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.     

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.     

Extensional flow behavior of polymer solutions and particle suspensions in a spinning motion

Mechanical spinning of fluid filaments was used to generate an extensional flow, in which rheological measurements were obtained for a Newtonian fluid, two aqueous polymer solutions, and two fluid suspensions of rod-shaped particles. The tensile stress was determined by measuring the tensile force of the fluid filament while the kinematics were determined from photographic measurement of the filament profile and the assumption of a flat velocity profile. The measured tensile stresses for the Newtonian fluid matched predicted stresses, thereby confirming the validity of the experimental technique.The spinning behavior of each polymer solution could be correlated as stress versus extension rate. The apparent “spinning viscosity” increased with increasing rate of extension, in contrast to shear-thinning behavior in viscometric flow. For the fluid suspensions, the presence of rod-shaped particles increased the apparent viscosity far more in extensional flow than in shear. Tensile stresses calculated from a theoretical formula for suspensions proposed by Batchelor agreed rather well with experiment. Some general criteria for the interpretation of the spinning experiment are proposed, and some microrheological implications of the present findings are discussed.     

Resistance to uniaxial extensional flow of fibre suspensions

The tensile stress due to resistance to uniaxial extensional flow of fibre suspensions in Newtonian and non-Newtonian fluids has been measured using the filament stretching technique. It has been found that addition of fibres to a Newtonian fluid increases the extensional viscosity. The steady state results agree with Bachelors theory and the stress growth behaviour is qualitatively predicted by the theory of Dinh and Armstrong. Experimental results from this work have also shown that the behaviour of a fibre suspension in viscoelastic fluid is qualitatively described by Fans equation. The added fibres increase the extensional stress growth coefficient of the viscoelastic fluid at low strain but have marginal effect on the fluid after the onset of strain-hardening.     

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.     

Viscoelastic effects in double‐pipe single‐pass counterflow heat exchangers

The conducto‐convective heat loss from a viscoelastic liquid, in the core of a double‐pipe heat exchanger arrangement, to a cooler Newtonian fluid flowing in the outer annulus is investigated with direct numerical simulations. A numerical algorithm based on the finite difference method is implemented in time and space with the Giesekus constitutive model for the viscoelastic liquids. The flow of both the annulus and core‐fluids is considered to be Poiseuille flow, driven by respective pressure gradients. In general, the results show that a viscoelastic core‐fluid leads to slightly lower (albeit comparable) attainable temperatures in the core‐fluid stream as compared with a corresponding Newtonian fluid. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element computations of viscoelastic two-phase flows using local projection stabilization

A three-field local projection stabilized (LPS) finite element method is developed for computations of a three-dimensional axisymmetric buoyancy driven liquid drop rising in a liquid column where one of the liquid is viscoelastic. The two-phase flow is described by the time-dependent incompressible Navier-Stokes equations, whereas the viscoelasticity is modeled by the Giesekus constitutive equation in a time-dependent domain. The arbitrary Lagrangian-Eulerian (ALE) formulation with finite elements is used to solve the governing equations in the time-dependent domain. Interface-resolved moving meshes in ALE allows to incorporate the interfacial tension force and jumps in the material parameters accurately. A one-level LPS based on an enriched approximation space and a discontinuous projection space is used to stabilize the numerical scheme. A comprehensive numerical investigation is performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column. The influence of the viscosity ratio, Newtonian solvent ratio, Giesekus mobility factor, and the Eötvös number on the drop dynamics are analyzed. The numerical study shows that beyond a critical Capillary number, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape and also exhibits a negative wake phenomena. However, a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape.     

Efficient microfluidic rectifiers for viscoelastic fluid flow

In this work we propose a new type of microfluidic rectifier, which is able to operate efficiently under creeping flow conditions. The flow of Newtonian and non-Newtonian fluids was investigated experimentally in different microchannels with triangular (nozzle/diffuser) and hyperbolic shapes in order to achieve high anisotropic flow resistance between the two flow directions. The Newtonian fluid used was de-ionized water and the viscoelastic fluids were aqueous solutions of polyacrylamide and polyethylene oxide with different molecular weights. Pressure drop measurements were performed in addition to visualizations of the flow patterns by streak line photography for a wide range of flow rates. For the Newtonian flows, inertia leads to the appearance of recirculations for both flow directions, but no significant rectification effects appear. For the viscoelastic fluids, two distinct behaviors are identified: at low flow rates, the pressure drops are similar in both flow directions; above a critical flow rate (or Deborah number), the flow patterns become quite different, leading to different flow rates in the forward and backward flow directions for the same pressure drop, i.e., rectification effects emerge. In particular, the viscoelastic fluid flow becomes unsteady in the forward direction, due to the presence of elastic instabilities, which leads to a significant increase in the flow resistance. Flow resistance ratios greater than three were achieved for the hyperbolic rectifier, clearly in excess of the value for the triangular-shaped rectifier and for other geometries proposed in the literature for operation in creeping flow conditions. This high diodicity is associated with the distinct nature of the extensional flows in the forward and backward directions of the hyperbolic-type microgeometry.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

Kinematics and normal stress differences of a viscoelastic fluid undergoing wiggle flow

An experimental program was carried out to determine the laminar regime kinematics and normal stress differences of a viscoelastic fluid in wiggle flow employing non-contact measurement techniques. The viscoelastic fluid was a 5% by weight solution of polyisobutylene dissolved in Primol 355, a high purity mineral oil.The kinematics were determined by Laser-Doppler Anemometry and compared with the data obtained for a Newtonian fluid, Primol 355, under identical flow conditions. It was found that the normalized axial velocity versus axial position curves along the centerline for both fluids superimposed at very low flow rates, an experimental verification that a viscoelastic fluid behaves like a Newtonian fluid under very low shear rates. However, at higher flow rates the behaviour of the viscoelastic fluid curves changed appreciably whereas the Newtonian fluid curves did not change at all. Thus, the effect of flow rate on viscoelastic fluid behaviour was also experimentally established.The normal stress differences were determined using a stress-birefringence apparatus. Data obtained along the centerline clearly exhibited a delayed growth of stress which should be attributed to the expected memory effects in viscoelastic fluid flow.     

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.     

Squeezing a viscoelastic liquid from a wedge: an exact solution

The paper reports an exact solution for the squeezing flow from a wedge of a general viscoelastic liquid. To obtain numerical values for the field variables, a network model that allows stress overshoot and shear-thinning in the start-up of a shear flow is adopted. It is found that both these features are important in this transient flow; stress overshoot is responsible for a stiffer response of the fluid (compared to the inelastic case) at moderate time —at large time, shear-thinning dominates and the fluid behaves like an inelastic fluid. On the other hand, the Oldroyd-B fluid always predicts a softer response than the Newtonian one. Furthermore, there is a limiting Weissenberg number above which one component of the stresses of the Oldroyd-B fluid increases unboundedly with time. This limiting Weissenberg number is approximately $\text{sol2}\text{3}$.     

The effect of viscoelasticity on a rising gas bubble

This paper investigates the role of viscoelasticity on the dynamics of rising gas bubbles. The dynamics of bubbles rising in a viscoelastic liquid are characterised by three phenomena: the trailing edge cusp, negative wake, and the rise velocity jump discontinuity. There is much debate in the literature over the cause of the jump discontinuity, which is observed once the bubble exceeds a certain critical volume. In this paper, the employment of some choice modelling assumptions allows insights into the mechanisms of the jump discontinuity which cannot be ascertained experimentally. The ambient fluid is assumed incompressible and the flow irrotational, with viscoelastic effects included through the stress balance on the bubble surface. The governing equations are solved using the boundary element method. Some Newtonian predictions are discussed before investigating the role of viscoelasticity. The model predicts the trademark cusp at the trailing end of a rising bubble to a high resolution. However, the irrotational assumption precludes the prediction of the negative wake. The corresponding absence of the jump discontinuity supports the hypothesis that the negative wake is primarily responsible for the jump discontinuity, as mooted in previous studies.     

A comparison of boundary element and finite element methods for modeling axisymmetric polymeric drop deformation

A modified boundary element method (BEM) and the DEVSS‐G finite element method (FEM) are applied to model the deformation of a polymeric drop suspended in another fluid subjected to start‐up uniaxial extensional flow. The effects of viscoelasticity, via the Oldroyd‐B differential model, are considered for the drop phase using both FEM and BEM and for both the drop and matrix phases using FEM. Where possible, results are compared with the linear deformation theory. Consistent predictions are obtained among the BEM, FEM, and linear theory for purely Newtonian systems and between FEM and linear theory for fully viscoelastic systems. FEM and BEM predictions for viscoelastic drops in a Newtonian matrix agree very well at short times but differ at longer times, with worst agreement occurring as critical flow strength is approached. This suggests that the dominant computational advantages held by the BEM over the FEM for this and similar problems may diminish or even disappear when the issue of accuracy is appropriately considered. Fully viscoelastic problems, which are only feasible using the FEM formulation, shed new insight on the role of viscoelasticity of the matrix fluid in drop deformation. Copyright © 2001 John Wiley & Sons, Ltd.     

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     

Surfactant spreading on a thin weakly viscoelastic film

A mathematical model is presented for surfactant-driven thin weakly viscoelastic film flows on a flat, impermeable plane. The Oldroyd-B constitutive relation is used to model the viscoelastic fluid. Lubrication theory and a perturbation expansion in powers of the Weissenberg number (We) are employed, which give rise to non-linear coupled evolution equations governing the transport of insoluble surfactant and thin liquid film thickness. Spreading on a Newtonian film is recovered to leading order and corrections to viscoelasticity are obtained at order We. These equations are solved numerically over a wide range of viscosity ratio (ratio of solvent viscosity to the sum of solvent and polymeric viscosities), pre-existing surfactant level and Peclet number (Pe). The effect of viscoelasticity on surfactant transport and fluid flow is investigated and the mechanisms underlying this effect are explored. Shear stress, streamwise normal stress and the temporal rate of change of extra shear stress generated from gradients in surfactant concentration dominate thin viscoelastic film flows whereas only shear stresses play a role in Newtonian thin film flows. Our results also reveal that, for weak viscoelasticity, the influence of viscosity ratio on the evolution of surfactant concentration and film thickness can be significant and varies considerably, depending on the concentration of pre-existing surfactant and surfactant surface diffusivity.     

Viscoelastic stresses in the stagnation flow of a dilute polymer solution

In this paper, we consider viscoelastic stresses T₁₁, T₁₂ and T₂₂ arising in the stagnation flow of a dilute polymer solution; in particular, we consider an upper convected Maxwell (UCM) fluid. We present exact solutions to the coupled partial differential equations describing the viscoelastic stresses and deduce the results for the stress T₂₂ of Becherer et al. [P. Becherer, A.N. Morozov, W. van Saarloos, Scaling of singular structures in extensional flow of dilute polymer solutions, J. Non-Newtonian Fluid Mech. 153 (2008) 183–190]. As we considered the viscoelastic stresses over two spatial variables, we are able to study the effect of variable boundary data at the inflow. As such, our results are applicable to a wider range of fluid flow problems.     

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.