Intermolecular And Surface Forces

Surface-tension-driven flows and, in particular, their tendency to decay spontaneously into drops have long fascinated naturalists, the earliest systematic experiments dating back to the beginning of the 19th century. Linear stability theory governs the onset of breakup and was developed by Rayleigh, Plateau, and Maxwell. However, only recently has attention turned to the nonlinear behavior in the vicinity of the singular point where a drop separates. The increased attention is due to a number of recent and increasingly refined experiments, as well as to a host of technological applications, ranging from printing to mixing and fiber spinning. The description of drop separation becomes possible because jet motion turns out to be effectively governed by one-dimensional equations, which still contain most of the richness of the original dynamics. In addition, an attraction for physicists lies in the fact that the separation singularity is governed by universal scaling laws, which constitute an asymptotic solution of the Navier-Stokes equation before and after breakup. The Navier-Stokes equation is thus continued uniquely through the singularity. At high viscosities, a series of noise-driven instabilities has been observed, which are a nested superposition of singularities of the same universal form. At low viscosities, there is rich scaling behavior in addition to aesthetically pleasing breakup patterns driven by capillary waves. The author reviews the theoretical development of this field alongside recent experimental work, and outlines unsolved problems.

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Nonlinear dynamics and breakup of free-surface flows

Electrohydrodynamics deals with fluid motion induced by electric fields. In the mid 1960s GI Taylor introduced the leaky dielectric model to explain the behavior of droplets deformed by a steady field, and JR Melcher used it extensively to develop electrohydrodynamics. This review deals with the foundations of the leaky dielectric model and experimental tests designed to probe its usefulness. Although the early experimental studies supported the qualitative features of the model, quantitative agreement was poor. Recent studies are in better agreement with the theory. Even though the model was originally intended to deal with sharp interfaces, contemporary studies with suspensions also agree with the theory. Clearly the leaky dielectric model is more general than originally envisioned.

ELECTROHYDRODYNAMICS:The Taylor-Melcher Leaky Dielectric Model

An Introduction to Magnetohydrodynamics

Bubbles in liquids, soft and squeezy objects made of gas and vapour, yet so strong as to destroy any material and so mysterious as at times turning into tiny light bulbs, are the topic of the present report. Bubbles respond to pressure forces and reveal their full potential when periodically driven by sound waves. The basic equations for nonlinear bubble oscillation in sound fields are given, together with a survey of typical solutions. A bubble in a liquid can be considered as a representative example from nonlinear dynamical systems theory with its resonances, multiple attractors with their basins, bifurcations to chaos and not yet fully describable behaviour due to infinite complexity. Three stability conditions are treated for stable trapping of bubbles in standing sound fields: positional, spherical and diffusional stability. Chemical reactions may become important in that respect, when reacting gases fill the bubble, but the chemistry of bubbles is just touched upon and is beyond the scope of the present report. Bubble collapse, the runaway shrinking of a bubble, is presented in its current state of knowledge. Pressures and temperatures that are reached at this occasion are discussed, as well as the light emission in the form of short flashes. Aspherical bubble collapse, as for instance enforced by boundaries nearby, mitigates most of the phenomena encountered in spherical collapse, but introduces a new effect: jet formation, the self-piercing of a bubble with a high velocity liquid jet. Examples of this phenomenon are given from light induced bubbles. Two oscillating bubbles attract or repel each other, depending on their oscillations and their distance. Upon approaching, attraction may change to repulsion and vice versa. When being close, they also shoot self-piercing jets at each other. Systems of bubbles are treated as they appear after shock wave passage through a liquid and with their branched filaments that they attain in standing sound fields. The N-bubble problem is formulated in the spirit of the n-body problem of astrophysics, but with more complicated interaction forces. Simulations are compared with three-dimensional bubble dynamics obtained by stereoscopic high speed digital videography.

Physics of bubble oscillations

The dynamic response of an initially spherical capsule subject to different externally imposed flows is examined. The neo-Hookean and Skalak et al. (Biophys. J., vol. 13 (1973), pp. 245–264) constitutive laws are used for the description of the membrane mechanics, assuming negligible bending resistance. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping-flow conditions are assumed to prevail. The capillary number , beyond the interval of stability, the membrane has two tips along the direction of elongation where the deformation is most severe, and no equilibrium shapes could be identified. For both regions outside the interval of stability, the membrane model is not appropriate and bending resistance is essential to obtain realistic capsule shapes. This pattern persists for the two constitutive laws that were used, with the Skalak et al. law producing a wider stability interval than the neo-Hookean law owing to its strain hardening nature.

Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling

The nonlinear radial oscillations of bubbles that are encapsulated in an elastic shell are investigated numerically subject to three different constitutive laws describing the viscoelastic properties of the shell: the Mooney–Rivlin MR, the Skalak SK, and the Kelvin–Voigt KV models are used in order to describe strain-softening, strain-hardening and small displacement Hookean behavior of the shell material, respectively. Due to the isotropic nature of the acoustic disturbances, the area dilatation modulus is the important parameter. When the membrane is strain softening MR the resonance frequency decreases with increasing sound amplitude, whereas the opposite happens when the membrane is strain hardening SK. As the amplitude of the acoustic disturbance increases the total scattering cross section of a microbubble with a SK membrane tends to decrease, whereas that of a KV or a MR membrane tends to increase. The importance of strain-softening behavior in the abrupt onset of volume pulsations, that is often observed with small insonated microbubbles at moderately large sound amplitudes, is discussed. © 2008 Acoustical Society of America. DOI: 10.1121/1.2909553

Nonlinear radial oscillations of encapsulated microbubbles subject to ultrasound: the effect of membrane constitutive law.

The transient deformation of an axisymmetric capsule freely suspended in a pure straining flow is studied, for sudden or periodic variations of the intensity of the rate of strain. The particle Reynolds number is supposed to be very small and the problem is solved numerically by means of the boundary integral method. In the case of a sudden start of flow, the time response of the capsule can be approximated by an exponential function, and is thus characterized by only two parameters: the equilibrium deformation D∞ and the characteristic response time τs. The respective influence of viscosity ratio, membrane elasticity, and initial particle geometry is analyzed. The dynamic response of the capsule subjected to periodic variations of the rate of strain is also studied. The response time τs appears to be an appropriate parameter to estimate the capsule adaptability to changing flow conditions.

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Transient response of a capsule subjected to varying flow conditions: Effect of internal fluid viscosity and membrane elasticity

It was shown by Rayleigh [Philos. Mag. 14, 184 (1882)] that a conducting spherical drop becomes unstable when the net dimensionless charge on its surface, Qc, exceeds the value of 4(π)1/2 . More recently, Tsamopoulos et al. [Proc. R. Soc. London Ser. A 401, 67 (1985)] have shown, both analytically and numerically, that at this point a transcritical bifurcation occurs. The finite element methodology that they employed is limited to cases where the drop shapes are not very deformed because of truncation problems with mesh representing the infinitely extending surrounding medium. This situation has now been rectified by employing the integral form of Laplace’s equation, which only requires discretization and solution on the surface of the drop. Thus a hybrid method results with the integral equations solved via boundary element techniques, while finite elements are still used for the remaining governing equations. Using this hybrid method, previous results have been reproduced much more accurately and efficiently. In addition, new solution families have been discovered. In particular, several shape families that are not symmetric about the equatorial plane were found to bifurcate from the families of two‐ and four‐lobed shapes. A disjoint family with saddle point shapes was found to extend to small values of charge. It corresponds to the Frankel–Metropolis family that is well known in nuclear physics (Cohen et al. [Ann. Phys. 82, 557 (1974)]). All newly discovered solution families are linearly unstable.

N. Pelekasis

Papers

Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling

Nonlinear radial oscillations of encapsulated microbubbles subject to ultrasound: the effect of membrane constitutive law.

Transient response of a capsule subjected to varying flow conditions: Effect of internal fluid viscosity and membrane elasticity

Equilibrium shapes and stability of charged and conducting drops

IEEE Ultrasonics Symposium Proceedings