Breakup of extending liquid threads

A theoretical and experimental study is made of the stability and breakup of an extending viscous liquid cylindrical thread suspended in an immiscible viscous liquid undergoing extensional flow.It is shown that disturbances initiated as the thread is formed will in general as time proceeds be damped, then amplified and finally damped again. By considering disturbances being continually given to the system it is thus found that a disturbance which dominates at one moment will be completely different from that at any other. Assuming that breakup occurs when the disturbance amplitude becomes equal to cylinder radius, results are obtained for the time to breakup and for the final drop size resulting from breakup in terms of fluid properties, extension rate and the amplitude of disturbance given to system.These results were confirmed by examining, with the aid of cinematography, the breakup of a liquid thread in hyperbolic flow.     

A new look at extensional rheology of low-density polyethylene

The nonlinear rheology of three selected commercial low-density polyethylenes (LDPE) is measured in uniaxial extensional flow. The measurements are performed using three different devices including an extensional viscosity fixture (EVF), a homemade filament stretching rheometer (DTU-FSR) and a commercial filament stretching rheometer (VADER-1000). We show that the measurements from the EVF are limited by a maximum Hencky strain of 4, while the two filament stretching rheometers are able to probe the nonlinear behavior at larger Hencky strain values where the steady state is reached. With the capability of the filament stretching rheometers, we show that LDPEs with quite different linear viscoelastic properties can have very similar steady extensional viscosity. This points to the potential for independently controlling shear and extensional rheology in certain rate ranges.     

Viscous flow through microfabricated hyperbolic contractions

We study the flow of a Newtonian fluid through microfabricated hyperbolic contractions followed by a sudden expansion, with the aim of investigating the potential of this geometry to serve as an extensional microrheometer. A set of planar converging geometries, with total Hencky strains ranging from 1.0 to 3.7, were fabricated in order to produce a homogeneous extensional flow field within the contraction. The velocity field in various planes of the hyperbolic contraction was quantified by means of microparticle image velocimetry (μPIV) and the pressure drop across the converging geometry was also measured and found to vary approximately linearly with the flow rate. Additionally, an extensive range of numerical calculations were carried out using a finite-volume method to help assess the performance of this geometry as a microfluidic elongational rheometer. The measured velocity fields in the contraction and associated pressure drops compare very well (to within 10%) with the numerical predictions. For the typical dimensions used in the microfluidic devices, the steady viscous flow through the contraction is shown to be three-dimensional and it is demonstrated that regions with nearly constant strain rate can only be achieved using geometries with large total Hencky strains under Hele–Shaw (potential-like) flow conditions.     

Mesoscopic dynamics of polymer chains in high strain rate extensional flows

We take a step towards accessing the physics of viscoelastic liquid breakup in high speed, high strain rate flows by performing Brownian dynamics computations of dilute uniaxial, equibiaxial, and ellipsoidal polymeric extensional flows. Our computational implementation of the bead-spring model, when tailored to the DNA molecule, consistently with recent works of Larson and co-workers, is shown: (a) to predict a coil-stretch transition at Deborah number De≈0.5, and (b) to reproduce the experimental longest relaxation time. Furthermore, after adapting the model parameters to represent the polyethylene oxide (PEO) chain (for M=10⁶ Da), we find it possible to reproduce our own experimental data of the longest relaxation time, the transient extensional viscosity of dilute solutions at small Deborah numbers, and a coil-stretch transition at Deborah number De≈0.5. Extended to large Deborah numbers, the model predicts that polymer stretching is controlled by: (a) the randomness of the initial conditions that, in combination with rapid kinematically imposed compression, lead to the formation of initially frozen chain-folds, and (b) the speed with which thermo-kinematic processes relax these folds. The slowest fold relaxation occurs during uniaxial extension. As expected, the introduction of stretching along a second direction enhances the efficiency of fold relaxation mechanisms. Even for Deborah numbers (based on the chain longest relaxation time) of the order of one thousand, there is a large variation in the time a polymer needs in order to extend fully, and the effects of Brownian motion cannot be ignored. The computed Trouton ratios and polymer contributions to the total stress as functions of Hencky strain provide information about the relative importance of elastic effects during polymeric liquid stretching. At high strain rates, the steady state elastic stresses increase linearly with the Deborah number, resembling at that stage an anisotropic Newtonian fluid (constant extensional viscosity).     

The effect of step-stretch parameters on capillary breakup extensional rheology (CaBER) measurements

Extensional rheometry has only recently been developed into a commercially available tool with the introduction of the capillary breakup extensional rheometer (CaBER). CaBER is currently being used to measure the transient extensional viscosity evolution of mid to low-viscosity viscoelastic fluids. The elegance of capillary breakup extensional experiments lies in the simplicity of the procedure. An initial step-stretch is applied to generate a fluid filament. What follows is a self-driven uniaxial extensional flow in which surface tension is balanced by the extensional stresses resulting from the capillary thinning of the liquid bridge. In this paper, we describe the results from a series of experiments in which the step-stretch parameters of final length, and the extension rate of the stretch were varied and their effects on the measured extensional viscosity and extensional relaxation time were recorded. To focus on the parameter effects, well-characterized surfactant wormlike micelle solutions, polymer solutions, and immiscible polymer blends were used to include a range of characteristic relaxation times and morphologies. Our experimental results demonstrate a strong dependence of extensional rheology on step-stretch conditions for both wormlike micelle solutions and immiscible polymer blends. Both the extensional viscosity and extensional relaxation time of the wormlike micelle solutions were found to decrease with increasing extension rate and strain of the step-stretch. For the case of the immiscible polymer blends, fast step-stretches were found to result in droplet deformation and an overshoot in the extensional viscosity which increased with increasing strain rates. Conversely, the polymer solutions tested were found to be insensitive to step-stretch parameters. In addition, numerical simulations were performed using the appropriate constitutive models to assist in both the interpretation of the CaBER results and the optimization of the experimental protocol. From our results, it is clear that any rheological results obtained using the CaBER technique must be properly considered in the context of the stretch parameters and the effects that preconditioning has on viscoelastic fluids.     

Extensional rheology of a shear-thickening cornstarch and water suspension

A filament-stretching rheometer is used to measure the extensional viscosity of a shear-thickening suspension of cornstarch in water. The experiments are performed at a concentration of 55 wt.%. The shear rheology of these suspensions demonstrates a strong shear-thickening behavior. The extensional rheology of the suspensions demonstrates a Newtonian response at low extension rates. At moderate strain rates, the fluid strain hardens. The speed of the strain hardening and the extensional viscosity achieved increase quickly with increasing extension rate. Above a critical extension rate, the extensional viscosity goes through a maximum and the fluid filaments fail through a brittle fracture at a constant tensile stress. The glassy response of the suspension is likely the result of jamming of particles or clusters of particles at these high extension rates. This same mechanism is responsible for the shear thickening of these suspensions. In capillary breakup extensional rheometry, measurement of these suspensions demonstrates a divergence in the extensional viscosity as the fluid stops draining after a modest strain is accumulated.     

Equibiaxial extensional flow of polymer melts via lubricated squeezing flow. II. Flow modeling

A model for lubricated squeezing flow of a viscoelastic fluid is developed in order to study the viability of this flow as a rheological technique for generating equibiaxial extensional deformations in polymer melts. In this simple flow model, the melt, described by an upper-convected Maxwell fluid, is squeezed between thin films of a Newtonian fluid. Comparisons of the model predictions for constant strain rate and constant stress flows are made with experimental results presented in the first paper. Predictions from the model are able to describe the effects of lubricant viscosity and experimental configuration and indicate the technique fails for these flows at Hencky strains of approximately one. The cause for this failure is lubricant thinning, which leads to significant errors in both the measured stress difference and the strain. Received: 31 January 2000 Accepted: 31 May 2000     

Can extensional viscosity be measured with opposed-nozzle devices?

Opposed-nozzle devices are widely used to try to measure the extensional viscosity of low-viscosity liquids. A thorough literature survey shows that there are still several unanswered questions on the relationship between the quantity measured in opposed-nozzle devices and the true extensional viscosity of the liquids. In addition to extensional stresses, opposed nozzle measurements are influenced by dynamic pressure, shear on the nozzles, and liquid inertia. Therefore the ratio of the apparent extensional viscosity that is measured to the shear viscosity that is independently measured is greater than three even for Newtonian liquids. The effect of inertia on the extensional measurements is analyzed by computer-aided solution of the Navier-Stokes system, and by experiments on low-viscosity Newtonian liquids(1 mPa sS 800 mPa s). The effect of nozzle separation-to-diameter ratio on the average residence time of the liquid is analyzed under the assumption of simple extensional flow kinematics. The average residence time of the liquid is independent of this ratio unless the radial inflow section of the extensional flow volume is related to the nozzle separation. Experiments indicate that in some cases widening the gap lowers the apparent extensional viscosity that is measured, whereas in other cases the opposite is true. In the light of these theoretical considerations and experimental observations, the use of systematic corrections to extensional viscosity measurements on non-Newtonian liquids is not recommended. Thus opposed nozzle devices should be considered as useful indexers rather than rheometers. Finally, measurements on a series of semi-dilute solutions of high molecular weight poly(ethylene oxide) in. water are also reported.Dedicated to the memory of Anastasios C. Papanastasiou     

Deformation and breakup of a non-Newtonian slender drop in an extensional flow

The deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid in a simple extensional and creeping flow has been theoretically studied. The power-law was chosen for the fluid inside the drop, and the deformation of the drop is described by a single ordinary differential equation, which was numerically solved. Asymptotic analytical expressions for the local radius were derived near the center and close to the end of the drop. The results for the shape of the drop and the breakup criterion are presented as a function of the capillary number, the viscosity ratio and type of non-Newtonian fluid inside the drop. An approximate analytical solution is also suggested which is in good agreement with the numerical results.     

Measurement of relaxation times in extensional flow of weakly viscoelastic polymer solutions

The characterization of the extensional rheology of polymeric solutions is important in several applications and industrial processes. Filament stretching and capillary breakup rheometers have been developed to characterize the extensional properties of polymeric solutions, mostly for high-viscosity fluids. However, for low concentration polymer solutions, the measurements are difficult using available devices, in terms of the minimum viscosity and relaxation times that can be measured accurately. In addition, when the slow retraction method is used, solvent evaporation can affect the measurements for volatile solvents. In this work, a new setup was tested for filament breakup experiments using the slow retraction method, high-speed imaging techniques, and an immiscible oil bath to reduce solvent evaporation and facilitate particle tracking in the thinning filament. Extensional relaxation times above around 100 μs were measured with the device for dilute and semi-dilute polymer solutions. Particle tracking velocimetry was also used to measure the velocity in the filament and the corresponding elongation rate, and to compare with the values obtained from the measured exponential decay of the filament diameter.     

Fibre spinning of a weakly elastic liquid

This paper presents a study of a silicone oil (poly(dimethyl siloxane)) in extensional deformation using an instrument developed recently by the authors. Data from steady shear and low amplitude sinusoidal deformation of this liquid clearly establish that it is weakly elastic. The viscometric data, for shear rates less than 100 s ⁻¹, are best represented by either the Maxwell model or the Jeffrey's model, the latter being marginally superior. The extensional data show that at low deformation rates, this fluid exhibits a Newtonian behavior with an apparent extensional viscosity equal to three times the shear viscosity. Under these conditions the velocity profiles along the spinline are also well represented by the Newtonian model. However, at higher deformation rates better predictions of the velocity profiles are obtained from the Jeffrey's and Maxwell models. At deformation rates above 100 s ⁻¹ none of these simple models is adequate. Under the conditions used in these experiments, the fractional increase in tensile stress along the fiber is shown both theoretically and experimentally to be a unique function of the total strain. Furthermore, the apparent extensional viscosity at any point on the spinline can be calculated from steady state expressions if allowance is made for the variation of stretch rates by defining a time averaged stretch rate.The results obtained here show that elasticity must be considered if these model liquids are used to conduct rheological experiments at high deformation rates. Additionally, it is found that elastic effects in extension can be predicted using simple constitutive equations provided viscometric data can be represented properly in the deformation rate range of interest. Finally, the present research further substantiates the utility of the extensional viscometer developed by the authors.     

A Rheotens-based setup for the constant strain rate uniaxial extension of uncured elastomers

A novel experimental setup for the uniaxial extension of uncured elastomers is presented, and room temperature experiments at constant Hencky strain rate are performed by means of a commercial Rheotens tensile tester originally designed for the determination of the melt strength of polymer melts. Successful results are obtained for materials related to many aspects of the elastomers production, namely, gum elastomers and carbon black compounds. Stress growth experiments are reported for filled and unfilled high-cis-polybutadiene, and the extensional behavior is related to the carbon black dispersion. Although originally thought as an experimental tool for polymer melts, the proposed Rheotens setup can also perform constant strain rate tensile testing on thermoplastic rubbers. Stress-strain experiments are performed on a microphase separated polystyrene-b-poly(ethylene butylene)-b-polystyrene (SEBS) copolymer and related blends with polypropylene, showing the effect of a constant deformation rate on the network response. Relaxation experiments after cessation of extensional flow are also reported for the investigated materials. With respect to commonly used tensile testing procedures for elastomers at constant elongation rate and time decreasing strain rate, a complete and accurate investigation of the extensional behavior of many uncured elastomers can be carried out with the additional advantage of using a reduced amount of material.     

Power requirement for the mixing of thixotropic liquids

A procedure has recently been proposed by Godfrey et al. [1] for the calculation of the variation of power consumption with time when a thixotropic liquid is agitated from rest using an impeller which rotates at constant speed. This procedure requires a knowledge of the power requirement for Newtonian and time-independent non-Newtonian liquids together with viscometric data for the thixotropic liquid obtained under constant shear rate conditions.Experimental work to test the procedure has been carried out in a 0.126-m-diameter cylindrical vessel with anchor, helical ribbon and helical screw impellers. Power consumption data were obtained for a range of Newtonian and time-independent non-Newtonian liquids and this was then used to make predictions of the measured power input to the thixotropic liquids: salad cream, tomato ketchup, yoghurt, paint and Laponite solutions.The agreement between experiment and theory was usually better than 10%. However, for the case of the helical screws rotating in Laponite it was observed that regions existed close to the vessel wall where there was no fluid circulation. In such cases, the predicted power input was greatly in excess of the measured value. This is not a serious limitation of the predictive procedure since efficient industrial mixers would keep the entire fluid in circulation.     

A simple extensional viscometer

An extensional viscometer is described in which the liquid filament leaving a capillary is subjected to a stretching deformation. In order to keep the flow rate through the capillary unaltered upon inception of stretching, the pressure head at the capillary entrance has to be reduced by an amount equal to the extensional viscoelastic stress at the capillary exit. This affords a simple means of measuring small fluid forces such as those that occur in the stretching of dilute polymer solutions. Since stretch rates can be obtained from a knowledge of the mass flow rate and the filament diameter profile, extensional viscosities can be computed. The efficacy of the technique is demonstrated by obtaining the anticipated results for Newtonian liquids.     

Uniaxial and biaxial extensional viscosity measurements of dilute and semi-dilute solutions of rigid rod polymers

Effective uniaxial extensional and biaxial extensional viscosities of dilute and semi-dilute solutions of collagen, a rigid rod molecule, have been measured with an opposing jet apparatus. The concentration of collagen in the glycerin/water solvent ranged from 50 to 2300 ppm. The data agree quantitatively with a theory developed by Batchelor describing the extensional viscosity of perfectly aligned rigid rods. The viscosity measured for the dilute rigid rod solutions is independent of the rate of strain as predicted by Batchelor's theory. Data taken on the semi-dilute, strain-thinning solutions at strain rates sufficiently high to align the rods in the extension direction also agree with the predictions of Batchelor's theory. The measured viscosity of semi-dilute solutions at low strain rates agree qualitatively with a theory developed by Doi and Edwards describing the strain-thinning behavior of semi-dilute rigid rod solutions.     

Filament-breaking length—a measure of elasticity in extension

A technique has been developed to assess elasticity in extension of fluids which are weakly elastic. The technique is based on stretching a fluid sample held between two small closely spaced co-axial disks until the fluid bridge or filament breaks. The distance between the disks on breakup, ‘the breaking length’, is measured. When the fluid is elastic, this length is greater than that of an equivalent inelastic fluid. An inelastic baseline was established by measuring the breaking lengths of a wide variety of Newtonian fluids. Measurements with weakly elastic fluids reveal that the extra breaking length increases with polymer type and concentration and with the rate of stretching, the expected behavior for elastic liquids. The breaking lengths of model paper coatings were measured and found to correlate with the degree of misting in a roll coating machine, indicating that droplet formation is related to extensional elasticity.     

The relevance of entry flow measurements for the estimation of extensional viscosity of polymer melts

The extensional viscosity of some flexible chain polymers and a thermotropic liquid crystalline polymer was measured in uniaxial extensional flow at constant extension rate. Power law functions were found for the dependence of the extensional viscosity at constant accumulated strain on strain rate. The stress growth curves were compared with measurements in axisymmetric entry flow, where both elongation and shear occur. The comparison showed that the values of the extensional viscosity calculated from the measurements in the entry flow correspond to the ones calculated from the viscosity growth measured in uniaxial elongation and averaged over extensional strain equal to what is accumulated on the fluid as it flows from the barrel into the capillary.     

An innovative extensional viscometer for low-viscosity and low-elasticity liquids

A spinline-type extensional viscometer is described in which an innovative method of tensile stress measurement is employed. A limited amount of liquid flows through a vertical capillary at a constant flow rate under the influence of a constant pressure head. The drainage time decreases when the liquid stream leaving the capillary is stretched by the application of vacuum. These drainage times are measured in a manner similar to that used for intrinsic viscosity measurements. The measured difference in drainage times, with and without stretching, is trivially related to the extensional stress at the capillary exit, and this provides a very simple method of accurately determining fluid stretching forces having a magnitude as low as 10^-4 N; stresses at other axial locations in the stretched liquid jet are obtained by means of a force balance in the usual manner. The validity of the proposed technique is demonstrated by obtaining the expected results for a Newtonian oil having a shear viscosity of 56.2 mPa-s. Also presented are preliminary data on polyethylene oxide-in-water solutions having an even lower shear viscosity.     

Dynamics and Breakup of Viscoelastic Liquids (A Review)

The phenomena of hydrodynamic breakup of liquid jets, drops, films, bridges, and filaments are reviewed for liquids with viscoelastic properties. The reasons for breakup are capillary instabilities, collisions with rigid obstacles, and other forms of dynamic action. The relationship between the properties of the liquids and the features of the breakup process is discussed.     

Dynamics of formation and dripping of drops of deformation-rate-thinning and -thickening liquids from capillary tubes

Dynamics of formation of drops of non-Newtonian liquids from capillary tubes is studied computationally. The rheology of the drop liquids is described by a constitutive relation that accounts for both deformation-rate-thinning and -thickening. The analysis is expedited by reducing the original system of three-dimensional but axisymmetric equations to a system of one-dimensional slender-jet equations. The slender-jet equations are solved by a method of lines using a finite element method for spatial discretization and an adaptive finite difference method for time integration. The simulations follow the formation in time of thousands of drops in sequence, including any satellites that may be produced upon the breakup of a thin thread connecting an about-to-form primary drop to the rest of the liquid attached to the tube. Rate-thickening is shown to produce bead-on-string patterns, which are typically attributed to viscoelastic effects, along the thin threads as they near pinch-off. Rate-thinning, on the other hand, is demonstrated to reduce the length of such thin threads. Simulations are used to identify conditions that may lead to minimization and/or elimination of unwanted satellites. Analysis of dripping or leaky faucets of non-Newtonian liquids reveals rich nonlinear dynamical behavior. As with Newtonian liquids, simple periodic or P-1, where P stands for period, dripping at low flow rates gives way to more complex responses as flow rate is increased. In addition to P-1, P-2, and P-4 responses seen in recent computational analyses of dripping faucets of Newtonian liquids, the new non-Newtonian simulations have also uncovered difficult-to-find P-3 responses as well as chaotic states. Rate-thinning and low viscosities are shown to enhance the complexity of observed responses. Rate-thickening, on the other hand, lowers the critical value of the flow rate for the onset of complexity but narrows the range of flow rates over which the dynamics is complex. The possibility of hysteresis is demonstrated and the effect of fluid rheology on the value of the flow rate for transition from dripping to jetting is determined.