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     

Observability of viscoelastic fluids

We apply the observability rank condition to study the observability of various viscoelastic fluids under imposed shear or extensional flows. In this paper the observability means the ability of determining the viscoelastic stress from the time history of the observations of the first normal stress difference. We consider four viscoelastic models: the upper convected Maxwell (UCM) model, the Phan–Thien–Tanner (PTT) model, the Johnson–Segalman (JS) model and the Giesekus model. Our study reveals that all of the four models have observability for all stress components almost everywhere under shear flow whereas under extensional flow most of the models have no observability for the shear stress component. More specifically, for UCM and JS models under imposed shear flow, the observations of the first normal stress difference allow the reconstruction of all components of viscoelastic stress. For UCM and JS models under extensional flow, the two normal stress components can be determined from the measurements of the first normal stress difference; the shear stress component does not affect the evolution of the normal stress components and consequently it cannot be extracted from the observations. Under shear flow, the PTT and Giesekus models have observability almost everywhere. That is, all components of the viscoelastic stress can be determined from the observations when the vector formed by the components of viscoelastic stress does not lie on a certain surface. Under extensional flow, the PTT model has observability almost everywhere for normal stress components whereas the Giesekus model has observability almost everywhere for all stress components. We also run simulations using the unscented Kalman filter (UKF) to reconstruct the viscoelastic stress from observations without and with noises. The UKF yields accurate and robust estimates for the viscoelastic stress both in the absence and in the presence of observation noises.     

Forced capillary breakup of liquid jets

The characteristics of the forced capillary breakup (FCB) of liquid jets have been investigated over a broad range of variation of the breakup parameters: jet orifice diameter (34–527 m), flow rate (10^–5–1 cm³/sec), and excitation amplitude and frequency. The theory is compared with experiment.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–61, March–April, 1988.The authors are grateful to E. V. Ametistov for his constant interest and assistance.     

Dynamics of vorticity stretching and breakup of isolated viscoelastic droplets in an immiscible viscoelastic matrix

The effect of droplet elasticity on transient deformation of isolated droplets in immiscible polymer blends of equal viscosity was investigated. In terms of the deformation parameter, Def*=a*–c/a*+c where a* and c are apparent drop principal axes, it undergoes two cycles of positive oscillations before reaching a negative value, followed by one cycle of oscillation before attaining a steady state negative value. This behavior was observed when Capillary number, Ca, was varied between 3 and 9 at a fixed Weissenber number, Wi, of 0.31, and when Ca number was held fixed at 8 and Wi number was varied between 0.21 and 0.40. In another blend of relatively lower Wi number of 0.21, one cycle of oscillation in Def* was observed before reaching steady state negative values when Ca number was varied between 3 and 14. The steady state Def* varies inversely with Ca number, with a stronger dependence for the blend with higher Wi number. The magnitude of oscillation increases with increasing Ca and Wi numbers. The critical Ca was found to be 12 and 14 for the two blends studied; these values are about 30 times greater than that of Newtonian blends.     

Nonlinear Taylor stability of viscoelastic fluids

Using Stuart's energy method, the torque on the inner cylinder, for a second order fluid, in the supercritical regime is calculated. It is found that when the second normal stress difference is negative, the flow is more stable than for a Newtonian fluid and the torque is reduced. If the second normal stress difference is positive, then the flow is more stable and there is no torque reduction. Experimental data related to the present work are discussed.Nomenclature a amplitude of the fundamentals - A _ij ⁽¹⁾ , A _ij ⁽²⁾ first and second Rivlin-Ericksen tensors - d r ₂–r ₁ - D d/dx - E - F - g _ij metric tensor - G torque on the inner cylinder in the supercritical regime - h height of the cylinders - k ₀ /d ² - k ₁ /d ² - I ₁ - I ₂ - I ₃ - I ₄ - r ₁, r ₂ radii of inner and outer cylinders respectively - r ₀ 1/2(r ₁+r ₂) - R Reynolds number ₁ r ₁ d/ ₀ - R _c critical Reynolds number - T Taylor number r _{1 1} ² d ^{3 2}/ ₀ ² *) - T _c critical Taylor number - u ₁, v ₁, w ₁ Fundamentals of the disturbance - u _i , v _i , w _i , (i>1) harmonics - mean velocity (not laminar velocity) - u –u ₁/ar _{1 1} - v v ₁/Rar _{1 1} - x (r–r ₀)/d - , material constants - 0 viscosity - wave number d - density - ₁ angular velocity of inner cylinder - tilde denotes complex conjugate     

Deformation and breakup of viscoelastic drops in planar extensional flows

A computer-controlled four-roll mill was used to investigate the deformation and break-up of polymeric drops in the well-characterized flow of an immiscible Newtonian fluid. Aqueous polymer solutions ranging in concentration from 160 ppm to 3% by weight were examined. For zero-shear-rate viscosity ratios greater than order 1, the deformation of the drops closely followed that of Newtonian fluids, irrespective of the droplet material. However, drops with viscosity ratios less than order 1 had significantly smaller critical deformations and the critical capillary number was found to be substantially smaller. Two modes of drop break-up were discovered that differed substantially from that observed for Newtonian drops in the inclusion of cusped ends and tip streaming.     

Numerical simulation of the nonuniform breakup of capillary jets

The problem of the evolution of the surface of a jet up to the stage at which it breaks up into droplets is solved numerically for two initial wave disturbances. The wave number of one of these coincides with the wave number of the disturbance that grows most strongly according to the linear theory, while the wave number of the other is varied. The effect of the wave numbers and the amplitude ratio of the initial disturbances on the breakup time and the appearance of nonuniformity is investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 12–17, March–April, 1993.     

Analysis of dynamical behavior of viscoelastic fluids

Summary A theoretical study is undertaken on the dynamical behaviour of viscoelastic fluids in simple shearing flow. In the present study differential-type constitutive equations are chosen in attempting to describe the dynamical behaviour of polymer solutions experimentally observed byPhilippoff. The dynamical system studied here takes into account shear rate and normal stresses as the dependent variables with shear stress kept constant during the transient period. It is found that the 3-constantOldroyd model describes qualitatively the experimentally observed behaviour.

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Zusammenfassung Es wird theoretisch das dynamische Verhalten viskoelastischer Flüssigkeiten in einer einfachen Scherströmung untersucht. Als Versuch zur Beschreibung des dynamischen Verhaltens von Polymerlösungen, die experimentell vonPhilippoff untersucht worden waren, werden hier Zustandsgieichungen vom Differentialtyp benutzt. Das hier untersuchte dynamische System berücksichtigt die Schergeschwindigkeit und die Normalspannungen als abhängige Variable von der Schubspannung, die während der Anlaufzeit konstant gehalten wird. Es zeigt sich, daß das Dreikonstantenmodell vonOldroyd das experimentell beobachtete Verhalten qualitativ beschreibt.

     

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.     

Laminar shear flow of plastic viscoelastic fluids

Summary The theory of plastic viscoelastic fluids was developed by the author to represent the rheological behavior of polymer melts and solutions with high loading of small particles. The present paper develops an asymptotic formulation of the general theory which applies to laminar shear flows. The formulation is analogous to Criminale, Ericksen and Filbey's theory for viscoelastic fluids. We apply this to study plane Poiseuille and Couette flow.With 2 tables     

The steady isothermal spinning of viscoelastic fluids

The flow problem given in the title has been considered for a modified Maxwell fluid. The resulting spin line equation is solved both numerically and analytically. It has been found that the results obtained by the above two methods are in agreement. This confirms the accuracy of the perturbation method which we adopted.     

Detecting instabilities in flows of viscoelastic fluids

There is a growing interest in developing numerical tools to investigate the onset of physical instabilities observed in experiments involving viscoelastic flows, which is a difficult and challenging task as the simulations are very sensitive to numerical instabilities. Following a recent linear stability analysis carried out in order to better understand qualitatively the origin of numerical instabilities occurring in the simulation of flows viscoelastic fluids, the present paper considers a possible extension for more complex flows. This promising method could be applied to track instabilities in complex (i.e. essentially non‐parallel) flows. In addition, results related to transient growth mechanism indicate that it might be responsible for the development of numerical instabilities in the simulation of viscoelastic fluids. Copyright © 2003 John Wiley & Sons, Ltd.     

On the dynamic response of viscoelastic fluids

Summary To define the dynamic shear behaviour of a viscoelastic fluid we require two functions — one elastic and one viscous. There are two simple alternatives based on the Voigt and Maxwell concepts which are mathematically interrelated. Current practice interprets the dynamic response of fluids as a function of angular frequency () in terms of the storage (G) and loss (G) moduli: the loss function is commonly converted to a viscosity = G/. As is well known the parameters andG are the elements of a Voigt model whereas it is near universal practice to interpret steady flow in terms of the Maxwell model. This paper shows how the interpretation of dynamic experiments on fluids in terms of the apparent Maxwell parameters is more simple, more sensitive, more consistent with steady flow behaviour, and physically more realistic.

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Zusammenfassung Zur Festlegung des dynamischen Scherverhaltens einer viskoelastischen Flüssigkeit benötigt man zwei Funktionen, eine für das elastische und eine für das viskose Verhalten. Dafür gibt es zwei einfache, mathematisch miteinander verknüpfte Alternativen, die entweder auf dem Voigtschen oder dem Maxwellschen Konzept gründen. Die übliche Praxis beschreibt das dynamische Verhalten der Flüssigkeiten als Funktion der Winkelgeschwindigkeit durch den SpeichermodulG und den VerlustmodulG bzw. die zugeordnete Viskosität = G/. Die Parameter undG sind bekanntlich die Elemente eines Voigt-Modells, wohingegen es nahezu allgemein üblich ist, das stationäre Fließen durch ein Maxwell-Modell zu beschreiben. Diese Arbeit zeigt nun, daß die Beschreibung dynamischer Versuche mit Hilfe von apparenten Maxwell-Parametern einfacher, empfindlicher, mit dem stationären Verhalten konsistenter und in physikalischer Hinsicht realistischer ist.

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Paper presented at British Society of Rheology conference. Rheometry: Methods of measurement and analysis of results, Shrivenham, April 8–11, 1975.

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With 13 figures and 2 tables     

Flow of viscoelastic fluids between rotating disks

Few boundary-value problems in fluid mechanics can match the attention that has been accorded to the flow of fluids, Newtonian and non-Newtonian, between parallel rotating disks rotating about a common axis or about distinct axes. An interesting feature which has been recently observed is the existence of solutions that are not axially symmetric even in the case of flow due to the rotation of disks about a common axis. In this article we review the recent efforts that have been expended in the study of both symmetric and asymmetric solutions in the case of both the classical linearly viscous fluid and viscoelastic fluids.The support of the Air Force Office of Scientific Research is gratefully acknowledged.     

A reduced simulation applied to the viscoelastic fatigue of polymers

The paper extends the use of the PGD method to viscoelastic evolution problems described by a large number of internal variables and with a large spectrum of relaxation times. The internal variables evolution is described by a set of linear differential equations that involve many time scales. The feasibility and the robustness of the method are discussed in the case of a polymer in a non-equilibrium state under creep and cyclic loading. The relationships between different time scales (loading and internal variables) are also discussed.     

Spectrum of developed turbulence of incompressible viscoelastic fluids

It is shown that fine-scale turbulent motions of a viscoelastic fluid damp out as in a viscous fluid with some effective viscosity dependent on the scale of the motion. The elasticity of deformation results in a diminution in the dissipativity of the turbulence, and hence, to an elongation of the high-frequency tail of the spectrum for a given energy influx.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–33, January–February, 1972.     

Nonmonodisperse breakup of capillary jets in a variable electric field

The electrical charging of capillary jets has a strong influence on their stability [1–10]. Well-known theoretical studies have been devoted to the linear [1–6], weakly linear [7], or finite-amplitude [10] stability of such jets in a constant electric field. In the present paper, an investigation is made in the framework of the full nonlinear equations. The main attention is devoted to effects associated with allowance for a time-variable electric field. It is shown that a sharp decrease of the surface charge may lead to an appreciable decrease in the size of the satellite droplets; allowance for the long-wavelength background also leads to a decrease in the size of the satellite droplets. In contrast, a sharp increase of the surface charge increases the relative contribution of the satellite droplets. At the same time, introduction of small-scale background perturbations can lead to a decrease in the contribution of the fine satellite droplets and to a weakening of their reaction to a rapidly increasing electric field. It is shown that the degree of monodisperseness can be increased by a relatively slowly varying electric field. An averaged effect of an electric field that varies rapidly in time is found. Appreciable increase of the initial perturbation amplitude in the case of a periodically varying electric field can lead to an appreciable increase in the degree of monodisperseness. The introduction of short-wavelength perturbations in a periodic electric field with large amplitude of the pulsations can lead to disappearance of the satellite droplets.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 55–62, March–April, 1991.     

Dynamic load balance applied to particle transport in fluids

This work presents a parallel numerical strategy to transport Lagrangian particles in a fluid using a dynamic load balance strategy. Both fluid and particle solvers are parallel, with two levels of parallelism. The first level is based on a substructuring technique and uses message passing interface (MPI) as the communication library; the second level consists of OpenMP pragmas for loop parallelisation at the node level. When dealing with transient flows, there exist two main alternatives to address the coupling of these solvers. On the one hand, a single-code approach consists in solving the particle equations once the fluid solution has been obtained at the end of a time step, using the same instance of the same code. On the other hand, a multi-code approach enables one to overlap the transport of the particles with the next time-step solution of the fluid equations, and thus obtain asynchronism. In this case, different codes or two instances of the same code can be used. Both approaches will be presented. In addition, a dynamic load balancing library is used on the top of OpenMP pragmas in order to continuously exploit all the resources available at the node level, thus increasing the load balance and the efficiency of the parallelisation and uses the MPI.