Showing papers in "Journal of Non-newtonian Fluid Mechanics in 1999"

The yield stress—a review or ‘παντα ρει’—everything flows?

Abstract: An account is given of the development of the idea of a yield stress for solids, soft solids and structured liquids from the beginning of this century to the present time. Originally, it was accepted that the yield stress of a solid was essentially the point at which, when the applied stress was increased, the deforming solid first began to show liquid-like behaviour, i.e. continual deformation. In the same way, the yield stress of a structured liquid was originally seen as the point at which, when decreasing the applied stress, solid-like behaviour was first noticed, i.e. no continual deformation. However as time went on, and experimental capabilities increased, it became clear, first for solids and lately for soft solids and structured liquids, that although there is usually a small range of stress over which the mechanical properties change dramatically (an apparent yield stress), these materials nevertheless show slow but continual steady deformation when stressed for a long time below this level, having shown an initial linear elastic response to the applied stress. At the lowest stresses, this creep behaviour for solids, soft solids and structured liquids can be described by a Newtonian-plateau viscosity. As the stress is increased the flow behaviour usually changes into a power-law dependence of steady-state shear rate on shear stress. For structured liquids and soft solids, this behaviour generally gives way to Newtonian behaviour at the highest stresses. For structured liquids this transition from very high (creep) viscosity (>106 Pa.s) to mobile liquid (

The rheology of fibre suspensions

Abstract: Theoretical and experimental work on the rheology of suspensions of slender fibres is reviewed. This is an area of enormous importance in applications such as the manufacture of fibre reinforced polymeric materials and the fabrication of articles out of reinforced plastic. The particular features of suspension rheology when the suspended particles are long and slender, notably the response in extensional flows, are also of fundamental scientific interest in their own right and the issues we shall be surveying are of interest to both theorists and experimenters.The published literature relevant to this topic is vast; one of us has listed well over 300 references [l]. In the rheological literature, attention is drawn to useful sections in a number of books [2, 3, 4]. Note that for suspensions of fibres and for solutions of long rigid polymers the theoretical approach, and indeed many theoretical results, are identical. Review articles which can be recommended include the comprehensive theoretical study by Brenner [5] and a review by Batchelor [6] which focus on the fundamental aspects of the topic. This contrasts with the more pragmatic approach of Jinescu [7]. A valuable contribution to bridging the gap between theory and practice was produced by Jeffrey and Acrivos [8]. More specifically devoted to fibre suspensions are the reviews by Maschmeyer and Hill [9] and by Ganani and Powell [10]. The behaviour of fibre suspensions in extensional flow has been recently reviewed in [11]. The starting point for theoretical work has to be the dynamics of a single particle suspended in a flowing fluid. From there we trace the steps necessary for the construction of a model from which the macroscopic behaviour of a suspension may be predicted. The influence of non-Newtonian behaviour of the suspending fluid has been studied to a limited extent. The main practical drawback of dilute suspension theory is that, for long slender fibres, diluteness means that fibres have to be far apart on a scale based on the length of the fibres rather than their diameter, and so dilute means extremely dilute in terms of the volume fraction of solids in the suspension. Consideration of the hydrodynamic interaction between particles is obviously necessary. In the extreme case of suspensions that axe so concentrated that, in effect, the fibres have no choice but to lie parallel to one another we find the regime of liquid crystalline behaviour. The issue of wall effects, i.e. hydrodynamic or direct mechanical interaction between the suspended fibres and the walls of the container within which the suspension is flowing, is extremely important. There is much published experimental work on suspensions both in Newtonian liquids, which is of fundamental scientific interest, and in molten polymers, with obvious relevance to the processing of fibre-reinforced plastics. Suspensions of fibres in wellcharacterized non-Newtonian liquids have perhaps been studied less until recently and we discuss this area critically. We shall present in tabular form references to papers on both these areas and attempt to make clear the rheological content of the reports. Studies on fibre orientation and attempts at flow visualization are also of considerable interest and we offer a further tabulation of work in this area. The contribution of C J S Petrie was made possible by a Visiting Research Fellowship of the University of Melbourne and by support from the Departments of Mathematics and Chemical Engineering of the University of Melbourne. This support and the granting of a period of study leave by the University of Newcastle upon Tyne are gratefully acknowledged. Research in non-Newtonian fluid mechanics at the University of Melbourne is supported by the Australian Research Council.

A consistent thin-layer theory for Bingham plastics

Abstract: Thin-layer theory is developed for Bingham plastic fluids; the specific case of a fluid flowing down an inclined plane is considered. In contrast to previous work it is indicated how the Bingham model leads directly to a self-consistent thin-layer theory; this does not rely upon adopting a bi-viscous approximation. The theory describes the fluid in terms of regions of fully plastic flow bounded by a `fake' yield surface. Above this fake yield surface are `pseudo-plugs' – regions in which the leading-order equations predict a plug, but which are seen to be weakly yielded at higher order.

Isothermal start-up of pipeline transporting waxy crude oil

Abstract: This paper is concerned with the start-up problem in pipeline transportation of gelled waxy crude oil after a period of shutdown. The analysis presented is based on a three-yield-stress model, which has been experimentally verified for waxy crude oil. Three possibilities of the start-up process are discussed according to the applied pressure relative to the complex yielding behaviour of the oil, which is represented by three characteristic yield stresses – an elastic-limit, a static and a dynamic yield stress. The physical model of the start-up assumes that the gelled oil is to be displaced by introducing another fluid under constant pump pressure and that the displacing fluid displays time-independent yield stress behaviour. Using rheological property data for a gelled crude oil from the North Sea and a time-dependent Bingham style equation, the flow after a successful start-up is simulated by computing changes in the oil flow rate with time, and the clearing time as function of the applied pressure and characteristics of the displacing fluid. Both laminar and turbulent flows of the two fluids are considered in the model. The results indicate that the yield stress and the time-dependent rheology of the gelled oil play an important role in determining the oil flow rate after start-up, and that the start-up computer model is highly sensitive to the rheological behaviour of the gelled oil. An appropriate method for measuring the rheological properties of waxy crude oil for this purpose is described.

Drag reduction in the turbulent pipe flow of polymers

Abstract: The paper concerns an experimental study of the fully developed turbulent pipe flow of several different aqueous polymer solutions: 0.25%, 0.3% and 0.4% carboxymethylcellulose (CMC), 0.2% xanthan gum (XG), a 0.09%/0.09% CMC/XG blend, 0.125% and 0.2% polyacrylamide (PAA). The flow data include friction factor vs. Reynolds number, mean velocity and near-wall shear rate distributions, and axial velocity fluctuation intensity u′ at a fixed radial location as a laminar/turbulent transition indicator. For each fluid we also include measurements of shear viscosity, first normal-stress difference and extensional viscosity. At high shear rates we find that the degree of viscoelasticity increases with concentration (0.3% CMC is an exception) for a given polymer, and in the sequence XG, CMC/XG, CMC, PAA, whilst at low shear rates the ranking changes to CMC, CMC/XG, XG, PAA. The extensional viscosity ranking is XG/CMC, XG, CMC, PAA at high strain rates and the same as that for the viscoelasticity at low shear rates. We find that the observed drag-reduction behaviour is consistent for most part with the viscoelastic and extensional-viscosity behaviour at the low shear and strain rates typical of those occurring in the outer zone of the buffer region. Although laminar/turbulent transition is practically indiscernible from the friction factor vs. Reynolds number plots, particularly for PAA and XG, the u′ level provides a very clear indicator and it is found that the transition delay follows much the same trend with elasticity/extensional viscosity as the drag reduction.

Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model

Abstract: A simple model consisting of the Upper Convected Maxwell constitutive equation and a kinetic equation for destruction and construction of structure, first proposed by Fredrickson in 1970, is used here to reproduce the complex rheological behavior of viscoelastic systems that also exhibit thixotropy and rheopexy under shear flow. The model requires five parameters that have physical significance and that can be estimated from rheological measurements. Several steady and unsteady flow situations were analyzed with the model. The model predicts creep behavior, stress relaxation and the presence of thixotropic loops when the sample is subjected to transient stress cycles. Such behavior has been observed with surfactant-based solutions and dispersions. The role of the characteristic time for structure built up, λ, in the extent and shape of the thixotropic loops is demonstrated.

Galerkin/least-square finite-element methods for steady viscoelastic flows

Abstract: The elastic viscous split stress formulation (EVSS) and the discrete EVSS formulation (DEVSS) are effective in stabilizing numerical simulations of viscoelastic flows and have been widely used. Following the concept of Galerkin least-square perturbations proposed by Hughes et al. [Comput. Meth. Appl. Mech. Eng. 73 (1989) 173–189] and Franca et al. [SIAM J. Numer. Anal. 28(6) (1991) 1680–1697; Comput . Meth. Appl. Mech. Eng. 99 (1992) 209–233; Ibid. 104 (1993) 31–48] we are able to give the DEVSS formulation a new explanation as a perturbation to the Galerkin method based on the strain-rate residual, and furthermore, introduce another stabilized formulation, here named as MIX1, based on the incompressibility residual of the finite element discretizations. The three formulations (EVSS, DEVSS, MIX1), combined with a h–p type finite element algorithm that employs the SUPG technique to solve the viscoelastic constitutive equations are then tested on three benchmark problems: the flow of the upper-convected Maxwell fluid between eccentric cylinders, the flow of the Maxwell fluid around a sphere in a tube and the flow of the Maxwell and Oldroyd-B fluids around a cylinder in a channel. The results are checked with previous published works; good agreement is observed. Our numerical experiments convincingly demonstrate that the MIX1 is an accurate algorithm and convergent in terms of the p-extension, it has the same level of stability and robustness as the DEVSS method and is superior to the EVSS method in some respects. More important is that with MIX1 method one needs not solve for the strain-rate tensor as in EVSS and DEVSS methods, therefore, the CPU time consumption in the MIX1 method especially when using a coupled iteration scheme can be radically reduced. The success of the MIX1 method presents a challenge to the widely accepted concept of making the momentum equation explicitly elliptic.

Extensional flow of a polystyrene Boger fluid through a 4 : 1 : 4 axisymmetric contraction/expansion

Abstract: The creeping flow of a dilute (0.025 wt%) monodisperse polystyrene/polystyrene Boger fluid through a 4:1:4 axisymmetric contraction/expansion is experimentally observed for a wide range of Deborah numbers. Pressure drop measurements across the orifice plate show a large extra pressure drop that increases monotonically with Deborah number above the value observed for a similar Newtonian fluid at the same flow rate. This enhancement in the dimensionless pressure drop is not associated with the onset of a flow instability, yet it is not predicted by existing steady-state or transient numerical computations with simple dumbbell models. It is conjectured that this extra pressure drop is the result of an additional dissipative contribution to the polymeric stress arising from a stress-conformation hysteresis in the strong non-homogeneous extensional flow near the contraction plane. Such a hysteresis has been independently measured and computed in recent studies of homogeneous transient uniaxial stretching of PS/PS Boger fluids. Flow visualization and velocity field measurements using digital particle image velocimetry (DPIV) show large upstream growth of the corner vortex with increasing Deborah number. At large Deborah numbers, the onset of an elastic instability is observed, first locally as small amplitude fluctuations in the pressure measurements, and then globally as an azimuthal precessing of the upstream corner vortex accompanied by periodic oscillations in the pressure drop across the orifice.

Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method : DAVSS-G/DG

Abstract: Accurate and robust finite element methods for computing flows with differential constitutive equations require approximation methods that numerically preserve the ellipticity of the saddle point problem formed by the momentum and continuity equations and give numerically stable and accurate solutions to the hyperbolic constitutive equation. We present a new finite element formulation based on the synthesis of three ideas: the discrete adaptive splitting method for preserving the ellipticity of the momentum/continuity pair (the DAVSS formulation), independent interpolation of the components of the velocity gradient tensor (DAVSS-G), and application of the discontinuous Galerkin (DG) method for solving the constitutive equation. We call the method DAVSS-G/DG. The DAVSS-G/DG method is compared with several other methods for flow past a cylinder in a channel with the Oldroyd-B and Giesekus constitutive models. Results using the Streamline Upwind Petrov–Galerkin method (SUPG) show that introducing the adaptive splitting increases considerably the range of Deborah number (De) for convergence of the calculations over the well established EVSS-G formulation. When both formulations converge, the DAVSS-G and DEVSS-G methods give comparable results. Introducing the DG method for solution of the constitutive equation extends further the region of convergence without sacrificing accuracy. Calculations with the Oldroyd-B model are only limited by approximation of the almost singular gradients of the axial normal stress that develop near the rear stagnation point on the cylinder. These gradients are reduced in calculations with the Giesekus model. Calculations using the Giesekus model with the DAVSS-G/DG method can be continued to extremely large De and converge with mesh refinement.

Rheological modeling of concentrated colloidal suspensions

Abstract: The use of concentrated colloidal suspensions is common in several industries such as paints, foodstuffs and pulp and paper. These suspensions are generally composed of strongly interactive particles. If the attractive forces dominate the repulsion and Brownian forces, the particles aggregate to form a three-dimensional network yielding a gel structure. Under flow, the micro-structure of suspensions can be drastically modified and the rheological properties are then governed by structure breakdown and build-up. In this work, we propose a structural network model based on a modified upper convected Jeffreys model with a single relaxation time and a kinetic equation to describe the flow-induced micro-structure evolution. Three distinct kinetic equations are tested for this purpose. The proposed model describes yield and thixotropic phenomena, nonlinear viscoelastic behavior and output signal distortions observed for relatively small strain amplitude during oscillatory measurements, and overshoots observed in stress growth experiments. A comparison of model predictions and experimental data for fumed silica and coating colors is also presented. However, different model parameters must be used to correctly predict the different flow properties indicating that a more versatile or generalized kinetic equation must be proposed.

Analytical solutions for squeeze flow with partial wall slip

Abstract: Squeeze flow between parallel plates of a purely viscous material is considered for small gaps both for a Newtonian and power law fluid with partial wall slip. The results for the squeeze force as a function of the squeezing speed reduce to the Stefan and Scott equations in the no slip limit, respectively. The slip velocity at the plate increases linearly with the radius up to the rim slip velocity v s . For small gaps H , the resulting apparent Newtonian rim shear rate—measured for a constant rim shear stress, i.e. an imposed force increasing proportional to 1/ H —yields a straight line if plotted versus 1/ H . The slope of the straight line is equal to 6 v s whereas the intersect with the ordinate yields the effective Newtonian rim shear rate to be converted into the true rim shear rate by means of the power law exponent. The advantage of the new technique is the separation of bulk shear and wall slip from a single test. A more general derivation for the Newtonian case being also valid for full lubrication and large gaps is used to explain the gap dependence of the squeeze modulus of an elastic material.

Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids

Abstract: We provide a mathematical analysis of the spectrum of the linear stability problem for one and two layer channel flows of the upper-convected Maxwell (UCM) and Oldroyd-B fluids at zero Reynolds number. For plane Couette flow of the UCM fluid, it has long been known (Gorodstov and Leonov, J. Appl. Math. Mech. (PMM) 31 (1967) 310) that, for any given streamwise wave number, there are two eigenvalues in addition to a continuous spectrum. In the presence of an interface, there are seven discrete eigenvalues. In this paper, we investigate how this structure of the spectrum changes when the flow is changed to include a Poiseuille component, and as the model is changed from the UCM to the more general Oldroyd-B. For a single layer UCM fluid, we find that the number of discrete eigenvalues changes from two in Couette flow to six in Poiseuille flow. The six modes are given in closed form in the long wave limit. For plane Couette flow of the Oldroyd-B fluid, we solve the differential equations in closed form. There is an additional continuous spectrum and a family of discrete modes. The number of these discrete modes increases indefinitely as the retardation time approaches zero. We analyze the behavior of the eigenvalues in this limit.

Maxwell fluid suction flow in a channel

Abstract: Two-dimensional, steady and incompressible suction flow of the upper-convected Maxwell fluid in a porous surface channel has been studied The combined effects of viscoelasticity and inertia are considered A similarity solution is assumed, resulting in a nonlinear system of ODEs that describes the relations between the two velocity components, the three deviatoric stresses and the pressure gradient This system is solved using two methods: an analytical solution, based on a power series method in terms of the transverse coordinate across the channel, and a fourth-order Runge–Kutta numerical integration scheme We first find the existing Newtonian flow solutions for suction and injection For the Maxwell fluid, the solutions of the power series and the numerical integration are in complete agreement in the range of Reynolds and Deborah numbers 0 ≤ Re ≤ 30 and 0 ≤ De ≤ 03 They show that the suction flow exhibits a flattening of the longitudinal velocity profile near the centerline and the establishment of boundary layers near the porous surfaces as Reynolds number increases It is also observed that when Deborah number increases, with a fixed Reynolds number, viscoelasticity affects the velocity profiles in the same way as inertia in a Newtonian fluid The application of the self-similar solution to the injection flow of the Maxwell fluid is also discussed

Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method

Abstract: A semi-Lagrangian finite volume scheme for solving viscoelastic flow problems is presented. A staggered grid arrangement is used in which the dependent variables are located at different mesh points in the computational domain. The convection terms in the momentum and constitutive equations are treated using a semi-Lagrangian approach in which particles on a regular grid are traced backwards over a single time-step. The method is applied to the 4 : 1 planar contraction problem for an Oldroyd B fluid for both creeping and inertial flow conditions. The development of vortex behaviour with increasing values of We is analyzed.

Numerical simulation of the transient flow of branched polymer melts through a planar contraction using the `pom–pom' model

Abstract: Calculations of the transient flow of branched polymer melts through a planar 4 : 1 contraction and along a channel are presented. The polymer is modelled as a monodisperse suspension of `pom–pom' molecules using the constitutive equation developed by McLeish and Larson [T.C.B. McLeish, R.G. Larson, J. Rheol. 42 (1998) 81–110], which provides a direct link between the molecular topology and the flow properties of the melt. The branching produces an enhancement in the size of the upstream vortex, similar to that observed experimentally with branched polymer melts. The effect arises from the widely differing responses in shear and extensional flow, and is predicted to be sensitive to time, degree of branching and Weissenberg number.

Thermoviscoelastic simulation of thermally and pressure-induced stresses in injection moulding for the prediction of shrinkage and warpage for fibre-reinforced thermoplastics

Abstract: In this paper we present a detailed thermoviscoelastic formulation for the simulation of thermally and pressure induced residual stresses in injection moulded short-fibre-reinforced thermoplastics. The computed residual stresses enable us to predict shrinkage and warpage in the finished products. We also apply an anisotropic version of a rotary diffusion equation to calculate the flow-induced fibre orientation distribution. The predicted fibre orientation state, together with micromechanical theories, allows the incorporation of anisotropy in material properties into the thermoviscoelastic model. Finally we report three numerical examples to indicate the success of the present model.

Large amplitude oscillatory shear of ER suspensions

Abstract: Electrorheological (ER) fluids are fascinating materials that undergo dramatic reversible changes in their rheological properties upon the application of electric fields. In many proposed applications, the fluids will be subjected to a dynamic stimulus with finite deformation. We use a particle-level simulation method to investigate the dynamic behavior of monolayer ER fluids. ER fluids are linear viscoelastic for only very small strain amplitudes. The transition to nonlinear deformation arises from very slight rearrangements of unstable structures. At large strain amplitudes, the behavior is viscoplastic, while at large dimensionless frequencies (∝ ω/E02, where ω is the oscillation frequency and E0 is the electric field strength), the response is Newtonian for all strain amplitudes. Simulation results agree qualitatively with experiments. The dependence of the flow behavior on the strain amplitude and dimensionless frequency is summarized in the form of a Pipkin diagram.

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Abstract: Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ -method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡ λV / R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De L c ≡ λV / L is shown to be an O (1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.

Non-linear flows in porous media

Abstract: A volume averaging approach is applied to derive governing equations for purely viscous homogeneous flows in porous media. Additional terms appear in the volume averaged governing equations related to porosity, tortuosity, shear factor, dispersion coefficient and macroscopic viscosity. Flow in porous media is non-linear in nature in that the shear factor, the dispersion coefficient and the macroscopic viscosity are functions of flow velocity. A shear factor model is proposed based on flow through orifice plates and the macroscopic viscosity is defined for purely viscous flows. The predictions of flow velocity profile and pressure drop using the proposed model agree well with published experimental results.

Modeling of viscoelastic lid driven cavity flow using finite element simulations

Abstract: In this study we have used a convergent and highly accurate mixed finite element technique to model the effect of fluid elasticity on the flow kinematics and the stress distribution in lid driven cavity flow. Our work is motivated by the desire to capture the important physical aspects of the basic flow and thus to better understand the purely elastic instability in recirculating flows which has been reported in the literature elsewhere [A.M. Grillet, E.S.G. Shaqfeh, Observations of viscoelastic instabilities in recirculation flows of Boger fluids, J. Non-Newtonian Fluid Mech. 64 (1996) 141–155; P. Pakdel, G.H. McKinley, Cavity flows of elastic liquids: purely elastic instablities, Phys. Fluids 10 (5) (1998) 1058–1070]. In our numerical investigations we have treated the corner singularities by incorporating a controlled amount of leakage which allows the computation of fully elastic mesh converged solutions. We begin by validating our Newtonian cavity results against previous work to show that the introduction of leakage does not appreciably modify the cavity recirculation flow. Then we examine the polymer stresses to understand how elasticity changes the flow kinematics, slowing the primary recirculation vortex and causing the vortex center to shift opposite of the direction of lid motion. Variations of the cavity aspect ratio are also explored. Focusing on the corners we find that the leakage relieves the corner singularities and moreover, finite leakage helps explain the unusual behavior seen in the radial velocity in experiments. Finally, we have reexamined the previously proposed mechanisms for elastic instability in this flow and put forth a new instability mechanism. Together, these mechanisms may better explain the complex aspect ratio dependence of the onset of elastic instability in lid driven cavity flow.

The FENE-L and FENE-LS closure approximations to the kinetic theory of finitely extensible dumbbells

Abstract: We address the closure problem for the non-linear kinetic model of a dilute polymeric solution known as the Warner Finitely Extensible Non-Linear Elastic (FENE) dumbbell model. This model cannot be translated into an equivalent constitutive equation for the polymer stress without invoking closure approximations which can have significant impact. For instance, the FENE-P macroscopic equation, based on the Peterlin closure approximation, has a dynamical behavior that is markedly different from the FENE model. The main qualitative difference occurs in start-up of elongation followed by relaxation, where the FENE theory exhibits hysteretic behavior for the stress versus average configuration, while the FENE-P does not (see Lielens et al. (1998) and Sizaire et al. (1998)). This hysteresis is observed experimentally by Doyle et al. (1998). The Peterlin approximation hence removes one of the FENE theory essential features. Our aim is to develop improved closure approximations capable of predicting hysteretic behavior and giving a rheological response closer to that of actual polymer solutions.

On the performance of enhanced constitutive models for polymer melts in a cross-slot flow

Abstract: A new class of viscoelastic constitutive equations is proposed that provides enhanced control of shear and elongational properties. Together with the widely used Giesekus and Phan-Thien Tanner model, these enhanced models are evaluated for a polymer melt (LDPE) in a cross-slot flow. This stagnation flow has a strong planar elongational deformation component. The material is characterized, in both viscometric simple shear flow and in uniaxial elongation. Velocities are measured with particle tracking velocimetry while field-wise flow-induced birefringence is used for correlation with stresses. The experimental results are compared with 2D viscoelastic simulations. With equal quality of describing viscometric data, the enhanced models can predict the flow properties of the cross-slot flow significantly better than the well known PTT and Giesekus model.

Variance reduction methods for CONNFFESSIT-like simulations

Abstract: The aim of this paper is to present simple but efficient variance reduction methods for CONNFFESSIT-like simulations, extending the ideas of Hulsen et al. [J. Non-Newtonian Fluid Mech. 70 (1997) 79–101] and Ottinger et al. [J. Non-Newtonian Fluid Mech. 70 (1997) 255–261]. Strongly correlated local ensembles of dumbbells were used, and equilibrium ensembles of dumbbells were subtracted. This idea is extended here to non-equilibrium ensembles of dumbbells. The methods are first presented in the frame of the plane Couette flow for Hookean and FENE dumbbells. Extensions to two-dimensional flows are discussed.

Atomisation of dilute polymer solutions in agricultural spray nozzles

Abstract: A series of standard fluids, characterised in terms of their steady shear and extensional behaviour, was investigated for the behaviour when atomised through a series of matched throughput agricultural spray nozzles The results show that there are complex, but qualitatively predictable, interactions between the rheological properties of the liquids and the atomisation behaviour Further, it is clearly demonstrated that low concentrations of very low molecular weight additives may have profound effects on the spray pattern produced by typical nozzles

The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a Newtonian fluid

Abstract: In this paper, we investigate the fluid motion and shape of a non-Newtonian drop, modeled as a Chilcott–Rallison fluid, in the steady, uniaxial extensional flow of an unbounded Newtonian liquid. We consider two values of the capillary number, Ca = 0.05 and 0.10, below the critical value for continuous elongation of a Newtonian drop in the same flow. Solutions are then obtained for two values of the Chilcott–Rallison extensibility parameter, L2 = 144 and 600, appropriate roughly for polymer solutions in the Boger fluid range of concentration, each for several values of the viscosity ratio λ, and the polymer concentration, c. It is shown that the main region of impact of the polymer on the flow inside the drop occurs on the central symmetry axis, and near the tips of the drop. The effect of polymer on the drop shape is complicated. In general, there is a balance between the influence of the viscoelastic tensile stress contribution in the direction of the symmetry axis, which tends to pull the interface inward and thus decrease the flow-induced deformation at the ends of the drop, and modifications of the viscous stress due to changes in the flow and the pressure due viscoelasticity, both of which tend to increase deformation. For L2 = 600, the tensile stress effect is dominant and the viscoelastic drop is less deformed overall and has lower curvature at the tip than for a Newtonian drop at the same capillary number. For L2 = 144, however, this trend is reversed and the effect of viscoelasticity is to increase deformation and increase the curvature at the end of the drop.

The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity vorticity (DAVSS-ω) formulation

Abstract: A parallel unstructured finite volume method (FVM) is developed and implemented under a distributed computing environment through the parallel virtual machine (PVM) libraries, and is used to simulate the channel flow of the Oldroyd-B fluid past a circular cylinder. Differing from our previous work [11, 12] , a discrete elastic viscous split stress (DEVSS) formulation together with an independent interpolation of the vorticity (DEVSS- ω ) is proposed in this paper. This method has almost the same stability behavior as the elastic viscous split stress (EVSS) formulation, and is suitable for complex constitutive models. To further improve the stability at high Deborah numbers, we combine the idea of the discrete adaptive elastic viscous split stress (DAVSS) formulation [7] with the independent interpolation of the vorticity to arrive at the DAVSS- ω method. The numerical implementation is based on the unstructured FVM method and the semi-implicit method for pressure-linked equations revised (SIMPLER) algorithm. The parallelization of the program is implemented by a domain decomposition strategy and using PVM software libraries. The results are compared with those by the EVSS, DEVSS, and the plain Oldroyd-B formulation (without splitting the stress). It is found that the drag coefficient first decreases and then increases with the De number, for a channel half width to cylinder radius ratio of h / R = 2. It is also confirmed that the drag enhancement at high Deborah number is due to the increasing extension effect in the regions near the front and the rear stagnation points.

Axisymmetric and non-axisymmetric elastic and inertio-elastic instabilities in Taylor–Couette flow

Abstract: Flow visualization is performed on two highly elastic, non-shear-thinning dilute polymer solutions (polyisobutylene/polybutene) in a wide gap Couette cell ( R 1 / R 2 =0.827) over a range of shear rates and choices of relative cylinder rotations. Axisymmetric structures of the type found in a narrow gap cell ( R 1 / R 2 =0.912) [B.M Baumert, S.J. Muller, Rheol. Acta 34 (1995) 147–159; B.M. Baumert, S.J. Muller, Phys. Fluids 9 (1997) 566–586] were reproduced. Additionally, helical and non-axisymmetric standing wave patterns were visualized. In the more viscous, more elastic fluid ( e =De/Re=15.0) transitions were found to be purely elastic. At the lowest shear rates at which transitions were observed, extremely slow-growing stationary axisymmetric counter-rotating vortices replace the purely azimuthal base flow. At rates more than twice the critical, axisymmetric oscillatory flow precedes the onset of steady vortices. Non-axisymmetric structures are first observed at rates more than 5 times the critical. The pattern is initially highly regular: two m =1 helices superpose to generate a non-axisymmetric standing wave pattern (`ribbons'). The ribbons quickly give way to intense merging structures with nearly axisymmetric cores. In the less viscous, less elastic fluid ( e =0.0562) the lowest rate transitions are to weak axisymmetric counter-rotating vortices. At progressively higher rates in the presence of centrifugal destabilization several types of axially translating axisymmetric vortices are generated. Co-rotation of the cylinders gives rise to translating vortices at lower shear rates than rotation of the inner cylinder alone. At somewhat more than twice the critical rate, robust and hysteretic m =1 ribbons are produced. Brief periods of unpaired m =1 helices are also detectable very near the critical rate for the ribbons. The spectrum of spatial and temporal frequencies of the ribbons is found to broaden with further increases of shear rate. At the highest shear rates examined for this fluid, the flow is non-axisymmetric and highly disordered.

An investigation of interfacial instabilities in the superposed channel flow of viscoelastic fluids

Abstract: Interfacial instabilities of superposed pressure driven channel flow of viscoelastic fluids are investigated theoretically using linear stability analysis. Nonlinear constitutive equations which accurately depict the steady as well as transient viscoelastic properties of typical polymeric melts and solutions with various degrees of flexibility and accuracy are used to assess the constitutive complexity required to accurately describe the stability characteristics of this class of flows by comparing the results of the stability analysis with the experimental results of Wilson and Khomami (J. Non. Newtonian Fluid Mech. 41 (1992) 255; J. Rheol. 37 (1993) 315) and Khomami and Ranjbaran (Rheol. Acta 36 (1997) 1–22). It is shown that the multimode Giesekus model, which can accurately describe the steady as well transient behavior of the polymeric test fluids used in the experiments, can quantitatively describe the interfacial instability phenomenon in terms of the neutral stability contour as well as the growth/decay rate behavior. A rigorous energy analysis based on a newly developed disturbance-energy equation for viscoelastic flows is performed to investigate the mechanism of the purely viscous and purely elastic interfacial instabilities in pressure and drag driven channel flows. The mechanism of shortwave purely viscous instability is found to be due to the viscosity mismatch and the subsequent perturbation vorticity mismatch at the interface (i.e. interfacial friction), whereas the mechanism of the longwave purely viscous instability is found to be due to the bulk Reynolds stresses. The mechanism of purely elastic instability is found to be due to the coupling between the perturbation velocity and the jump in normal stresses across the interface at longwaves as well as shortwaves. An examination of perturbation velocity field reveals that for purely elastic longwave instability the jump in the normal stresses across the interface leads to a perturbation back flow in the bulk resulting in either accumulation (destabilizing) or depletion (stabilizing) of the fluid below the crest of the perturbed interface. In the case of purely elastic shortwave instability, coupling of the jump in the normal stresses across the interface and the perturbation velocity leads to perturbation vorticities adjacent to the interface which drive the instability.

A first-order model for bubble dynamics in a compressible viscoelastic liquid

Abstract: The radial dynamics of a spherical bubble in a compressible viscoelastic liquid is studied by means of a simplified singular-perturbation method to first-order in the bubble-wall Mach number. The three-parameter linear Oldroyd model is adopted to describe the viscoelastic properties of the liquid. The equation of motion for the bubble radius and the pressure equation are derived and numerical calculations are conducted for the case of bubble collapse in a constant-pressure field. It is concluded that the rheology of the liquid strongly influence the behaviour of bubbles only for values of the Reynolds number ( Re =R 0 p ∞ ρ ∞ /η where R 0 is the initial radius, p ∞ is the undisturbed liquid pressure, ρ ∞ is the liquid density and η is the liquid viscosity) smaller than 10 2 while for Re ≥ 10 2 , sound emission is the main damping mechanism in spherical bubble dynamics. In both cases, the 1/ r law of pressure attenuation through the liquid is not affected by the viscoelastic properties of the liquid. The effect of polymer additives on spherical bubble collapse is also discussed.

Mechanics of the squeeze flow of planar fibre suspensions

Abstract: This paper discusses the axisymmetric squeeze flow of concentrated transversely isotropic fibre suspensions in a power-law matrix and relates to the processing of composite materials such as sheet moulding compounds (SMCs) and glass mat thermoplastics (GMTs). A solution to the squeeze flow problem for a transversely isotropic power-law fluid is presented first, followed by a more detailed micromechanical analysis. In the first part of the paper a variational approach is applied to the interpretation of squeeze flow behaviour. This gives a simple expression for the total pressure, which enables the contributions due to extension and shear to be separated. Applying the procedure to GMT data suggests that the dissipation is predominantly extensional, except at very low plate separations. In the second part, a non-local constitutive equation is derived based on a simple drag law for hydrodynamic interactions. This is then used to model the pressure distribution when the effective length of the fibres is comparable to or determined by the dimensions of the squeeze flow plates. The model is shown to describe the observed squeeze flow stresses in both long and short fibre systems and to relate behaviour to the underlying resin flow properties.