations, these states differ only by a rotation. However, correlations between them are not well defined in the case of large deformations. The main goal of this study is to compare these two states of deformation by means of experimental and theoretical approaches. An incompressible isotropic hyperelastic material was used. The experimental procedures were performed using digital image correlation (DIC). The simple shear deformation was obtained by single lap joint testing, while the pure shear was achieved by means of planar tension testing. Classical hyperelastic constitutive equations available in the literature were used. As a consequence, the results indicate that simple shear cannot be considered as pure shear combined with a rotation when large deformation is assumed, as widely considered in literature.     

Formation of droplets and bubbles in a microfluidic T-junction-scaling and mechanism of break-up

This article describes the process of formation of droplets and bubbles in microfluidic T-junction geometries. At low capillary numbers break-up is not dominated by shear stresses: experimental results support the assertion that the dominant contribution to the dynamics of break-up arises from the pressure drop across the emerging droplet or bubble. This pressure drop results from the high resistance to flow of the continuous (carrier) fluid in the thin films that separate the droplet from the walls of the microchannel when the droplet fills almost the entire cross-section of the channel. A simple scaling relation, based on this assertion, predicts the size of droplets and bubbles produced in the T-junctions over a range of rates of flow of the two immiscible phases, the viscosity of the continuous phase, the interfacial tension, and the geometrical dimensions of the device.     

Shear-Induced "Melting" of an Aqueous Foam

We present diffusing-wave spectroscopy measurements of bubble dynamics in a continuously sheared aqueous foam. At slow strain rates, isolated clusters of bubbles intermittently rearrange from one solidly packed configuration to another, even though the macroscopic flow appears continuous. At fast rates, bubbles instead move smoothly and continuously throughout the entire foam. In other words, shear flow that appears macroscopically laminar is similarly laminar down to the bubble scale; effectively the foam "melts." The crossover to this regime can be understood in terms of elastic energy accumulation and viscous dissipation mechanisms. In particular, the strain rate needed for shear-induced melting to occur is set by the ratio of a yield strain to the rearrangement event duration. To explore the implications for macroscopic flow, we compare these bubble-scale dynamics directly with viscosity measurements. Copyright 1999 Academic Press.     

Electric phenomena in multi-bubble cavitation fields

The processes of formation and accumulation of electric charges in the splitting and deformation of cavitation bubbles in an ultrasonic wave field are considered in terms of the local electrification theory. The influence of different factors on the electrification of the bubble-liquid interface is discussed. It is established that, in the splitting of a cavitation bubble and, possibly, in its deformation, the local field strength near the bubble surface dramatically depends on the radius of the neck formed in the bubble. It is shown that, although the stationary concentration of cavitation bubbles may be very high (～10⁴–10⁵ cm^?3), the probability for several deformed cavitation bubbles of “required size” to emit luminescence at a given instant of time depends on the ultrasound intensity and other test conditions, a conclusion supported by experimental data.     

A finite-temperature density functional study of electron self-trapping in 3He and 4He

We introduce a compact finite-temperature density functional model to study electron self-trapping in both liquid and vapor (3)He and (4)He. This model can quantitatively reproduce the most essential thermodynamic properties of (3)He and (4)He along their liquid-vapor coexistence lines. The structures and energetics of self-trapped electron bubbles on the 1S ground state and 1P excited state are particularly investigated. Our results show that 1S and 1P bubbles exist in liquid at any temperature, whereas 1S bubbles exist in vapor only above 1.6 K in (3)He and above 2.8 K in (4)He, 1P bubbles exist in vapor only above 2.5 K in (3)He and 4.0 K in (4)He. An initially spherical 1P bubble is unstable against deformation towards a peanut shape. In liquid, a peanut-shaped 1P bubble is held from fission by surface tension until reaching the liquid-vapor critical point, whereas in vapor it always splits into two smaller bubbles. The existence of 1P bubbles in finite-temperature liquid helium and their fission instability in helium vapor reveal interesting physics in this system.     

Thermocapillary Alignment of Gas Bubbles Induced by Convective Transport

The effect of a weak convective heat transfer on the thermocapillary interaction of two bubbles with an arbitrary orientation relative to an externally imposed temperature gradient is examined. Asymptotic analysis of the case of large separation distances, Z, suggests that the corrections to the bubbles' velocities are of (Pe/Z²), rather than (Pe²) previously found for an isolated bubble. Equal-sized bubbles are known to move with the same velocities, as if they were isolated, when heat conduction is the only transport mechanism. However, the convective transport results in a relative motion of the bubbles. The tendency of equal bubbles to line up in a plane perpendicular to the applied thermal gradient is shown analytically in the weakly nonlinear limit of small Pe numbers, and an interesting interaction behavior in the case of unequal bubbles is discussed.     

Bubbles no more: in-plane trapping and removal of bubbles in microfluidic devices

Gas bubbles present a frequent challenge to the on-chip investigation and culture of biological cells and small organs. The presence of a single bubble can adversely impair biological function and often viability as it increases the wall shear stress in a liquid-perfused microchannel by at least one order of magnitude. We present a microfluidic strategy for in-plane trapping and removal of gas bubbles with volumes of 0.1-500 nL. The presented bubble trap is compatible with single-layer soft lithography and requires a footprint of less than ten square millimetres. Nitrogen bubbles were consistently removed at a rate of 0.14 μL min(-1). Experiments were complemented with analytical and numerical models to comprehensively characterize bubble removal for liquids with different wetting behaviour. Consistent long-term operation of the bubble trap was demonstrated by removing approximately 4000 bubbles during one day. In a case study, we successfully applied the bubble trap to the on-chip investigation of intact small blood vessels. Scalability of the design was demonstrated by realizing eight parallel traps at a total removal rate of 0.9 μL min(-1) (measured for nitrogen).     

Liquid crystalline cellulose derivative elastomer films under uniaxial strain

Mesogenic cellulose derivative chains cross-linked into free-standing thin films were prepared by a shear-casting technique from anisotropic precursor solutions of thermotropic (acetoxypropyl)cellulose. After shear cessation a macroscopically oriented serpentine structure with the director in average along the shear direction is locked resulting in anisotropic optical and mechanical properties of the material. These films were submitted to an external uniaxial mechanical field perpendicular and parallel to the shear direction. Stretching perpendicular to the shear direction produced significant director rotations and a reset of order of the director order parameter for a deformation in the range 2–3 as detected by X-rays and optical microscopy. The different response found for strains imposed parallel and perpendicular to the initial average director orientation indicates that even though our system shows a serpentine director modulation that is either attenuated or reinforced by deformations parallel or perpendicular to the shear direction, its behaviour is similar to theoretical predictions for monodomain nematic elastomers described in the literature.