An experimental investigation of the motion of long bubbles in inclined tubes

The relative motion of single long air bubbles suspended in a constant liquid flow in inclined tubes has been studied experimentally. Specially designed instrumentation, based on the difference in refractive properties of air and liquid with respect to infrared light, has been constructed and applied to measure bubble propagation rates.A series of experiments were performed to determine the effect of tube inclination on bubble motion with liquid Reynolds and Froude numbers, and tube diameter as the most important parameters.Particular aspects of the flow are described theoretically, and model predictions were found to compare well with observations. A correlation of bubble and average liquid velocities, based on a least squares fit to the data is suggested. Comparisons with other relevant models and data are also presented.     

The motion of long bubbles through viscoelastic fluids in capillary tubes

The penetration of long gas bubble through a viscoelastic fluid in a capillary tube has been studied in order to investigate the influence of viscoelastic material properties on the hydrodynamic coating thickness and local flow kinematics. Experiments are conducted for three tailored ideal elastic (Boger) fluids, designed to exhibit similar steady shear properties but substantially different elastic material functions. This allows for the isolation of elastic and extensional material effects on the bubble penetration process. The shear and extensional rheology of the fluid is characterized using rotational and filament stretching rheometers (FSR). The fluids are designed such that the steady-state extensional viscosity measured by the FSR at a Deborah number (De) greater than 1 differs over three orders of magnitude (Trouton ratio = 10³–10⁶). The experiment set up to measure the hydrodynamic coating thickness is designed to provide accurate data over a wide range of capillary numbers (0.01 < Ca < 100). The results indicate that the coating thickness in this process increases with an increase in the extensionally thickening nature of the fluid. Experiments are also conducted using several different capillary tube diameters (0.1 < D < 1 cm), in order to compare responses at similar Ca but different flow De. Suitable scaling methods and nonlinear viscoelastic constitutive equations are explored to characterize the displacement process for polymeric fluids. Bubble tip shapes at different De are recorded using a CCD camera, and measured using an edge detection algorithm. The influence of the mixed flow field on the bubble tip shape is examined. Particle tracking velocimetry experiments are conducted to compare the influence of viscoelastic properties on the velocity field in the vicinity of the bubble tip. Local shear and extension rates are calculated in the vicinity of the bubble tip from the velocity data. The results provide quantitative information on the influence of elastic and extensional properties on the bubble penetration process in gas-assisted injection molding. The bubble shape and velocity field information provides a basis for evaluating the performance of constitutive equations in mixed flow. Received: 19 January 1999 Accepted: 30 June 1999     

The movement of single large liquid bubbles in vertical porous tubes

The motion of single large liquid bubbles (exhibiting micropolar behaviour) surrounded by a liquid under gravity in closed vertical porous tube has been studied. The effect of slip parameter and porosity parameter has been studied on pressure gradient, velocity distribution for given film thickness. A possible application of the result for detecting the pressure fluctuations of a human body has been suggested.     

On the motion of thin vortex tubes

The motion of a tube of vorticity with a cross sectional radius that is everywhere small compared to local radius of curvature of the tube is considered. In particular, we determine the inviscid motion of the 3D space curve that traces the centerline of the tube for an arbitrary distribution of axial vorticity within the core.     

On the interaction between two consecutive elongated bubbles in a vertical pipe

The motion of elongated air bubbles in a vertical pipe filled with water is studied quantitatively using video imaging of the flow and subsequent digital image processing of the recorded sequence of images. Experiments are carried out to determine the influence of the separation distance between two consecutive bubbles (liquid slug length) upon the behavior of the trailing bubble in vertical slug flow. The details of the trailing bubble acceleration and merging process are observed and the instantaneous parameters of the trailing bubble, such as its shape, velocity, acceleration, etc., are measured as a function of the separation distance. The leading bubble is found to be unaffected by the trailing elongated bubble.     

On the equations of motion of a fluid with suspended particles or gas bubbles

Summary Linear equations have been derived for the motion of a fluid with suspended impurities. The nature of the impurities is characterised by a continuous parameter which varies over a finite interval. It is proved that, under certain assumptions, the velocity fields can be described by a family of potential functions. An application is given by considering a sound wave passing through a water-airbubble mixture.     

On the interaction of Taylor bubbles rising in two-phase co-current slug flow in vertical columns: turbulent wakes

An experimental study on the interaction between Taylor bubbles rising through a co-current flowing liquid in a vertical tube with 32 mm of internal diameter is reported. The flow pattern in the bubble's wake was turbulent and the flow regime in the liquid slug was either turbulent or laminar. When the flow regime in the liquid slug is turbulent (i) the minimum distance between bubbles above which there is no interaction is 5D-6D; (ii) the bubble's rising velocity is in excellent agreement with the Nicklin relation; (iii) the experimental values of the bubble length compare well with theoretical predictions (Barnea 1990); (iv) the distance between consecutive bubbles varied from 13D to 16D and is insensitive to the liquid Reynolds number. When the flow regime in the liquid slug is laminar (i) the wake length is about 5D-6D; (ii) the minimum distance between bubbles above which there is no interaction is higher than 25D; (iii) the bubble's rising velocity is significantly smaller than theoretical predictions. These results were explained in the light of the findings of Pinto et al. (1998) on coalescence of two Taylor bubbles rising through a co-current liquid. Received: 2 February 2000 / Accepted: 15 March 2001     

Countercurrent gas-liquid flow in parallel vertical tubes

The subject of this paper is the flow between an upper reservoir, containing a liquid, and a lower reservoir, containing a gas, interconnected by parallel vertical tubes. The characteristics of the combined system are predicted from a knowledge of the behavior of flow in individual tubes. Numerous modes of possible operation are described analytically and demonstrated experimentally. The effects of system geometry, changes in gas supply characteristics, operating procedure and two-phase flow regimes on the transitions between modes and system stability are presented. Predictions are made for the limiting case of a large number of identical parallel channels.     

On the slow motion of a cluster of bubbles under the combined action of gravity and thermocapillarity

The slow migration of N spherical bubbles under combined buoyancy and thermocapillarity effects is investigated by appealing solely to $3N+1$ boundary-integral equations. In addition to the theory and the associated implementation strategy, preliminary numerical results are both presented and discussed for a few clusters involving 2, 3, 4 or 5 bubbles with a special attention paid to the case of rigid configurations. To cite this article: A. Sellier, C. R. Mecanique 333 (2005).     

The behavior of gas-liquid interfaces in vertical tubes

Experimental studies of interface behavior when a gas flow, confined in a vertical tube, flows past a stationary body of liquid are presented. Critical conditions necessary for the interface to become unstable or break up are investigated. Specific phenomena studied include: penetration of liquid from a reservoir into the top open end of a vertical tube from which gas is emerging, flow of gas past a liquid ring maintained on the inside wall of the tube, conditions for the support of a “hanging film” on the tube wall, formation of droplets and establishment of a continuous upwards-flowing liquid film. A general mathematical formulation of this problem is presented and used to derive the set of relevant dimensionless parameters. Solutions are obtained to certain simple cases and are shown to be consistent with experiment in the limits in which one or more of the variables exerts negligible influence.     

Modelling the initial motion of large cylindrical and spherical bubbles

A two-dimensional, transient, finite difference technique based on a volume fraction specification of the free surface position and accounting for the effects of surface tension is shown to accurately predict the initial motion of large cylindrical and spherical bubbles. The predictions compare very favourably with the experimental data of Walters and Davidson. The initial acceleration of cylindrical and spherical bubbles is properly predicted as g and 2g respectively. The penetration of a tongue of liquid from below is the dominant process by which large deformations from the original shape take place and is well predicted by the model in both cases. For the spherical case the eventual transition into a toroidal bubble is demonstrated and the circulation associated with a rising toroidal bubble as a function of its volume upon release is shown to agree very well with experiments. Iterative linear equation-solving techniques applicable to the special nature of the linear system resulting from such a free surface specification are surveyed and a simple Jacobi iteration based on red-black ordering is found to perform well. The impact of the free surface on the relaxation of the linear system and the convergence criteria is also explored.     

Modeling of the motion of light-weight particles and bubbles in turbulent flows

The aim of the paper is to present a statistical model of motion and dispersion of light-weight particles and bubbles in turbulent flows. The model is based on a kinetic equation for the probability density function of particle velocities. The results of modeling the parameters of particles and bubbles in a plane-channel flow are compared with known data of direct numerical simulation.     

Kinetic model for the motion of compressible bubbles in a perfect fluid

Collective behavior of compressible gas bubbles moving in an inviscid incompressible fluid is studied. A kinetic approach is employed, based on an approximate calculation of the fluid flow potential and formulation of Hamilton's equations for generalized coordinates and momenta of bubbles. Kinetic equations governing the evolution of a distribution function of bubbles are derived. These equations are similar to Vlasov's equations. Conservation laws which are direct consequences of the kinetic system are found. It is shown that for a narrowly peaked distribution function they form a closed system of hydrodynamical equations for the mean flow parameters. The system yields the analogue of Rayleigh–Lamb's equation governing oscillations of bubbles. A variational principle for the hydrodynamical system is established and the linear stability analysis is performed.     

On the pressure of cavitation bubbles

Shock wave emission upon the collapse of a cavitation bubble attached to a rigid wall is investigated using high-speed photography with 200 million frames/s and 5 ns exposure time. At a distance of 68 μm from the bubble wall, the shock pressure is 1.3 ± 0.3 GPa. The shock pressure decays proportionally to r^−1.5 with increasing distance from the bubble. An estimation of the peak pressure at the bubble wall reveals a pressure of about 8 GPa. A major part of the shock wave energy is dissipated within the first 100 μm from the bubble wall.     

An efficient method for the prediction of the motion of individual bubbles

We present a method for the simulation of two-phase flows which can be applied to problems characterised by the presence of up to several hundreds of gas bubbles. The bubble model is kept simple, requiring only six parameters to describe the shape of a single bubble. The model is coupled to a conventional time discrete finite-volume scheme for the solution of the Navier–Stokes equations by the density field which is calculated on basis of the information on the positions and the shapes of the bubbles before each time step. The motion of the bubbles is in turn calculated from an analysis of the computed flow field. Systematical errors due to simplifications are eliminated by the introduction of correction factors. For a selection of fluid dynamical problems, the results of simulations using the method are compared to experimental data. Good quantitative agreement could be found.     

Forced convective film condensation inside vertical tubes

A theoretical study of forced convective film condensation inside vertical tubes is presented. We propose a unified procedure for predicting the pressure gradient and condensation heat transfer coefficient of a vapor flowing turbulently in the core and associated with laminar or turbulent film on the tube wall. The analysis for the vapor flows is performed under the condition that the velocity profiles are locally self-similar. The laminar and turbulent film models equate the gravity, pressure and viscous forces, and consider the effect of interfacial shear. The transition from laminar to turbulent film depends not only on the liquid Reynolds number but also on the interfacial shear stress. In this work we also proposed a new eddy viscosity model which is divided into three regions: the inner region in liquid condensate near the wall; the interface region including both liquid and vapor; and the outer region for the vapor core. Comparisons of the theory with some published experimental data showed good agreement.     

The motion of gas bubbles rising through stagnant liquid

The scope of the present over-all investigation comprises the determination of rise velocity, path, shape and drag characteristics of gas bubbles rising freely in water in terms of bubble size.The material presented in this first report is limited to the measurement of the rise velocity of air bubbles of various sizes and the determination of their paths.A new experimental technique is introduced which consists essentially of measuring the time interval when the bubble traverses two light beams. An electronic system including two phototubes, a pulse amplifier and shaper, and an electronic timer has been used. Also a photographic technique has been used to determine the bubble's size, shape and path.The results show that the type of motion may be predicted from the value of the Reynolds number at which the motion takes place. Five types of motion were observed in the experiments: 1. Rectilinear motion, 2. motion on a helical path, 3. first plane then helical motion, 4. plane motion, 5. rectilinear motion with rocking.The rise velocity of the bubbles which move on those paths varies with the heightz. All the bubbles studied are accelerated to a maximum velocity in a few cm above the capillary tube. For the bubbles smaller than 4.2 mm, this maximum velocity decreased to an asymptotic value far from the capillary tube.

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Zusammenfassung Das Ziel dieser gesamten Untersuchungen ist die Bestimmung der Steiggeschwindigkeit sowie charakteristischer Daten der Bahn, der Form und des Widerstandes von frei in Wasser aufsteigenden Gasblasen in Abhängigkeit von der Blasengröße. Dieser erste Bericht handelt nur von der Messung der Aufstiegsgeschwindigkeit von Luftblasen verschiedener Größe und der Bahnbestimmung.Eine neue, hier angewandte Experimentiertechnik besteht im wesentlichen in der Messung des Zeitintervalls zwischen den Durchgängen der Blase durch zwei Lichtstrahlen. Es wurde ein elektronisches System benutzt, bestehend aus zwei Photozellen, Verstärker, Umformer und Zeitnehmer. Größe, Form und Bahn der Blasen wurde photographisch bestimmt.Die Ergebnisse zeigen, daß die Bewegungsform sich im wesentlichen aus dem Wert der Reynolds-Zahl voraussagen läßt. Fünf verschiedene Formen wurden beobachtet: 1. gradlinige Bewegung, 2. spiralige Bewegung, 3. anfangs ebene, später spiralige Bewegung, 4. ebene Bewegung, 5. gradlinige Schaukelbewegung.Die Aufstiegsgeschwindigkeit längs dieser Bahnen ändert sich mit der Höhe. Alle beobachteten Blasen werden innerhalb einiger Zentimeter oberhalb der Kapillare auf eine Maximalgeschwindigkeit beschleunigt. Für Blasen unter 4,2 mm Durchmesser sinkt diese Maximalgeschwindigkeit auf einen asymptotischen Wert ab, der in größerer Entfernung von der Kapillare erreicht wird.

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Nomenclature C _D Drag coefficient - D _h Helix diameter - R _h Helix radius - R Radius of curvature of the spherical cap - R _e Reynolds number determined in terms of the equivalent bubble diameter (=(vd _e)/ ₁) - d _e Equivalent bubble diameter - g Acceleration of gravity - h Helix pitch - m Mass - r Bubble radius - v Bubble velocity - v _z Bubble rise velocity - Acceleration - ₁ Dynamic viscosity of the liquid - _g Dynamic viscosity of the gas - ₁ Kinematic viscosity of the liquid - ₁ Density of the liquid - _g Density of the gas - Helix slope defined by tg =h/D _h     

Asymptotic analysis of the non-linear behavior of long anisotropic tubes

An asymptotically correct beam model is obtained for a long, thin-walled, circular tube with circumferentially uniform stiffness (CUS) and made of generally anisotropic materials. By virtue of its special geometry certain small parameters cause unusual non-linear phenomena, such as the Brazier effect, to be exhibited. The model is constructed without ad hoc approximations from 3D elasticity by deriving its strain energy functional in terms of generalized 1D strains corresponding to extension, bending, and torsion. Large displacement and rotation are allowed but strain is assumed to be small. Closed-form expressions are provided for the 3D non-linear warping and stress fields, the 1D non-linear stiffness matrix and the bending moment–curvature relationship. In bending, failure could be caused by limit-moment instability, local buckling or material failure of a ply. A procedure to determine the failure load is provided based on the non-linear response, neglecting micro-mechanical failure modes, post-failure behavior, and hygrothermal effects. Asymptotic considerations lead to the neglect of local shell interlaminar and transverse shear stresses for the thin-walled configuration. Results of the theory are illustrated for a few symmetric, antisymmetric angle-ply and unsymmetric layups and show that some previously published theories are not asymptotically correct.