Bubble Chain Research Articles

Bubbles' rising dynamics in polymeric solutions

The dynamics of a chain of bubbles rising in polymeric solutions was investigated by stimulus-response simulation, birefringence and particle image velocimetry. Two aspects were identified as central to interactions and coalescence: stress creation by bubble passages and their relaxation. This competition displays complex nonlinear dynamics and can be described by a simple but explicit mathematical model.

Bubbles' rising dynamics in polymeric solutions

The dynamics of a chain of bubbles rising in polymeric solutions was investigated by stimulus-response simulation, birefringence and particle image velocimetry. Two aspects were identified as central to interactions and coalescence: stress creation by bubble passages and their relaxation. This competition displays complex nonlinear dynamics and can be described by a simple but explicit mathematical model.

Anisotropy in the sound field generated by a bubble chain

A vertical chain of rising bubbles represents a transition from individual to continuum behaviour in a compressible gas–liquid flow. Experiments on the distribution of acoustic pressure around a bubble chain revealed a strong anisotropy in the acoustic field in the frequency band generated by individual bubbles. Sound appeared to propagate much more efficiently along the chain than normal to it. A simple theoretical model using a linear coupled-oscillator approximation was developed to explain this result. Although comparison with experiments is inherently qualitative, the model clearly demonstrated the anisotropy. The model also reproduced the change in pulse waveform along the chain. The results suggest that the enhancement of sound intensity along the chain can to some extent be explained by bubbles acting as resonant amplifiers re-transmitting vibrations.

Chain of Bubbles in Polymer Solution

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Chaos in bubbling—nonlinear analysis and modelling

In the present paper, nonlinear features and analytical results for the chaotic bubbling from a submerged orifice are described. A chain of air bubbles was produced from the single orifice of 2 mm in diameter and micro-convection induced by the bubble generation was recorded using hot-probe anemometer located close to the orifice. The air flow rate was varied widely from q=100 to 2000 cc/ min and the aspects of bubbling were observed by high-speed video. The nonlinear analysis is performed for the time series data of hot-probe anemometer especially in the range of q=435– 1500 cc/ min . The calculated largest Lyapunov exponent shows that with increase of air volume flow rate, the time period for the process of liquid flow to lose stability becomes shorter and at high air flow rate such as q=1500 cc/ min , it is shorter than the time period between subsequent bubbles. To explain such chaotic behaviors of bubbling, a simple model has been proposed. The model simulates the process of interaction between the elastic bubble wall and liquid. Simulation results compared well with the analytical results of experimental data. Summarizing, it is concluded that one of the reasons for chaos appearance is the nonlinear character of interaction between an elastic bubble wall and the liquid stream.

Single Bubble Chain Behavior and Liquid Flow Structures in the Vicinity of a Nozzle (Liquid Jet Developments by Single Bubble Chains)

The behavior of single nitrogen gas bubble chains and the structures of flow field in the vicinity of a nozzle were experimentally studied. Both bubble diameters and bubble generation frequencies were precisely controlled independently to be 2.2 mm, and from 1 Hz to 10 Hz (corresponding bubble distance ; 300∼30 mm), respectively, by using innovative acoustic device. Careful investigations of flow field in the vicinity of a nozzle by using Particle Image Velocimetry enabled us to investigate the following two significant physical phenomena. First, The lift forces acting on bubbles in the bubble chains were observed to increase as bubble generation frequency increases, and were largely affected by the preceding bubbles. Second, developments of characteristic liquid jet were identified to play a dominant role in generation of lift forces on each bubble in single bubble chains, and were theoretically investigated by applying the conservation of liquid momentum.

Effects of Generation Frequency and Bubble Diameter on the Behaviors of Single Bubble Chains.

Effects of Generation Frequency and Bubble Diameter on the Behaviors of Single Bubble Chains.

209 単一気泡列中の気泡挙動および周囲液体流れ場構造 : 気泡列による液ジェットの生成

209 単一気泡列中の気泡挙動および周囲液体流れ場構造 : 気泡列による液ジェットの生成

Continuous chain of bubbles in concentrated polymeric solutions

Motion of bubbles in a liquid is a ubiquitous yet very nontrivial phenomenon. The available body of knowledge and even its language connotation consider bubbles only as a discrete, detached from each other, units. It turns out that in concentrated polymer solutions bubbles may form a qualitatively new structure. The experimental study reported below shows that bubbles may form a very stable, continuous, slowly rising, connected long chain similar to beads, or bubble “sausage.”

Resummation of QCD corrections to theηcdecay rate

We examine the ratio of the decay rate of the ${\ensuremath{\eta}}_{c}$ into light hadrons to the decay rate into photons and find that most of the large next-to-leading-order (NLO) correction is associated with running of the strong coupling ${\ensuremath{\alpha}}_{s}.$ We resum such contributions by analyzing final-state chains of vacuum-polarization bubbles. We show that the nonperturbative parts of the bubble chains can be absorbed into a color-octet matrix element, once one has used contour deformations of the phase-space integrals to cancel certain contributions. We argue that these contributions are incompatible with the uncertainty principle. We also argue that perturbation theory is reliable only if one carries out the phase-space integrations before the perturbation summation. Our results are in good agreement with experiment and differ considerably from those that one obtains by applying the scale-setting method of Brodsky, Lepage, and Mackenzie to the NLO result.

Towards the understanding of bubble interactions and coalescence in non-Newtonian fluids: a cognitive approach

The present work provides new insights into the behavior of air bubbles in non-Newtonian fluids. The interactions and coalescence between bubbles rising in non-Newtonian fluids were simultaneously investigated by means of birefringence measurements and particle image velocimetry for a chain of bubbles formed from a submerged orifice. Two aspects are identified for the first time as central to interactions and coalescence: (i) the stress creation by the passage of bubbles, and (ii) their relaxation due to the fluid's memory. This competition displays complex nonlinear dynamics, from periodic phenomena to deterministic chaos. From these fundamental mechanisms, a cognitive model based on behavioral rules has been developed to describe collective behaviors of a group of bubbles. By simulating bubbles as adaptive agents with their fluid via residual stresses, model predictions for consecutive coalescence between a great number of bubbles compare very satisfactorily with the experimental investigation.

On bubbles rising in line

The dynamic behaviour of equal-sized spherical gas bubbles rising in vertical line was studied numerically at Reynolds number ( Re) 50–200. A force-law model was suggested for hydrodynamic interactions within the bubble chain. Three forces acted on each bubble: buoyancy, viscous drag, and inviscid inertia forces. Both local (nearest-neighbour approximation) and non-local (distant effects) interactions between bubbles were considered. The viscous non-local interactions consisted in creation of velocity disturbances by the passage of bubbles and resulted in progressive drag reduction down the chain. Due to the balance between creation and decay of the disturbances, the distant interactions affected only a certain number of the anterior bubbles until the limit chain drag was reached. This drag then applied to all remaining bubbles independent of their positions and the distant effects were thus eliminated. The inviscid non-local interactions consisting in inertial coupling between distant bubbles were found weaker than the viscous non-local interactions and were, therefore, neglected. Two kinds of bubble chain with different boundary conditions were distinguished: the free-end chain and the fixed-end chain. The free-end chain tended to split into smaller bubble groups that subsequently interacted and produced a highly fragmented long-time chain structure. The typical phenomena in bubble interactions were the following: merger, separation, pairing-off, re-pairing, and oscillation. The fixed-end chain allowed for uniform spacing, rose faster than an isolated bubble, and supported propagation of concentration disturbances. Uniform spacing became unstable at low Re and bubbles displayed chaotic behaviour. The results produced by the model were compared with data published in the literature. The model was able to predict and explain the key phenomena observed in experiments. This correspondence was obtained in the nearest-neighbour approximation (the distant effects may not be very strong in real systems). Intensive non-local interactions affected the chain behaviour substantially. The analogy between hydrodynamic and mechanical ‘masses-on-spring’ systems was pointed out.

Validation of a Two-Dimensional Hyperbolic System Modelling a Gas Fluidized Bed

Numerical solutions of a gas fluidized bed model in two space dimensions are presented. This model is hyperbolic and contains particle pressure, but no particle viscosity. The results are compared with experimental data available in the literature for a wide variety of phenomena. Investigated are: the rise velocity of a single, isolated bubble; the frequency of variation of bubble diameter with time; bubble splitting; bubble frequency and the coalescence of a bubble chain formed by gas injected through a single orifice; analysis of the coalescence of bubbles aligned vertically, as well as that of those not in vertical alignment; the formation of slugs in narrow beds; and, eruption at the bed surface. The simulation results show both qualitative and quantitative agreement with the literature.

Evidence for in-line bubble interactions in non-Newtonian fluids

This work reports a study of in-line bubble interactions in non-Newtonian fluids by varying the injection period between bubbles. The experimental data reveal that the bubble rise velocity depends upon the injection period, obviously due to in-line interactions. The original rheological simulation, based on the close relationship between the rise of a chain of bubbles and consecutive shear deformations, proves for the first time that in-line interactions are mainly governed by a dynamical competition between the creation and relaxation of stresses in fluids. This interaction mechanism is clearly corroborated by a newly developed visualisation method based on birefringence. With the decrease of the injection period, such interactions become strong enough and coalescence occurs. The spectral analysis of the bubble passage at different heights in the column demonstrates that the interactions are no longer linear accompanied by successive bifurcations leading to chaotic coalescence.

Air bubble migration in a granular porous medium: Experimental studies

To provide insight into the mechanism of bubble migration in in situ air sparging in a coarse porous medium, the rise of air bubbles through a granular porous medium is studied through video photography enhanced with image analysis. The porous medium is simulated by a fully saturated cylindrical glass column filled with 4‐mm glass beads in random packing order. The experimental procedure for introducing and visualizing a single air bubble is discussed, and the average rise velocity as a function of the emerging bubble radius is directly measured. Compressed air is injected into the base of the column through the use of an electronically controlled solenoid valve. The resulting bubble motion is recorded by camcorder, and still frames are captured and enhanced from the videotape with an image analyzer. The vertical rise of a bubble through the porous medium displays a linear dependence on time. This is the first study to quantify the terminal velocity of an air bubble rising in a stationary porous medium. Comparison is made to experimental data on air bubbles in water and Soltrol obtained using the same video photography and image analysis techniques. Although a porous medium is sometimes represented as a bundle of capillary tubes, an air bubble rising in a granular porous medium is shown to exhibit significantly different behavior from a bubble rising in either a capillary tube or a tube with a diameter comparable to the equivalent diameter of the bubble. The effect of a bubble chain, a series of single bubbles separated vertically from each other by a few centimeters, on air bubble migration in a granular porous medium is also discussed.

The Mechanism of Buffeting Force on Tubes Immersed in Gas-Fluidized Beds.

Key mechanisms relating to buffeting forces on tubes in fluidized beds have been identified by analyzing results from simultaneous measurement of force and pressure on a horizontal tube. Careful experiments have been carried out in a two-dimensional fluidized bed with an accurate sensor tube for single bubble passage, bubble chains and free-bubbling. It is shown that the buffeting force in fluidized beds consists of the net pressure force, i.e., form drag and lift, and additional forces induced by particles become so large for high-velocity free-bubbling beds that it is impossible to estimate the total force only from the net pressure force, while for low velocity beds the total force is mostly accounted for by the net pressure force. The wake particles play an important role for large bubbles, and therefore the details of this component have also been investigated for single bubbles as well as for bubbles undergoing splitting and coalescence.

Foam patterning in porous media

We consider a model of patterning of one-dimensional foam--bubble chain confined in a bamboolike capillary. The discrete model of such a foam describes a distribution of foam films---lamellae that, like ``bridges,'' span a capillary. This model is a kind of Ulam map, which admits many metastable distributions of lamellae in a bamboolike capillary as governing parameters (external pressure drop, lattice parameter, lamella tension, and gas compressibility) overcome certain barriers. In particular, some random distributions of bubble sizes over the chain are suited to solutions of the proposed discrete deterministic model. Randomization of lamella positions speaks in favor of the possibility of the glasslike patterning of foam in a bamboolike capillary. For such ``chaotic'' foam structures, the admissible pressure drop that the bubble chain can sustain, i.e., the so-called start-up, yield pressure drop, rises. We show that the start-up pressure drop depends upon the length of the chain nonlinearly. Only for short chains does it linearly depend upon the number of bubbles in the chain. For infinitely long chains, a saturation effect is observed; i.e., the critical pressure drop becomes independent of the chain length.

The Use of Power Expansions in Quantum Field Theory

Methods of summation of power series relevant to applications in quantum theory are reviewed, with particular attention to expansions in powers of the coupling constant and in inverse powers of an energy variable. Alternatives to the Borel summation method are considered and their relevance to different physical situations is discussed. Emphasis is placed on quantum chromodynamics. Recent applications of the renormalon language to perturbation expansions (resummation of bubble chains) in various QCD processes are reported and the importance of observing the full renormalization-group invariance in predicting observables is emphasized. News in applications of the Borel-plane formalism to phenomenology are conveyed. The properties of the operator-product expansion along different rays in the complex plane are examined and the problem is studied as to how the remainder after subtraction of the first n terms depends on the distance from the Euclidean region. Estimates of the remainder are explicitly calculated and their strong dependence on the nature of the discontinuity along the cut is shown. Relevance of this subject to calculations of various QCD effects is discussed.

Chaotic bubble coalescence in non-Newtonian fluids

This work aims at studying in-line bubble coalescence in non-Newtonian fluids. The visualisation and power spectrum of time series data, recorded via an optical sensing device, confirm that the bubble formation at the orifice is perfectly periodic under a constant gas flowrate. However, the separation interval between bubbles becomes irregular during rise, until, at a certain height above the orifice, the coalescence occurs. An original approach is elaborated by relating the rise of a chain of bubbles to consecutive shear deformations. A series of measurements on a rheometer proves for the first time that the bubble coalescence is mainly governed by the dynamical competition between the creation and relaxation of shear stresses. The time delay embedding method of reconstructing the phase-space diagram is applied to time series data recorded at different heights in the bubble column. The calculation of several parameters: the largest Lyapunov exponent, the correlation dimension, the power spectrum, and the phase portraits, reveals that the coalescence between bubbles obeys a chaotic and deterministic mechanism.

Renormalons beyond one loop

Higher order renormalons beyond the chain of one-loop bubbles are discussed. A perturbation method for the infrared renormalon residue is found. The large order behavior of the current-current correlation function due to the first infrared renormalon is determined in both QED and QCD to the first three orders.