Abstract: Most previous theoretical investigations of gas bubble dynamics have assumed an uncontaminated gas–liquid interface. Recently, however, the potential importance of layers of surface active agents on bubble dynamics has been increasingly recognized. In this work it is assumed that a continuous layer of incompressible, solid elastic material separates the gas from the bulk Newtonian liquid. Elasticity is modeled to include viscous damping. A Rayleigh–Plesset‐like equation describing the dynamics of such surface‐contaminated gas bubbles is derived. The equation predicts that the surface layer supports a strain that counters the Laplace pressure and thereby stabilizes the bubble against dissolution. An analytical solution to this equation which includes both the fundamental and second‐harmonic response is presented. The dispersion relation describing the propagation of linear pressure waves in liquids containing suspensions of these bubbles also is presented. It is found that (1) the resonance frequencies of ...

Abstract: Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure–velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The ‘piston’ is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments.Part 1 of this work studies the fluid films deposited on capillary walls in the limit Ca → 0 (Ca ≡ μU/σ, where μ is the fluid viscosity, U the bubble velocity, and σ the surface tension). Part 2 (Wong et al. 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x. For 1 [Lt ] x [Lt ] Ca−1, the film profile is frozen with a thickness of order Ca2/3 at the centre and order Ca at the sides. For x ∼ Ca−1, surface tension rearranges the film at the centre into a parabolic shape while the film at the sides thins to order Ca4/3. For x [Gt ] Ca−1, the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x ∼ Ca−5/3, the height of the parabola is order Ca2/3. Finally, for x [Gt ] Ca−5/3, the height decreases as Ca1/4x−1/4.

Abstract: This work determines the pressure–velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke & Morris 1995), we find that the drag scales as Ca2/3 in the limit Ca → 0 (Ca μU/σ, where μ is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca2/3. The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary.The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure–velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by σa2/μ, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Qc, whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Qc [Lt ] 10−6. For Qc [Lt ] Q [Lt ] 1, the plug flow dominates, and the gradient in liquid pressure varies with Q2/3. For Q [Lt ] Qc, the corner flow dominates, and the pressure gradient varies linearly with Q. A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media.

Abstract: A technique for measuring the ζ-potential of a gas-liquid interface that overcomes the valid criticisms of previous attempts is proposed and demonstrated for the case of an air bubble in deionized water. The ζ-potential is found to be -65 mV with a reliability exceeding any previously reported results. The apparatus used is a horizontal tube that is rapidly rotating about its axis. The tube contains the liquid and small gas bubbles. An electric field imposed along the tube axis causes the bubble to move and the ζ-potential is related to the electrophoretic mobility.

Abstract: An experimental study was made of the effect of a periodic velocity perturbation on the separation bubble downstream of the sharp-edged blunt face of a circular cylinder aligned coaxially with the free stream. Velocity fluctuations were produced with an acoustic driver located within the cylinder and a small circumferential gap located immediately downstream of the fixed separation line to allow communication with the external flow. The flow could be considerably modified when forced at frequencies lower than the initial Kelvin-Helmholtz frequencies of the free shear layer, and with associated vortex wavelengths comparable to the bubble height. Reattachment length, bubble height, pressure at separation, and average pressure on the face were all reduced. The effects on the large-scale structures were studied on flow photographs obtained by the smoke-wire technique. The forcing increased the entrainment near the leading edge. It was concluded that the final vortex of the shear layer before reattachment is an important element of the flow structure. There are two different instabilities involved, the Kelvin-Helmholtz instability of the free shear layer and the “shedding” type instability of the entire bubble. A method of frequency scaling is proposed which correlates data for a variety of bubbles and supports an analogy with Karman vortex shedding.

Abstract: A small atmospheric bubble was introduced into a surfactant suspension in a captive bubble surfactometer. After film formation to the equilibrium surface tension at the bubble air-liquid interface, th

Abstract: The parameters describing the birth of film droplets originating from bubbles bursting on seawater surfaces are presented. Results are given for bubble sizes Db from 2 to 14.6 mm equivalent volume diameter. It is shown, contrary to earlier reports, that the films of all bubbles with Db up to at least 14.6 mm burst in an orderly manner in which a hole appears at a well-defined location, usually the film's edge, and propagates from there gathering up the film's mass into a toroidal ring as it progresses. This process is enabled because surface tension provides the force required to sustain the centripetal accelerations. Film drops are created when beads, of sufficient size, form along the length of the toroidal ring and surface tension is insufficient to maintain the centripetal accelerations at these accumulation points. Pieces of the ring break loose and leave the toroidal ring along paths tangential to the bubble's cap. It is shown that only bubbles larger than 2.4 mm diameter can launch film droplets by this means and that this begins when the film has rolled up through an angle of about 31° independent of both bubble size and (theoretically) surface tension. Film drop spray patterns recorded on MgO-coated cylindrical shells surrounding the burst bubbles yield film drop numbers and trajectories. In addition, film drop size distributions, their speed of launch, and the speed at which the film opens have been determined as a function of bubble size. The droplet sizes cited here are substantially larger than most previous estimates, and with a high probability, these droplets follow downward trajectories which lead them to impact the surface. A strong inference may be drawn that these impacts give birth to secondary droplets that are smaller than their parents and which have upward velocity components.

Abstract: Experiments and numerical simulations on lateral migration of a single bubble in stagnant liquids and laminar flows were conducted in the present study to examine the effects of the Eotvos number Eo and dimensionless liquid volumetric flux on lateral forces. It was confirmed that (1) a lateral force due to the existence of the wall acts on a bubble and (2) a lift force due to the net circulation of liquid strongly depends on Eo. Empirical models for the wall and lift forces were also proposed and their validity was confirmed by comparing measured and calculated bubble trajectories.

Abstract: A simple model of bubble dynamics is constructed in which effects of thermal conduction and those of evaporation and condensation of water vapor are included. Results of numerical calculations when a bubble is irradiated by an ultrasonic wave reveal that the effect of thermal conduction is considerable. The calculated results fit with the experimental data of the radius–time curve much more satisfactorily than those by a model without thermal conduction. It is also concluded that a bubble transduces the energy of the acoustic wave into heat. Numerical calculations also reveal that the partial pressure of water vapor in a bubble differs considerably from the saturated vapor pressure especially at collapse of the bubble. Connection with sonoluminescence is also discussed.

Abstract: When an air bubble rises in a viscoelastic fluid there is a critical capillary number for cusping and jump in velocity: when the capillary number is below critical, which is about 1 in our data, there is no cusp at the tail of a (smooth) air bubble. For larger volumes, a two-dimensional cusp, sharp in one view and broad in the orthogonal view, is in evidence. Measurements suggest that the cusp tip is in the generic form y = ax2/3 satisfied by analytic cusps. The intervals of volumes for which dramatic changes in air bubble shape take place is very small and the two to ten fold increase in the rise velocity which accompanies the small change of volume could be modelled as a discontinuity. A second drag transition and an orientational transition occurred when U/c > 1 where U is the rise velocity of an air bubble and c is the shear wave speed. For U/c 1 then U is proportional to d and the Deborah number does not change with U. Moreover the bubble shapes when U/c 1 when inertia is dominant. The formation of cusps occurs in all kinds of columns of different sizes and shapes. Cusping is generic but the orientation of the broad edge with respect to the sidewalls is an issue. There is no preferred orientation in columns with round cross-sections, or in the case of walls far away from the rising bubble. In columns with rectangular cross-sections, three relatively stable configurations can be observed: the cusp can be observed in the wide window and the broad edge in the narrow window; the cusp can be observed in the narrow window and the broad edge in the wide window or, less frequently, the broad edge lies along a diagonal. These orientational and drag alternatives are directly analogous to those which are observed in the settling of long or broad solid bodies (Liu & Joseph 1993).

Abstract: The capillary tension developed during drying of a gel is so large that cavitation (homogeneous nucleation of bubbles of vapor) could occur. It is shown that homogeneous nucleation is possible only if a certain characteristic pore size (the breakthrough radius) is small enough to generate capillary tension exceeding a critical value predicted by classical nucleation theory. Since the criterion for bubble growth is the same as for advance of menisci into the drying surface, bubbles can reach macroscopic dimensions only after the critical point of drying (CPD). Heterogeneous nucleation can occur at any time, but growth of the bubbles is still possible only after the CPD is reached. Nucleation is facilitated when: (a) the pore entry radius is much smaller than the pore interior; (b) there is a strong gradient in composition, so that the liquid/vapor surface tension is highest near the drying surface; or (c) there is a spatial gradient in pore size, with the smallest pores near the drying surface. Nitrogen desorption measurements can be used to determine the highest tension reached during drying of a xerogel; in this way it is shown that capillary tension can approach the values necessary to produce homogeneous nucleation.

Abstract: Slag foams have been investigated with smaller bubbles than those used in the previous studies.[5,6,7] The bubbles were generated by argon gas injection with the nozzle of multiple small orifices and by the slag/metal interfacial reaction of FeO in the slag with carbon in the liquid iron. The foam stability in terms of the foam index for a bath-smelting type of slag (CaO-SiO2-Al2O3-FeO) was determined for different bubble sizes. The average diameter of bubbles in the foam was measured by an X-ray video technique. When the foam was generated by the slag/metal interfacial reaction at 1450 °C, it was found that the average bubble diameter varied from less than 1 to more than 5 mm as a function of the sulfur activity in the carbon-saturated liquid iron. The foam index was found to be inversely proportional to the average bubble diameter. A general correlation is obtained by dimensional analysis in order to predict the foam index from the physical properties of the liquid slag and the average size of the gas bubbles in the foam.

Abstract: The structure of the flow around an oblate ellipsoidal bubble of fixed shape is studied by means of direct numerical simulation for Reynolds numbers Re up to 103. In agreement with a previous study by Dandy and Leal [Phys. Fluids 29, 1360 (1986)] the computations demonstrate that if the bubble aspect ratio χ is high enough a standing eddy can exist at the rear of the bubble in an intermediate range of Re. This eddy disappears beyond a certain Reynolds number and it is shown that its existence is governed by the competition between accumulation and evacuation of the vorticity in the flow. The range of Re where the eddy exists increases very rapidly with χ meaning that this structure is certainly present in many experimental situations. The evolution of the drag coefficient with Re reveals that the oblateness has a dramatic influence on the minimum value of Re beyond which Moore’s theory [J. Fluid Mech. 23, 749 (1965)] can be used to predict the rise velocity of a bubble of fixed shape. In contrast, owing to the shape of the vorticity distribution at the surface of the bubble, no noticeable influence of the standing eddy on the drag is found. A quantitative comparison between the present results and those of previous authors shows that the computational description of the boundary layer around curved free surfaces is not a trivial matter since a strong influence of the numerical method is observed.

Abstract: We have investigated encounters between two bubbles rising at high Reynolds number in pure water. The purity of the water was verified by measuring the rise velocity of one bubble. If the equivalent radius is larger than 0.91 mm the trajectorie becomes unstable and the bubble performs a zig-zag motion, most likely caused by viscous effects. Above a critical Weber, based on the approach velocity bubbles were found to bounce in pure water. If their size and initial separation do not exceed a critical value bubbles approach again and coalesce after bouncing, which is in agreement with potential flow calculations. Above a mimimum equivalent radius of ≈ 0.86 mm, which is near the onset of path instability, the behaviour of the bubble pair becomes totally different. The bubbles then perform a symmetrical zig-zag motion with respect to each other, which can not be described by potential flow theory.

Abstract: Bubble nucleation and bubble growth in porous media is an important problem encountered in processes, such as pressure depletion and boiling. To understand its basic aspects, experiments and numerical simulations in micromodel geometries were undertaken. Experiments of bubble growth by pressure depletion were carried out in 2-D etched-glass micromodels and in Hele-Shaw cells. Nucleation of bubbles and the subsequent growth of gas clusters were visualized. Contrary to the bulk or to Hele-Shaw cells, gas clusters in the micromodel have irregular and ramified shapes and share many of the features of an external invasion process (e.g. of percolation during drainage). A pore network numerical model was developed to simulate the growth of multiple gas clusters under various conditions. The model is based on the solution of the convection-diffusions equation and also accounts for capillary and viscous forces, which play an important role in determining the growth patterns. Numerical simulation resulted in good agreement with the experimental results.

Abstract: Sonoluminescence (SL), the phenomenon of light emission associated with the collapse of bubble oscillating under ultrasonic pressure field has been studied by solving the conservation equations for the gas inside bubble analytically. Heat transfer in the liquid layer adjacent to the bubble wall has also been considered in this analysis. It has been found that the gas behavior is neither adiabatic nor isothermal for a bubble under ultrasound conditions. In this analysis, the launch condition and the Hugoniot curve for the shock propagation have been identified, and the shock duration of 2.7 to 17 ps, which is comparable to experimental result, has been obtained with the use of a similarity solution for converging spherical shock. For SL, the gas temperature after the shock focusing has been found to be 7000 K∼ 44000 K, depending on the equilibrium bubble radius and the driving amplitude of ultrasound. It has also been found that the heat flux at bubble collapse is as large as 47 GW/m 2 , which could be mor...

Abstract: This paper describes the use of a transputer-based 8-electrode capacitance tomography system for imaging gas bubbles in a fluidized bed in the vicinity of an air distributor plate. The quantitative results show how the solid concentration distribution varies as a function of time for three different flow regimes: bubbling, slugging and the transition to turbulent. Bubble shape, length and coalescence can be observed.

Abstract: A significant degree of cell damage is observed during suspension cell culture with air sparging. Protective agents can be added to the culture medium to protect the cells from damage. It has been observed that cells tend to adhere to air-medium interfaces and cell damage is mainly due to this cell-bubble interaction; protective additives have been found to prevent this cell adhesion to the bubble surfaces. In this article, it is demonstrated that the interfacial tension between the air and medium is related to the effectiveness of the protective additives to prevent adhesion of cells to this interface. Five different types of additives (Pluronic F-68, Methocels, dextran, Polyvinyl alcohol, and polyethylene glycols) were studied in an effort to determine their protective characteristics. Liquid-vapor interfacial tensions of the culture medium, with and without the additives, were measured by two different techniques (maximum bubble pressure method and Wilhelmy plate method). In addition, visualization techniques showed that in the presence of certain protective additives cells do not adhere to the bubble surface. Results obtained from these experiments indicate that the additives which rapidly lower the liquid-vapor interfacial tension of the culture medium also prevent adhesion of cells to the bubble surface. Experiments have also been conducted to determine the number of cells killed due to bubble rupture, and it was observed that this number is related to the amount of cells adhering to the bubble surface.

Abstract: The dynamics of a large number of bubbles separated by distances of the order of their radii in highly viscous fluids with specific application to foams is investigated. The growth of bubbles is due to diffusion of gas from the fluid and the momentum transfer between the fluid and the bubbles. Equations governing the growth of a single bubble in a shell of fluid containing limited dissolved agas are coupled with the transport equations for the fluid under non-isothermal conditions. The resulting set of equations are solved numerically for a system of bubbles growing along the axial direction in a mold. The results predict a bubble size distribution along the axial direction with large bubbles close to the melt front and smaller bubbles close to the gate, which results in a density distribution in the molded article. Experimental studies on structural foam under nonisothermal conditions are performed. The transient bulk foam density is measured by monitoring the melt front as the foam expands. The predicted values of the foam density are compared with the experimental results and the sources of error are discussed.

Abstract: We have studied the effects of solute, in particular aqueous electrolyte, on bubble formation at capillary orifices (diameters from 50 μm to 1 mm) and frits at varying gas flow rates. Using a stroboscope, video microscope, and rotating mirror, we have obtained pictures which show how bubble formation involves the interaction of bubbles at the orifice. These interactions depend on the value of the surface elasticity E (proportional to c (dγ/dc)2) due to positively (e.g., ethanol) or negatively (e.g., NaCl ) adsorbed solute. At low flow rates consecutive bubbles do not interact. Each bubble detaches and leaves the orifice region before the next one starts forming. At intermediate flow rates the more closely spaced, consecutive bubbles begin to interact. In pure liquids there is no barrier to bubble coalescence and the detached bubble is "fed" by the subsequent bubble as this starts to grow. The process may be repeated several times before the original bubble has risen out of range. In solutions where E is large enough bubble coalescence is inhibited. Instead of "feeding" into the detached bubble the following bubble pushes it aside, and the bubbles appear to "bounce" off each other. Bouncing may give rise to a characteristic sequence of larger and smaller bubbles (often as sidestreams) if the emerging bubbles break off prematurely from the orifice due to the inertia of the original bubble. The transition from feeding to bouncing depends critically on E of the solution and leads to a smaller average bubble size for larger E values. At high flow rates detached bubbles are invariably fed by several subsequent ones, regardless of the surface elasticity. At very high flow rates the bubbling becomes chaotic, but the interaction of bubbles after leaving the orifice area produces smaller bubbles in solutions. In general, bouncing is more likely to occur with narrow and irregular capillaries. The dramatically different appearance of gas-sparged columns in salt water and freshwater has its origin in the differences between assemblies of pores showing mainly feeding (freshwater) or bouncing (salt water).

Abstract: A processor that can execute both CISC and RISC instructions has an integer pipeline and a floating point pipeline. RISC instructions are sent to the floating point pipeline at the beginning of the integer pipeline, but CISC instructions re-align the floating point pipeline. CISC instructions are sent to the floating point pipeline near the end of the integer pipeline to allow the integer pipeline to fetch memory operands for the floating point pipeline. Thus the floating point pipeline relies on the memory operand fetch facilities of the integer pipeline. Complex CISC fetch-operate instructions pass through the integer pipeline first to fetch a floating point operand, and then begin the floating point pipeline for execution of a floating point operation. However, RISC instructions only use register operands and can begin the floating point pipeline earlier, reducing latency until the floating point result is produced. Rapid re-configuration of the pipeline alignment between a pipeline optimized for RISC instructions and one optimized for CISC instructions is possible with muxes and a mode register. Exception handling and pipeline coordination are also described.