Motion of a bubble in a viscous liquid

The discussion concerns steady-state flow of a viscous fluid around a spherical bubble at small Reynolds number R. Asymptotic matching [1] provides a way of calculating the resistance force, which agrees well with the measured force for R < 5. The rate of growth or dissolution of the bubble is calculated on the assumption that the Péclet number is large.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 107–111, January–February, 1971.We are indebted to V. G. Levich for a discussion.     

Numerical study of bubble rise and interaction in a viscous liquid

In this article, the flow instabilities during the rise of a single bubble in a narrow vertical tube are studied using a transient two-dimensional/axisymmetric model. To predict the shape of the bubble deformation, the Navier-Stokes equations in addition to an advection equation for liquid volume fraction are solved. A modified volume-of-fluid technique based on Youngs' algorithm is used to track the bubble deformation. To validate the model, the results of simulations for terminal rise velocity and bubble shape are compared with those of the experiments. The effect of different parameters such as initial bubble radius, channel height, liquid viscosity and surface tension on the shape and rise velocity of the bubble is investigated.     

Interactions between bubble formation and heating surface in nucleate boiling

The heat transfer and bubble formation is investigated in pool boiling of propane. Size distributions of active nucleation sites on single horizontal copper and steel tubes with different diameter and surface finishes have been calculated from heat transfer measurements over wide ranges of heat flux and selected pressure. The model assumptions of Luke and Gorenflo for the heat transfer near growing and departing bubbles, which were applied in the calculations, have been slightly modified and the calculated results have been compared to experimental investigations by high speed video techniques. The calculated number of active sites shows a good coincidence for the tube with smaller diameter, while the results for the tube with larger diameter describe the same relative increase of the active sites. The comparison of the cumulative size distribution of the active and potential nucleation sites demonstrates the same slope of the curve and that the critical radius of a stable bubble nuclei is smaller than the average cavity size.     

Numerical study of single bubble motion in liquid metal exposed to a longitudinal magnetic field

The paper presents numerical simulations modeling the ascent of an argon bubble in liquid metal with and without an external magnetic field. The governing equations for the fluid and the electric potential are discretized on a uniform Cartesian grid and the bubble is represented with a highly efficient immersed boundary method. The simulations performed were conducted matching experiments under the same conditions so that sound validation is possible. The three-dimensional trajectory of the bubble is analyzed quantitatively and related to the flow structures in the wake. Indeed, the substantial impact of the magnetic field in the bubble trajectory results from its influence on the wake. Quantitative data describing the selective damping of vortex structures are provided and discussed. As a result of applying a longitudinal field, the time-averaged bubble rise velocity increases for large bubbles, it reaches a maximum and then decreases when further increasing the magnetic interaction parameter. For small bubbles, the time-averaged bubble rise velocity decreases when increasing the magnetic field. The bubble Strouhal number as a dimensionless frequency is reduced with the application of a magnetic field for all bubbles considered and the zig–zag trajectory of the bubble becomes more rectilinear.     

Numerical simulation of shock propagation in a polydisperse bubbly liquid

The effect of distributed bubble nuclei sizes on shock propagation in a bubbly liquid is numerically investigated. An ensemble-averaged technique is employed to derive the statistically averaged conservation laws for polydisperse bubbly flows. A finite-volume method is developed to solve the continuum bubbly flow equations coupled to a single-bubble-dynamic equation that incorporates the effects of heat transfer, liquid viscosity and compressibility. The one-dimensional shock computations reveal that the distribution of equilibrium bubble sizes leads to an apparent damping of the averaged shock dynamics due to phase cancellations in oscillations of the different-sized bubbles. If the distribution is sufficiently broad, the phase cancellation effect can dominate over the single-bubble-dynamic dissipation and the averaged shock profile is smoothed out.     

Influence of heat transfer on the dynamic response of a spherical gas/vapour bubble

The standard approach to analyse the bubble motion is the well known Rayleigh–Plesset equation. When applying the toolbox of nonlinear dynamical systems to this problem several aspects of physical modelling are usually sacrificed. Particularly in vapour bubbles the heat transfer in the liquid domain has a significant effect on the bubble motion; therefore the nonlinear energy equation coupled with the Rayleigh–Plesset equation must be solved. The main aim of this paper is to find an efficient numerical method to transform the energy equation into an ODE system, which, after coupling with the Rayleigh–Plesset equation can be analysed with the help of bifurcation theory. Due to the strong nonlinearity and violent bubble motions the computational effort can be high, thus it is essential to reduce the size of the problem as much as possible. In the first part of the paper finite difference, Galerkin and spectral collocation methods are examined and compared in terms of efficiency. In the second part free and forced oscillations are analysed with an emphasis on the influence of heat transfer. In the case of forced oscillations the unstable branches of the amplification diagrams are also computed.     

Dissolution of an ellipsoidal bubble in a slightly viscous liquid

The steady-state velocity, the degree of deformation, and the convective-diffusion-limited rate of quasisteady-state growth (or dissolution) are considered for gas bubbles having shapes close to those of spheres or disks. It is assumed that there are no surface-active substances in the liquid. A qualitative agreement is found between the calculated dissolution rate and the experimental data.Notation a radius of the sphere of equivalent volume - u bubble velocity with respect to the still liquid at infinity - kinematic viscosity of the liquid - liquid density - D gas diffusion coefficient in the liquid - surface tension - g gravitational acceleration - d [R=2au/]-Reynolds number - e [P=2au/D]-Peclet number - f [W=2au²/]-Weber number The author thanks V. G. Levich for a discussion of these results.     

A numerical investigation of electrohydrodynamic (EHD) effects on bubble deformation under pseudo-nucleate boiling conditions

In this article, the electrohydrodynamic (EHD) effects on nucleate boiling are studied by developing a numerical modelling of EHD effect on bubble deformation in pseudo-nucleate boiling conditions. The volume of fluid (VOF) method is employed to track the interface between the gas–liquid two phases; the user-defined code is written and added to the commercial software FLUENT to solve the electric field and the corresponding electric body force. On this basis, the model is applied to study the EHD effects on heat transfer and fluid flows. An initial air bubble surrounded by liquid CCl₄ and attached to a horizontal superheated wall under the action of electric field is studied. The results of the EHD effect on bubble shape evolution are compared with those of available experiments showing good agreement. The mechanism of EHD enhancement of heat transfer and the EHD induced phenomena including bubble elongation and detachment are analyzed in detail.     

Stability of a numerical algorithm for gas bubble modelling

A free-surface-tracking algorithm based on the SOLA-VOF method is analysed for numerical stability when modelling gas bubble evolution in a fluid. It is shown that an instability can arise from the fact that the bubble pressure varies with its volume. A time step stability criterion is introduced which is a function of the natural oscillation period but does not depend on the mesh size. This dependence suggests that the instability is likely to arise in the case of a general motion of a bubble, especially if break-up occurs. The effect is shown using linear Fourier analysis of the discretized equation for radial bubble oscillation and demonstrated numerically using a CFD code FLOW-3D. One- and three-dimensional situations are considered: a bubble in a fluid bounded by two concentric surfaces and a bubble floating in a fluid chamber with and without gravity. In cases where no analytical solution is available, a numerical method for the stability time step limit calculation is suggested based on finding the natural oscillation frequency. The nature of the instability suggests that it can be a feature of any numerical algorithm which models transient fluid flow with a free surface.     

Bubble dynamics in a two-dimensional gas-solid fluidized bed

Related referential studies on gas-solid two-phase flows were briefly reviewed. Bubble ascending in a two-dimensional （2D） gas-solid fluidized bed was studied both experimentally and numerically. A modified continuum model expressed in the conservation form was used in numerical simulation. Solid-phase pressure was modeled via local sound speed; gas-phase turbulence was described by the K-ε two-equation model. The modified implicit multiphase formulation （IMF） scheme was used to solve the model equations in 2D Cartesian/cylindrical coordinates. The bubble ascending velocity and particle motion in the 2D fluidized bed were measured using the photochromic dye activation （PDA） technique, which was based on UV light activation of particles impregnated with the dye. Effects of bed height and superficial gas velocity on bubble formation and ascent were investigated numerically. The numerically obtained bubble ascending velocities were compared with experimental measurements. Gas bubble in jetting gas-solids fluidized bed was also simulated numerically.     

A rising bubble in a polymer solution

We propose a boundary integral method to study the shape of a bubble rising under gravity in a dilute polymer solution. Constitutive properties are modelled using a FENE model [M.D. Chilcott, J.M. Rallison, J. Non-Newtonian Fluid Mech. 29 (1988) 381] with a pure surface tension interface. We employ a birefringent strand representation [O.G. Harlen, J.M. Rallison, M.D. Chilcott, High-Deborah-number flows of dilute polymer, J. Non-Newtonian Fluid Mech.34 (1990) 319–349] of the wake to simulate the shape and the time-dependent motion of the bubble. Steady and non-steady solutions reproduce qualitatively the bubble deformation seen in experiment with a small region of very high curvature near the rear stagnation point of the bubble. We find a limit point for steady axisymmetric solutions if the polymer concentration is increased or the surface tension is decreased. Rise speed jump discontinuities were not found.     

The buoyancy of a gas bubble in a tube filled with a viscous liquid

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 30, No. 6, pp. 98–105, November–December, 1989.     

Quasi-steady deformation of a two-dimensional bubble placed within a potential viscous flow

The shape evolution of a two-dimensional bubble, bounded by a simple closed curve, which is initially placed within a potential viscous flow, is analysed. It is assumed that the influence of gravity and inertia forces is negligible, so the quasi-steady approximation can be applied. Reformulating the problem for Stokes equations with relevant boundary conditions at the free surface in terms of the bianalytic stress-stream function, and using the time-dependent conformal mappingz(,t) of a unit disk onto an unbounded flow domain sought, an infinite system of ordinary differential equations for the Laurent coefficients ofz(,t) is derived. A class of exact solutions is found for the case when the principal part of the complex velocity of the dominant flow at infinity is a polynomial, and the problem of formation of a pointed bubble is discussed.

---

Sommario E analizzata l'evoluzione di una bolla bi-dimensionale, limitata da una curva chiusa semplice, inizialmente posta in un flusso potenziale viscoso. Si assume che l'influenza della gravità e delle forze inerziali sia trascurabile, cosicchè si può applicare l'approssimazione quasi-stazionaria. Riformulando il problema per le equazioni di Stokes con le opportune condizionial contorno sulla superficie libera in termini della funzione stress-stream bianalitica, e usando la tecnica delle trasformazioni conformi dipendenti dal tempoz(,t) di un disco unitario su un dominio di flusso non limitato incognito, viene derivato un sistema infinito di equazioni differenziali ordinarie per i coefficienti di Laurent diz(,t). Viene trovata una classe di soluzioni esatte per il caso in cui la parte principale della velocità complessa del flusso dominante all'infinito è una polinomiale, ed è discusso il problema della formazione di una bolla lenticolare.

     

Hybrid analytical/numerical modeling of nanosecond laser-induced micro-jets generated by liquid confining devices

The generation of micro-jets with pulsed laser irradiation is a key enabling technique for microfluidic devices, printers and needle-free drug injectors. Modeling approaches for such devices are essential to optimize their design and performance. Here we present a hybrid analytical/numerical model to simulate nanosecond laser-induced micro-jets generated by a dual-chamber liquid confining device. The simulated device consists of two chambers; the first one is closed and filled with a propellant liquid and the second is filled with the liquid to be ejected and equipped with a nozzle. Laser-induced cavitation is generated in the first chamber, which is separated by an elastic membrane from the second one, to reduce the thermo-mechanical impact of the absorbed laser energy on the liquid to be ejected. By modifying the generalized form of the Rayleigh–Plesset equation to account for the pressure variation inside the chamber, we show that the geometry of the liquid confining device affects drastically laser-induced bubble dynamics and the resulting jet ejection dynamics. We also demonstrate the effect of the membrane size, laser energy and nozzle size variation on the micro-jet dynamics. We found that such devices can generate micro-jets (velocity: 0.93 m/s to 48.39 m/s) suitable for micro-drop printing (volume: 0.097 nL to 7.68 nL). Although we focused on printing applications, the modeling approach presented here can be widely adapted for designing and optimizing needle-free drug injectors and microfluidic devices.     

Numerical study of flow and heat transfer during a high-speed micro-drop impact on thin liquid films

Numerical simulation of high-speed micro-droplet impingement on thin liquid film covering a heated solid surface has been carried out. Effect of droplet Weber number and liquid film thickness on the characteristics of flow and heat transfer has been investigated using the coupled level set and volume of fluid method. The code is validated against both the experimental and numerical results from the literature. Results show that the crown dynamics is mostly affected by variations in the initial film thickness but is weakly influenced by changes in the Weber number. The liquid within the film can be categorized as three regions based on the heat transfer distribution: the static film region, the transition region, and the impact region. The transient local wall temperature shows three stages: first stage when the temperature decreases rapidly, followed by a second stage in which the temperature starts to rise and then becomes almost constant in the third stage. After drop impact, the local Nusselt number continuously increases until reaching a maximum value, and then decreases approaching the initial impact stage. Our analysis of the change in Weber number shows that larger Weber number contributes to intense temperature variation at the crater core relative to other radial locations. Lastly, the results reveal that the thinner liquid film leads to lower wall temperature and hence, higher average Nusselt number.     

Analysis of the behavior of a bubble in a viscous incompressible liquid by finite element method

Summary The finite element method is used for the analysis of the behavior of a nonspherical bubble in a viscous incompressible liquid under axial conditions. The finite element approximations of the Navier Stokes equations are formulated by taking velocity and pressure as unknown variables. The flow field is discretized by triangular elements, and the bubble surface is represented by isoparametric elements with curved sides. This numerical technique is applied to the simulation of an initially spherical vapor bubble collapsing near a plane solid wall. It is made clear that a jet formed on the bubble is decelerated by the effect of liquid viscosity.

---

Übersicht Mit dem Verfahren der Finiten Elemente wird das Verhalten einer nichtkugelförmigen Blase in einer zähen inkompressiblen Flüssigkeit bei rotationssymmetrischen Bedingungen untersucht. Die Navier-Stokesschen Gleichungen werden mit der Methode der Finiten Elemente näherungsweise gelöst. Das Strömungsfeld um die Blase wird in kleine dreieckige Elemente zerlegt, wobei die Oberfläche der Blase durch isoparametrische Elemente mit gekrümmten Seiten ersetzt wird. Die numerischen Rechnungen werden für den Zusammenbruch einer Blase, die anfangs kugelförmig war, in der Nähe einer ebenen festen Wand durchgeführt. Das Ergebnis zeigt, daß sich die Geschwindigkeit des in der Blase entstehenden Flüssigkeitsstrahls wegen des Einflusses der Zähigkeit vermindert.

     

Effect of bubble size on void fraction fluctuations in dispersed bubble flows

It is known that bubble size affects seriously the average void fraction in bubbly flows where buoyant velocities vary considerably with bubble size. On the contrary, there is no systematic literature report about bubble size effects on the intensity and frequency of void fraction fluctuations around the average void fraction. This work aims to provide such information. An electrical impedance technique is employed along with non-intrusive ring electrodes to register void fraction fluctuations down to 10⁻⁵. Bubble size fluctuations are estimated from high resolution optical images. Experiments are conducted in co-current upward dispersed bubble flow inside a 21 mm tube with average bubble size between ∼50 and ∼700 μm. Water and blood simulant are used as test liquids with velocity from ∼3 to ∼30 cm s⁻¹. The above resemble conditions of Decompression Sickness (DCS) in the bloodstream of human vena cava. It is found that the intensity and frequency of void fraction fluctuations vary appreciably with bubble size at constant gas and liquid flow rates. Moreover, these variations are not random but scale with bubble size. As a first step to quantify this effect, an empirical expression is derived that relates average bubble size to the ratio standard deviation/average value of void fraction.