Effect of viscoelastic properties of a liquid on the dynamics of small oscillations of a gas bubble

A series of papers has been devoted to questions of gas bubble dynamics in viscoeiastic liquids. Of these papers we mention [1–4]. The radial oscillations of a gas bubble in an incompressible viscoeiastic liquid have been studied numerically in [1, 2] using Oldroyd's model [5]. Anexact solution was found in [3], and independently in [4], for the equation of small density oscillations of a cavity in an Oldroyd medium when there is a periodic pressure change at infinity. The analysis of bubble oscillations in a viscoeiastic liquid is complicated by properties of limiting transitions in the rheological equation of the medium. These properties are of particular interest for the problem under investigation. These properties are discussed below, and characteristics of the small oscillations of a bubble in an Oldroyd medium are investigated on the basis of a numerical analysis of the exact solution obtained in [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 82–87, May–June, 1976.The authors are grateful to V. N. Nikolaevskii for useful advice and for discussing the results.     

Analysis of surface tension driven leveling in viscoelastic films

We study the surface tension driven leveling of surface irregularities in viscoelastic films deposited on a flat surface. Analytical results are presented for the generalized Maxwell model in the limit of infinitesimal surface disturbances. The results show that elasticity retards the leveling process. In some cases, the retardation effect is dramatic. Keunings' finite element algorithm for solving viscoelastic moving boundary problems is used to analyze the case of finite-amplitude surface disturbances. Results for an Oldroyd B fluid display significant elastic effects which are consistent with the asymptotic predictions.     

Surface tension driven flows in micro-gravity conditions

The importance of convective flows generated by surface tension gradients, in comparison with the ones generated by other driving forces, has been investigated in connection with space technological applications involving fluid processes. A theoretical model of the boundary conditions at the interface, considered free and diffusive, has been derived in general tensor form to allow for the use of non orthogonal curvilinear co-ordinates. For the study of flow fields contained in enclosures, these co-ordinates are more suitable to fit all teh boundaries, in particular near the contact angle between the interface and the solid walls, thus giving more accurate numerical solutions. A computational procedure to solve the complete set of bulk and surface equations is proposed and applied to a simplified two dimensional flow in a rectangular enclosure with a temperature gradient between the lateral walls. The numerical results show the importance of considering the interface to be deformable and diffusive for an accurate evaluation of the convective flow in the fluid bulk.     

Forced oscillations of a gas bubble in a spherical volume of a compressible liquid

A spherically symmetric problem of oscillations of a single gas bubble at the center of a spherical flask filled with a compressible liquid under the action of pressure oscillations on the flask wall is considered. A system of differential-difference equations is obtained that extends the Rayleigh-Plesset equation to the case of a compressible liquid and takes into account the pressure-wave reflection from the bubble and the flask wall. A linear analysis of solutions of this system of equations is performed for the case of harmonic oscillations of the bubble. Nonlinear resonance oscillations and nearly resonance nonharmonic oscillations of the bubble caused by harmonic pressure oscillations on the flask wall are analyzed. Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 111–118, March–April, 1999.     

Nonlinear analysis of the surface tension driven breakup of viscoelastic filaments

The surface tension driven breakup of viscoelastic filaments into droplets is qualitatively different from that of Newtonian liquid filaments. Disturbances on filaments of dilute polymer solutions often grow to a configuration consisting of nascent droplets connected by a thin ligament; the breakup time for this configuration is much longer than that predicted by extensions of Rayleigh's linear stability theory. We present here a nonlinear analysis of surface tension driven breakup of viscoelastic filaments using two complementary approaches that given equivalent results: a transient finite element solution and a one-dimensional thin filament approximation. We show that significant nonlinear effects lead to the experimentally-observed nascent droplet-ligament configuration, and we predict the entire evolution of the filament profile. Agreement with available experimental data for profile evolution and breakup of jets of Newtonian fluids and dilute polymer solutions is excellent.     

Free nonlinear oscillations of a thermally relaxing spherical gas bubble in an incompressible liquid

The quasi-adiabatic regime of free oscillation of a bubble in the presence of irreversible interphase heat transfer between the bubble and the ambient liquid is studied. On the basis of simplified model equations of a rarefield bubble mixture, a nonlinear-oscillation equation of the relaxation type is obtained. In constructing an exact particular solution of this equation, the heat transfer law associated with bubble compression is established. For studying the harmonic oscillations, the Krylov-Bogolyubov-Mitropol’skii asymptotic method is used. It is shown that, for a small bubble, the viscosity and heat transfer effects are of the same order. For a small bubble, the influence of these effects on the formation of the natural-oscillation frequency, which is small in the linear approximation, may be significant in the nonlinear formulation. For a large bubble, the influence of these effects is negligible in both approximations. For the approximate solution of the nonlinear equation, a uniformly valid second-order expansion is constructed.     

Surface tension and buoyancy-driven flow in a non-isothermal liquid bridge

The Navier–Stokes–Boussinesq equations governing the transport of momentum, mass and heat in a non-isothermal liquid bridge with a temperature-dependent surface tension are solved using a vorticity-stream-function formulation together with a non-orthogonal co-ordinate transformation. The equations are discretized using a pseudo-unsteady semi-implicit finite difference scheme and are solved by the ADI method. A Picard-type iteration is adopted which consists of inner and outer iterative processes. The outer iteration is used to update the shape of the free surface. Two schemes have been used for the outer iteration; both use the force balance normal to the free surface as the distinguished boundary condition. The first scheme involves successive approximation by the direct solution of the distinguished boundary condition. The second scheme uses the artificial force imbalance between the fluid pressure, viscous and capillary forces at the free surface which arises when the boundary condition for force balance normal to the surface is not satisfied. This artificial imbalance is then used to change the surface shape until the distinguished boundary condition is satisfied. These schemes have been used to examine a variety of model liquid bridge situations including purely thermocapillary-driven flow situations and mixed thermocapillary- and bouyancy-driven flow.     

On the influence of buoyancy on the surface tension driven flow around a bubble on a heated wall

The surface tension driven flow in the liquid vicinity of gas bubbles on a heated solid wall has been investigated both, in a reduced gravity environment aboard a sounding rocket, and in an earth-bound experiment. Both experiments deal with temperature gradients within the liquid surrounding of a bubble which cause variations of the surface tension. These, in turn, lead to a liquid flow around the bubble periphery termed thermocapillary or thermal Marangoni-convection. On Earth, this phenomenon is widely masked by buoyancy. We therefore carried out an experiment under reduced gravitational acceleration. In order to simultaneously observe and record the flow field and the temperature field liquid crystal tracers have been applied. These particles offer the feature of selectively reflecting certain wavelengths of incident white light depending on the crystals temperature. Although the bubble injection system did not perform nominally during the flight experiment, some interesting flow characteristics could be observed. Comparison of results obtained in microgravity to data measured on Earth reveal that due to the interaction of thermocapillarity and buoyancy a very compact vortex flow results on ground, while in microgravity the influence on the surface tension driven flow penetrates much deeper into the bulk. This result is of special interest regarding the production of materials in space. Dedicated to Professor Dr. Julius Siekmann on the occasion of his 70th birthday The work described herein was supported by the German space agency DARA (Deutsche Agentur für Raumfahrtangelegenheiten GmbH) through DARA Grant 50 WM 9434. The authors thank the European Space Agency (ESA) for the opportunity to conduct the TEXUS 33 sounding rocket experiment. The flight hardware has been partly built by Daimler-Benz-Aerospace which is gratefully acknowledged. Also, the authors are indebted to Mr. H.-H. Wolf for his careful evaluation of the particle images     

Surface tension driven jet break up of strain-hardening polymer solutions

Experimental studies attempting to ascertain the influence of viscoelasticity on the atomization of polymer solution are often hindered by the inability to decouple the effect of shear thinning from the effect of extensional hardening. Here, the influence of viscoelasticity on the jet break up of a series of non-shear-thinning viscoelastic fluids is quantified. Previous characterization using an opposed-nozzle rheometer identified the critical extensional rates for strain hardening of these model fluids. The strain hardening fluids exhibit a beads-on-string structure with reduction or elimination of satellite drops. Capillary instabilities grow on the filaments connecting the spheres and eventually break the filaments up into a string of very small drops about one order of magnitude smaller than the satellite drops formed by a Newtonian fluid with the same shear viscosity, surface tension, and density. These results confirm that strain hardening is the key rheological property in jet break up and that the critical extensional rate of a fluid is pertinent in determining the final characteristics of break up. Results suggest that the opposed-nozzle rheometer does probe extensional behavior in the range of extensional rates that are relevant to jet break up, providing a tool to roughly predict jet break up.     

Diffusion-induced growth of a gas bubble in a viscoelastic fluid

The diffusion-induced growth of a spherical gas bubble surrounded by a thin shell of viscoelastic fluid containing a limited amount of dissolved gas is analyzed. This is representative of a situation when a large number of bubbles grows in close proximity in a viscoelastic medium. The upper-convected Maxwell model is employed to describe the rheology of the fluid. Limited quantities of the dissolved gas available in the liquid shell mandates solution of the convection-diffusion equation, as opposed to using similarity solutions or polynomial profiles to describe the mass transport across the interface. Utilizing the properties of a potential field and a Lagrangian transformation, a new approach is introduced to solve the coupled system of integro-differential equations governing the bubble growth. The results indicate that, at the early stages of the growth, bubbles in a viscoelastic fluid grow faster than in a Newtonian fluid. However, eventually they attain the same steady-state configuration.     

Spherical bubble collapse in viscoelastic fluids

The collapse of a spherical bubble in an infinite expanse of viscoelastic fluid is considered. For a range of viscoelastic models, the problem is formulated in terms of a generalized Bernoulli equation for a velocity potential, under the assumptions of incompressibility and irrotationality. The boundary element method is used to determine the velocity potential and viscoelastic effects are incorporated into the model through the normal stress balance across the surface of the bubble. In the case of the Maxwell constitutive equation, the model predicts phenomena such as the damped oscillation of the bubble radius in time, the almost elastic oscillations in the large Deborah number limit and the rebound limit at large values of the Deborah number. A rebound condition in terms of ReDe is derived theoretically for the Maxwell model by solving the Rayleigh–Plesset equation. A range of other viscoelastic models such as the Jeffreys model, the Rouse model and the Doi-Edwards model are amenable to solution using the same technique. Increasing the solvent viscosity in the Jeffreys model is shown to lead to increasingly damped oscillations of the bubble radius.     

Evolution of the structure of viscoelastic liquid media upon pulse volume tension

It is shown experimentally that the unbounded bubble cavitation in a highly viscous liquid (glycerin) can develop only in a regime of very slow volume tension of a liquid sample. Upon pulse volume tension, cavitation in the sample damps at the initial stage (the bubbles do not reach pronounced sizes) and fragmentation occurs because of perturbations generated on its free surface. The mechanism of bubble growth from micropores in a thixotropic medium (gel) is explained based on experimental results and theoretical estimates. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 105–111, January–February, 2000.     

Methods of calculating the roots of the dispersion equation for free oscillations of a gas bubble in a liquid

The problem of free radial oscillations of gas bubbles in a liquid is considered. The structure of the roots of the dispersion equation in the presence of heat transfer between the phases is studied in detail. It is shown that this equation has two complex-conjugate roots and an infinite number of real roots; all of the roots lie in the left complex half-plane, providing damping of radial oscillations. Approximate expressions for these roots are obtained.     

Thermal convection in a viscoelastic liquid

The onset of convection in a layer of viscoelastic liquid heated from below is investigated. It is shown that the nature of the convective solution depends strongly on the particular constitutive relation used to characterize the viscoelasticity. For certain models and certain parameter ranges the convection is supercritical and stable, while for other models and parameter ranges it can be subcritical and unstable. It is suggested that observations of convective behavior can provide a test for constitutive relations proposed for a particular liquid. A Fourier representation of the solution to the nonlinear problem is developed which is shown to admit aperiodic, or chaotic, solutions in a specific truncation that generalizes the classical Lorenz system for the Newtonian Bénard problem.     

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.     

Longitudinal and torsional oscillations of a rod in a viscoelastic fluid

The flow of a viscoelastic fluid due to the torsional and longitudinal oscillations of an infinite circular rod is examined. The idealized equation of state to characterise this liquid is of the implicit Oldroyd-B model, for which momentum equations are solved analytically. The effect of the Weissenberg number and the viscosity ratio on the flow field are discussed. Also, the axial shear force and torque on the rod are computed. Received: 14 March 1997 Accepted: 14 May 1998     

Analysis of diffusion-induced bubble growth in viscoelastic liquids

Transport models of diffusion-induced bubble growth in viscoelastic liquids are developed and evaluated. A rigorous model is formulated that can be used to describe bubble growth or collapse in a non-linear viscoelastic fluid, and takes into account convective and diffusive mass transport as well as surface tension and inertial effects. Predictions for bubble growth dynamics demonstrating the importance of fluid elasticity are presented. These predictions indicate that for diffusion-induced bubble growth in viscoelastic liquids, the lower bound for growth rate is given by growth in a Newtonian fluid and the upper bound by diffusion-controlled growth. The influence of non-linear fluid rheology on bubble growth dynamics is examined and found to be relatively minor in comparison to fluid elasticity. It is shown how previously published models employing various approximations can be derived from the rigorous model. Comparisons of predicted bubble growth dynamics from the rigorous and approximate models are used to establish the ranges of applicability for two commonly-used approximations. These comparisons indicate that models using a thin boundary layer approximation have a rather limited range of applicability. An analysis of published experimental bubble growth data is also carried out using appropriate transport models.     

Break-up of a viscoelastic liquid jet

Applying Green's continuum theory of a slender body, the process of liquid jet break-up is analysed for a viscoelastic upper-convected Jeffreys fluid. In contrast to a Newtonian liquid an enforced growth of the perturbation is received from a linear analysis. A non-linear numerical analysis shows the viscosity-dependent filament formation between growing droplets of the viscoelastic liquid. The radius of these filaments decreases in an uniaxial extensional flow.