Spectral element method for viscoelastic flows in a planar contraction channel

A new algorithm, which combines the spectral element method with elastic viscous splitting stress (EVSS) method, has been developed for viscoelastic fluid flows in a planar contraction channel. The system of spectral element approximations to the velocity, pressure, extra stress and the rate of deformation variables is solved by a preconditioned conjugate gradient method based on the Uzawa iteration procedure. The numerical approach is implemented on a planar four‐to‐one contraction channel for a fluid governed by an Oldroyd‐B constitutive equation. The behaviour of the Oldroyd‐B fluids in the contraction channel is investigated with various Weissenberg numbers. It is shown that numerical solutions obtained here agree well with experimental measurements and other numerical predictions. Copyright © 2003 John Wiley & Sons, Ltd.     

Comparison of a quasi-Newtonian fluid with a viscoelastic fluid in planar contraction flow

The flow of a 5.0 wt.% solution of polyisobutylene in tetradecane through a planar 4 : 1 contraction exhibiting a shear thinning viscosity is simulated using the flow-type sensitive quasi-Newtonian fluid model. The shear viscosity is fitted by the Giesekus model, which, with the chosen parameters, leads to an extension thickening elongational viscosity. The stress and velocity fields of the numerical simulations are compared with the experimental results of Quinzani et al. [J. Non-Newtonian Fluid Mech. 52 (1994) 1–36] and the numerical results of the viscoelastic simulation using the Giesekus model of Azaiez et al. [J. Non-Newtonian Fluid Mech. 62 (1996) 253–277]. It can be shown that the quasi-Newtonian fluid qualitatively predicts the essential features of the flow in the vicinity of the contraction.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

Viscoelastic flow analysis using the software OpenFOAM and differential constitutive equations

Viscoelastic fluids are of great importance in many industrial sectors, such as in food and synthetic polymers industries. The rheological response of viscoelastic fluids is quite complex, including combination of viscous and elastic effects and non-linear phenomena. This work presents a numerical methodology based on the split-stress tensor approach and the concept of equilibrium stress tensor to treat high Weissenberg number problems using any differential constitutive equations. The proposed methodology was implemented in a new computational fluid dynamics (CFD) tool and consists of a viscoelastic fluid module included in the OpenFOAM, a flexible open source CFD package. Oldroyd-B/UCM, Giesekus, Phan-Thien–Tanner (PTT), Finitely Extensible Nonlinear Elastic (FENE-P and FENE-CR), and Pom–Pom based constitutive equations were implemented, in single and multimode forms. The proposed methodology was evaluated by comparing its predictions with experimental and numerical data from the literature for the analysis of a planar 4:1 contraction flow, showing to be stable and efficient.     

Simulation of viscoelastic fluids in a 2D abrupt contraction by spectral element method

This study presents the vortex structure and numerical instability increase occurring when the level of elasticity is enhanced in inertial flows in planar contraction configuration for finitely extensible nonlinear elastic model by Peterlin (FENE‐P) fluid 1 . The re‐entrant corner effect on corner vortices is also considered. The calculations are performed using extended matrix logarithm formulation described in a previous paper: A. Jafari et al. A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements. Computer & Fluids 2010; 39 (9):1425–1438. In that reference, the proposed algorithm has been tested for simple geometry such as Poiseuille flow. In this study, we are interested in the capability of this algorithm for more complex geometry. This formulation helps to reach higher values of the Weissenberg number when compared with the classical one. Copyright © 2015 John Wiley & Sons, Ltd.     

Modeling and simulation of three‐dimensional extrusion swelling of viscoelastic fluids with PTT,Giesekus and FENE‐P constitutive models

The investigation of the extrusion swelling mechanism of viscoelastic fluids has both scientific and industrial interest. However, it has been traditionally difficult to afford theoretical and experimental researches to this problem. The numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three‐dimensional extrusion swelling of viscoelastic fluids flowing through out of a circular die. The rheological responses of viscoelastic fluids are described by using three kinds of differential constitutive models including the Phan‐Thien Tanner model, the Giesekus model, and the finite extensible nonlinear elastic dumbbell with a Peterlin closure approximation model. A streamface‐streamline method is introduced to adjust the swelling free surface. The calculation stability is improved by using the discrete elastic‐viscous split stress algorithm with the inconsistent streamline‐upwind scheme. The essential flow characteristics of viscoelastic fluids are predicted by using the proposed numerical method, and the mechanism of swelling phenomenon is further discussed.Copyright © 2013 John Wiley & Sons, Ltd.     

Two-dimensional planar flow of a viscoelastic plastic medium

The present study is concerned with finite element simulation of the planar entry flow of a viscoelastic plastic medium exhibiting yield stress. The numerical scheme is based on the Galerkin formulation. Flow experiments are carried out on a carbon black filled rubber compound. Steady-state pressure drops are measured on two sets of contraction or expansion dies having different lengths and a constant contraction or expansion ratio of 4:1 with entrance angles of 90, 45 and 15 degrees. The predicted and measured pressure drops are compared. The predicted results indicate that expansion flow has always a higher pressure drop than contraction flow. This prediction is in agreement with experimental data only at low flow rates, but not at high flow rates. The latter disagreement is possibly an indication that the assumption of fully-developed flow in the upstream and downstream regions is not realistic at high flow rates, even for the large length-to-thickness ratio channels employed. The evolution of the velocity, shear stress, and normal stress fields in the contraction or expansion flow and the location of pseudo-yield surfaces are also calculated.     

Extracting extensional properties through excess pressure drop estimation in axisymmetric contraction and expansion flows for constant shear viscosity,extension strain-hardening fluids

In this study, hyperbolic contraction–expansion flow (HCF) devices have been investigated with the specific aim of devising new experimental measuring systems for extensional rheological properties. To this end, a hyperbolic contraction–expansion configuration has been designed to minimize the influence of shear in the flow. Experiments have been conducted using well-characterized model fluids, alongside simulations using a viscoelastic White–Metzner/FENE-CR model and finite element/finite volume analysis. Here, the application of appropriate rheological models to reproduce quantitative pressure drop predictions for constant shear viscosity fluids has been investigated, in order to extract the relevant extensional properties for the various test fluids in question. Accordingly, experimental evaluation of the hyperbolic contraction–expansion configuration has shown rising corrected pressure drops with increasing elastic behaviour (De=0～16), evidence which has been corroborated through numerical prediction. Moreover, theoretical to predicted solution correspondence has been established between extensional viscosity and first normal stress difference. This leads to a practical means to measure extensional viscosity for elastic fluids, obtained through the derived pressure drop data in these HCF devices.     

VISCOELASTIC MODELLING OF ENTRANCE FLOW USING MULTIMODE LEONOV MODEL

A simulation of planar 2D flow of a viscoelastic fluid employing the Leonov constitutive equation has been presented. Triangular finite elements with lower-order interpolations have been employed for velocity and pressure as well as the extra stress tensor arising from the constitutive equation. A generalized Lesaint–Raviart method has been used for an upwind discretization of the material derivative of the extra stress tensor in the constitutive equation. The upwind scheme has been further strengthened in our code by also introducing a non-consistent streamline upwind Petrov–Galerkin method to modify the weighting function of the material derivative term in the variational form of the constitutive equation. A variational equation for configurational incompressibility of the Leonov model has also been satisfied explicitly. The corresponding software has been used to simulate planar 2D entrance flow for a 4:1 abrupt contraction up to a Deborah number of 670 (Weissenberg number of 6·71) for a rubber compound using a three-mode Leonov model. The predicted entrance loss is found to be in good agreement with experimental results from the literature. Corresponding comparisons for a commercial-grade polystyrene, however, indicate that the predicted entrance loss is low by a factor of about four, indicating a need for further investigation. © 1997 by John Wiley & Sons, Ltd.     

Solving viscoelastic free surface flows of a second‐order fluid using a marker‐and‐cell approach

This work is concerned with the numerical simulation of two‐dimensional viscoelastic free surface flows of a second‐order fluid. The governing equations are solved by a finite difference technique based on the marker‐and‐cell philosophy. A staggered grid is employed and marker particles are used to represent the fluid free surface. Full details for the approximation of the free surface stress conditions are given. The resultant code is validated and convergence is demonstrated. Numerical simulations of the extrudate swell and flow through a planar 4:1 contraction for various values of the Deborah number are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical simulation of contraction and expansion flows of Langmuir monolayers

Langmuir monolayers consist of amphiphilic molecules at the air–water interface and can be modeled as two-dimensional fluids. Earlier experiments [D.J. Olson, G.G. Fuller, J. Non-Newtonian Fluid Mech. 89 (2000) 187–207] on 4:1 contraction and 4:1 expansion flows have been simulated using an integral constitutive equation of the K-BKZ type, suitably modified to account for strain-thickening in the planar extensional viscosity. The model has been used to fit linear viscoelastic data (G′ and G″) and the shear viscosity (η_S), while the amount of strain-hardening is assumed, due to lack of experimental data. The simulations are in good agreement with the experiments on Newtonian monolayers, which show no vortices in the contraction but large inertial vortices in the expansion. For the viscoelastic monolayer (a poly-octadecyl methacrylate or PODMA), the opposite is true. The contraction flow shows vortices, while in the expansion flow the vortex activity is substantially reduced compared with the Newtonian one. The viscoelastic behavior is well captured by the model, provided that substantial strain-thickening is exhibited by the monolayer in planar extension. The latter behavior is very much like that for a branched LDPE melt, which also shows big vortices due to strain-hardening in planar as well as in uniaxial extension.     

Transonic Euler computation in streamfunction co-ordinates

Numerical simulation by a finite element method is used to examine the problem of the rotating flow of a viscoelastic fluid in a cylindrical vessel agitated with a paddle impeller. The mathematical model consists of a viscoelastic constitutive equation of Oldroyd B type coupled to the hydrodynamic equations expressed in a rotating frame. This system is solved by using an unsteady approach for velocity, pressure and stress fields. For Reynolds numbers in the range 0.1–10, viscoelastic effects are taken into account up to a Deborah number De of 1.33 and viscoelasticity and inertia cross-effects are studied. Examining the velocity and stress fields as well as the power consumption, it is found that their evolutions are significantly different for low and moderate inertia. These results confirm the trends of experimental studies and show the specific contribution of elasticity without interference of the pseudoplastic character found in actual fluids.     

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first‐order upwind approximation for the viscoelastic stress. A non‐uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non‐linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd‐B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.     

Effect of viscoelasticity on the rotation of a sphere in shear flow

When particles are dispersed in viscoelastic rather than Newtonian media, the hydrodynamics will be changed entailing differences in suspension rheology. The disturbance velocity profiles and stress distributions around the particle will depend on the viscoelastic material functions. Even in inertialess flows, changes in particle rotation and migration will occur. The problem of the rotation of a single spherical particle in simple shear flow in viscoelastic fluids was recently studied to understand the effects of changes in the rheological properties with both numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346] and experiments [Snijkers et al., J. Rheol. 53 (2009) 459–480]. In the simulations, different constitutive models were used to demonstrate the effects of different rheological behavior. In the experiments, fluids with different constitutive properties were chosen. In both studies a slowing down of the rotation speed of the particles was found, when compared to the Newtonian case, as elasticity increases. Surprisingly, the extent of the slowing down of the rotation rate did not depend strongly on the details of the fluid rheology, but primarily on the Weissenberg number defined as the ratio between the first normal stress difference and the shear stress.In the present work, a quantitative comparison between the experimental measurements and novel simulation results is made by considering more realistic constitutive equations as compared to the model fluids used in previous numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346]. A multimode Giesekus model with Newtonian solvent as constitutive equation is fitted to the experimentally obtained linear and nonlinear fluid properties and used to simulate the rotation of a torque-free sphere in a range of Weissenberg numbers similar to those in the experiments. A good agreement between the experimental and numerical results is obtained. The local torque and pressure distributions on the particle surface calculated by simulations are shown.     

Two-dimensional viscoelastic flows: experimentation and modeling

Extensive experimental data on the birefringence in converging and diverging flows of a polymeric melt have been obtained. The birefringence and pressure drop measurements were carried out in working cells of planar geometry having different contraction angles and contraction ratios. For investigation of diverging or abrupt expansion flow, the direction of flow in the cells was reversed. The theoretical predictions are based upon the Leonov constitutive equation and a finite element scheme with streamwise integration.In contrast to Newtonian and second-order fluids, viscoelastic fluids at high shear rates show significant differences in pressure drop and birefringence (i.e. stresses) in converging and diverging flows. For a constant flow rate, the pressure drop is higher and the birefringence smaller in diverging flows than in converging flows. This difference increases with increasing flow rate. Further, for the same contraction ratio but different contraction angles, the birefringence maximum increases considerably with contraction angle. In addition, an increase in contraction ratio has the same effect.The viscoelastic constitutive equation of Leonov has been shown to describe all the above viscoelastic effects observed in the experiments. In general, a reasonable agreement between theory and experiment has been obtained, which shows the usefulness of the Leonov model in describing actual flows.     

Numerical modeling and investigation of viscoelastic fluid–structure interaction applying an implicit partitioned coupling algorithm

We investigate the interaction between a viscoelastic Oldroyd-B fluid and an elastic structure via simulations applying an implicit partitioned coupling algorithm. Simulations are done for a flow through a channel with a flexible wall and a lid-driven cavity flow with flexible bottom. In addition, we make use of a mass–spring–dashpot prototype model to study the dynamic interaction problem. Both the simulation results and the analysis of the prototype model show that there are obvious differences in the fluid–structure interaction if the fluids are viscoelastic instead of purely viscous. These differences appear in the deformation of the solid at stationary state and in the equilibrium position, amplitude, frequency as well as phase shift of the oscillation. Moreover, we investigate the influence of numerical and physical parameters on the implicit partitioned coupling algorithm for simulation of viscoelastic fluid–structure interaction problems.     

A numerical study of constitutive models endowed with Pom-Pom molecular attributes

The branched polymer melts are modeled respectively in this investigation by the existing XPP and PTT–XPP models, along with the proposed S-MDCPP (Single/Simplified Modified Double Convected Pom-Pom) model developed on the basis of the existing MDCPP model. A pressure stabilized mass equation is formulated with the finite increment calculus (FIC) process to restrain and further eliminate spurious oscillations of pressure field due to the incompressibility of fluids. The discrete elastic viscous stress splitting (DEVSS) technique is employed, in order to retain an elliptic contribution in the weak form of the momentum equation. An inconsistent streamline-upwind (SU) method is applied to spatially discretize the constitutive equations. The mass, momentum conservation and constitutive equations are discretized and solved by the iterative stabilized fractional step algorithm along with the Crank–Nicolson implicit difference scheme. Thus the finite elements with equal low-order interpolation approximations for velocity–pressure–stress variables can be devised to numerically simulate the viscoelastic contraction flows for branched LDPE melts. The influences of the three viscoelastic constitutive models and the branched arms at the end of the Pom-Pom molecule on the rheological behaviors occurring in this complex flow are discussed. The numerical results demonstrate that the proposed S-MDCPP model is capable of reproducing some properties similar to those predicted by the XPP model in high shear flow and, on the other hand, reproducing some properties similar to those predicted by the PTT–XPP model in high elongational flow. Furthermore, the proposed S-MDCPP model is capable of well identifying the macromolecule topological structures of branched polymer melts.     

Velocity overshoots in gradual contraction flows

This study reports the results of a systematic numerical investigation, using the upper-convected Maxwell (UCM) and Phan-Thien–Tanner (PTT) models, of viscoelastic fluid flow through three-dimensional gradual planar contractions of various contraction ratios with the aim of investigating experimental observations of extremely large near-wall velocity overshoots in similar geometries [R.J. Poole, M.P. Escudier, P.J. Oliveira, Laminar flow of a viscoelastic shear-thinning liquid through a plane sudden expansion preceded by a gradual contraction, Proc. Roy. Soc. Lond. Ser. A 461 (2005) 3827]. We are able to obtain good qualitative agreement with the experiments, even using the UCM model in creeping-flow conditions, showing that neither inertia, second normal-stress difference nor shear-thinning effects are required for the phenomenon to be observed. Guided by the numerical results we propose a simple explanation for the occurrence of the velocity overshoots and the conditions under which they arise.     

A numerical method for steady and nonisothermal viscoelastic fluid flow for high Deborah and Péclet numbers

A combined finite element/streamline integration method is presented for nonisothermal flows of viscoelastic fluids. The attention is focused on some characteristic problems that arise for numerical simulation of flows with high Deborah and Péclet numbers. The two most important problems to handle are the choice of an outflow boundary condition for not completely developed flow and the treatment of the dissipative term in the temperature equation. The ability of the numerical method to handle high Deborah and Péclet numbers will be demonstrated on a contraction flow of an LDPE melt with isotropic and anisotropic heat conductivity. The influence of anisotropic heat conduction and the difference between the stress work and mechanical dissipation will be discussed for contraction flows. Received:  4 February 1997 Accepted: 23 October 1997