Viscoelastic Flow Problems Research Articles

Computing flow-induced stresses of injection molding based on PTT model

A numerical approach is introduced to solve the viscoelastic flow problem of filling and post-filling in injection molding. The governing equations are in terms of compressible, non-isothermal fluid, and the constitutive equation is based on the PTT model. By introducing some hypotheses according to the characteristics of injection molding, a quasi-Poisson type equation about pressure is derived with part integration. Besides, an analytical form of flow-induced stress is also generalized by using Undermined Coefficient Method. The conventional Galerkin approach is employed to solve the derived pressure equation, and the 'upwind' difference scheme is used to discrete the energy equation. Coupling is achieved between velocity and stress by Super Relax Iteration Method. The flow in the test mold is investigated by comparing the numerical results and photoelastic photos for polystyrene, showing flow-induced stresses are closely related to melt temperatures. The filling of a two-cavity box is also studied to investigate the viscoelastic effects on real injection molding.

A monolithic FEM approach for the log-conformation reformulation (LCR) of viscoelastic flow problems

In this work, we discuss special numerical techniques for viscoelastic flow problems given in log-conformation reformulation (LCR). In particular, we consider Oldroyd-B and Giesekus-type fluids. We utilize a fully coupled monolithic finite element approach that treats all the numerical variables simultaneously. Thus, it is possible to do a direct steady approach and to avoid pseudo-time stepping with correspondingly small time step sizes in the case of a nonsteady approach. The Newton method handles the discrete nonlinear system, which results from the FEM discretization with consistent edge-oriented FEM stabilization techniques. In each nonlinear step, a direct sparse solver or a geometrical multigrid solver with special Vanka smoother deals with the resulting linear subproblems. Moreover, local grid refinement helps to reduce the computational efforts and to increase the accuracy of functional values. The merit of the presented methodology, for the well-known ‘flow around cylinder’ benchmark problem, is that we can obtain the discrete approximations by using a direct steady approach. Thus, the numerical effort can be rather independent of the examined We numbers. Furthermore, the ‘black box’ techniques can deal with any given viscoelastic models easily, hereby showing the same advantageous numerical convergence behaviour of the above mentioned fluids.

On error estimates of the penalty method for the viscoelastic flow problem I: Time discretization

In this paper, we investigate the two-dimensional viscoelastic fluid motion problem arising in the Oldroyd model. By applying the dual method, the Helmholtz decomposition and some other techniques, we deduce the long-time optimal error estimates for its penalty method. Furthermore, the global optimal error bounds in the L 2 and H 1-norms for the time semi-discrete of the penalty system are derived under the backward Euler implicit scheme, which improves the best estimate available.

An optimization strategy for die design in the low-density polyethylene annular extrusion process based on FES/BPNN/NSGA-II

An optimization strategy for die design in the polymer extrusion process is proposed in the study based on the finite element simulation, the back-propagation neural network, and the non-dominated sorting genetic algorithm II (NSGA-II). The three-dimensional simulation of polymer melts flow in the extrusion process is conducted using the penalty finite element method. The model for predicting the flow patterns in the extrusion process is established with the artificial neural network based on the simulated results. The non-dominated sorting genetic algorithm II is performed for the search of globally optimal design variables with its objective functions evaluated by the established neural network model. The proposed optimization strategy is successfully applied to the die design in low-density polyethylene (LDPE) annular extrusion process. A constrained multi-objective optimization model is established according to the characteristics of annular extrusion process. The minimum of velocity relative difference, δu, and the minimum of swell ratio, Sw, that, respectively, ensure the extrinsic feature, mechanical property, and dimensional precision of the final products are taken as optimization objectives with a constrained condition on the maximum shear stress. Three important die structure parameters, including the die contraction angle α, the ratio of parallel length to inner radius L/Ri, and the ratio of outer to inner radius Ro/Ri, are taken as design variables. The Phan-Thien–Tanner constitutive model is adopted to describe the viscoelastic rheological characteristics of LDPE whose parameters are fitted by the distributions of material functions detected on the strain-controlled rheometer. The penalty finite element model of polymer melts flowing through out of the extrusion die is derived. A decoupled method is employed to solve the viscoelastic flow problem with the discrete elastic-viscous split-stress algorithm. The simulated results are selected and extracted to constitute the learning samples according to the orthogonal experimental design method. The back propagation algorithm is adopted for the training and the establishment of the predicting model for the optimization objective. A Pareto-optimal set for the constrained multi-objective optimization is obtained using the constrained NSGA-II, and the optimal solution is extracted based on the fuzzy set theory. The optimization for die parameters in the annular extrusion process of low-density polyethylene is performed and the optimization objective is successfully achieved.

Heat transfer in a viscoelastic boundary layer flow over a stretching sheet

Abstract Studies are made on the viscoelastic fluid flow and heat transfer characteristics over a stretching sheet with frictional heating and internal heat generation or absorption. The heat transfer analysis has been carried out for the cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF). The momentum equation is decoupled from the energy equation for the present incompressible boundary layer flow problem with constant physical parameters. Exact solution for the velocity field and the skin-friction are obtained. Also, the solutions for the temperature and heat transfer characteristics are obtained in terms of Kummer’s function. The work due to deformation in energy equation, which is essential while formulating the viscoelastic boundary layer flow problems, is considered. This paper examines the effect of viscoelastic parameter, Eckert number, Prandtl number and non-uniform heat source/sink parameter on temperature distribution, wall temperature gradient for PST-case and wall temperature for PHF-case.

Numerical modeling of viscoelastic flows using equal low-order finite elements

A mixed finite element scheme abbreviated as I_PS_DEVSS_CNBS scheme for modeling viscoelastic flow problems is presented. The finite incremental calculus (FIC) pressure stabilization process and the discrete elastic-viscous stress-splitting method (DEVSS) are introduced into the general framework of the iterative version of the fractional step algorithm with the use of the Crank–Nicolson-based splitting. Inconsistent streamline upwinding method (SU) is employed to spatially discretize the constitutive equation of viscoelastic fluids. Equal low-order finite elements which violate the LBB compatibility conditions are successfully used in the proposed scheme. In addition, the Oldroyd-B and the PTT models have been integrated into the proposed scheme to solve the 4:1 sudden contraction flow problem. The numerical results demonstrate prominent stability and accuracy of both pressure and stress distributions over the flow domains provided by the proposed scheme within the Weissenberg number range studied in the present work, as compared with the reference solutions reported in the literatures.

A comparison of four implementations of the log-conformation formulation for viscoelastic fluid flows

We compare four finite element implementations of a log-conformation formulation for the numerical solution of viscoelastic fluid flow problems. These formulations essentially differ the treatment of the advective term. The first formulation was introduced in Fattal and Kupferman (2005) [1] while the second one, much simpler to implement, was proposed by Coronado et al. (2007) [2]. We then introduce the two variants of this last formulation. In the first one, a special treatment is done to the computation of the advective term u ⋅ ∇ e s which is more accurate than the one described in Coronado et al. (2007) [2]. In the second variant, we show how e s and ∇ e s can be computed in order to allow a complete linearization of the constitutive equation and its solution by Newton’s method. A comparative study of these four formulations is done on the benchmark problem of the flow of an Oldroyd-B fluid past a circular cylinder in a rectangular channel.

Heat transfer in a Walter’s liquid B fluid over an impermeable stretching sheet with non-uniform heat source/sink and elastic deformation

In this present article an analysis is carried out to study the boundary layer flow behavior and heat transfer characteristics in Walter’s liquid B fluid flow. The stretching sheet is assumed to be impermeable, the effects of viscous dissipation, non-uniform heat source/sink in the presence and in the absence of elastic deformation (which was escaped from attention of researchers while formulating the viscoelastic boundary layer flow problems)on heat transfer are addressed. The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. Analytical solutions are obtained for the resulting boundary value problems. The effects of viscous dissipation, Prandtl number, Eckert number and non-uniform heat source/sink on heat transfer (in the presence and in the absence of elastic deformation) are shown in several plots and discussed. Analytical expressions for the wall frictional drag coefficient, non-dimensional wall temperature gradient and non-dimensional wall temperature are obtained and are tabulated for various values of the governing parameters. The present study reveals that, the presence of work done by deformation in the energy equation yields an augment in the fluid’s temperature.

A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes

We present special numerical simulation methods for non-isothermal incompressible viscous fluids which are based on LBB-stable FEM discretization techniques together with monolithic multigrid solvers. For time discretization, we apply the fully implicit Crank–Nicolson scheme of 2nd order accuracy while we utilize the high order Q2P1 finite element pair for discretization in space which can be applied on general meshes together with local grid refinement strategies including hanging nodes. To treat the nonlinearities in each time step as well as for direct steady approaches, the resulting discrete systems are solved via a Newton method based on divided differences to calculate explicitly the Jacobian matrices. In each nonlinear step, the coupled linear subproblems are solved simultaneously for all quantities by means of a monolithic multigrid method with local multilevel pressure Schur complement smoothers of Vanka type. For validation and evaluation of the presented methodology, we perform the MIT benchmark 2001 [M.A. Christon, P.M. Gresho, S.B. Sutton, Computational predictability of natural convection flows in enclosures, in: First MIT Conference on Computational Fluid and Solid Mechanics, vol. 40, Elsevier, 2001, pp. 1465–1468] of natural convection flow in enclosures to compare our results with respect to accuracy and efficiency. Additionally, we simulate problems with temperature and shear dependent viscosity and analyze the effect of an additional dissipation term inside the energy equation. Moreover, we discuss how these FEM-multigrid techniques can be extended to monolithic approaches for viscoelastic flow problems.

An artificial‐dissipation‐based fractional step scheme for upper‐convected Maxwell (UCM) fluid flow past a circular cylinder

AbstractThe fully explicit characteristic‐based split (CBS) scheme is employed to solve viscoelastic flow problems. The upper‐convected Maxwell (UCM) model is employed in the present study. In addition to allowing equal‐order interpolations for pressure and velocity, the proposed method along with an appropriate artificial damping scheme is able to produce stable solutions for different Deborah numbers (De). The higher‐order time terms introduced by the simplified characteristic Galerkin approximation are sufficient to obtain stable solution at very low De, but an additional damping is essential to maintain positive definitiveness of the conformation tensor at higher De. We demonstrate the need for an additional damping by analysing the basic forward time central space scheme applied to the constitutive equations. A second‐order artificial damping is employed to counteract the negative dissipation introduced by the explicit time discretization. The example studied in this paper is the widely used and difficult problem of flow past a circular cylinder. The results presented show that the velocity and extra stresses converge easily to a steady state at lower De values. At higher De values, the convergence to steady state is slow due to the incremental way in which the artificial damping is added. Copyright © 2008 John Wiley & Sons, Ltd.

A direct simulation method for flows with suspended paramagnetic particles

A direct numerical simulation method based on the Maxwell stress tensor and a fictitious domain method has been developed to solve flows with suspended paramagnetic particles. The numerical scheme enables us to take into account both hydrodynamic and magnetic interactions between particles in a fully coupled manner. Particles are assumed to be non-Brownian with negligible inertia. Rigid body motions of particles in 2D are described by a rigid-ring description implemented by Lagrange multipliers. The magnetic force, acting on the particles due to magnetic fields, is represented by the divergence of the Maxwell stress tensor, which acts as a body force added to the momentum balance equation. Focusing on two-dimensional problems, we solve a single-particle problem for verification. With the magnetic force working on the particle, the proper number of collocation points is found to be two points per element. The convergence with mesh refinement is verified by comparing results from regular mesh problems with those from a boundary-fitted mesh problem as references. We apply the developed method to two application problems: two-particle interaction in a uniform magnetic field and the motion of a magnetic chain in a rotating field, demonstrating the capability of the method to tackle general problems. In the motion of a magnetic chain, especially, the deformation pattern at break-up is similar to the experimentally observed one. The present formulation can be extended to three-dimensional and viscoelastic flow problems.

Application of a Modified Unsteady Term for Improving Convergence Behavior of Steady-State Finite Element Analysis of Viscoelastic Fluids

To improve the convergence behavior of the decoupled finite element method for the steady-state simulation of viscoelastic flow problems, we introduced a modified unsteady term into the momentum and constitutive equations, respectively. To make sure an advantage of the new technique, the eccentric flow problem of viscoelastic fluid was chosen by comparing stability of the simulations in the lower and higher Weissenberg numbers. It was shown that (1) the simulation results obtained with and without the new technique are the same, (2) the new technique is more effective for calculating the viscoelastic flows in the higher Weissenberg numbers. Furthermore, to investigate the flexibility of this technique, we applied it to the die-swell flow analysis of viscoelastic fluid. It was shown that the technique also improves convergence behavior in the die-swell flow simulation.

Asymptotic behavior of linearized viscoelastic flow problem

In this article, we provide some asymptotic behaviors of linearized viscoelastic flows in a general two-dimensional domain with certain parameters small and the time variable large.

An adaptive remeshing strategy for viscoelastic fluid flow simulations

In the last few years, we have developed a fairly general adaptive finite element solution procedure which can be applied to a large variety of problems. In this paper, this strategy is briefly recalled and applied to the solution of two-dimensional viscoelastic fluid flow problems. A log-conformation formulation recently introduced by Fattal and Kupferman [R. Fattal, R. Kupferman, Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation, J. Non-Newtonian Fluid Mech. 126 (2005) 23-37] was implemented in order to improve the convergence properties of the numerical scheme. We confirm some results obtained in Hulsen, Fattal and Kupferman [M. Hulsen, R. Fattal, R. Kupferman, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithm, J. Non-Newtonian Fluid Mech. 127 (2005) 27-39] and in some instances, we show that mesh adaptation allows to almost automatically reproduce accurate results obtained on very fine structured meshes.

Computing flow-induced stresses of injection molding based on the Phan–Thien–Tanner model

A numerical approach is introduced to solve the viscoelastic flow problem of filling and post-filling in injection molding. The governing equations are in terms of compressible, non-isothermal fluid, and the constitutive equation is based on the Phan–Thien–Tanner model. By introducing some hypotheses according to the characteristics of injection molding, a quasi-Poisson type equation about pressure is derived with part integration. Besides, an analytical form of flow-induced stress is also generalized by using the undermined coefficient method. The conventional Galerkin approach is employed to solve the derived pressure equation, and the ‘upwind’ difference scheme is used to discrete the energy equation. Coupling is achieved between velocity and stress by super relax iteration method. The flow in the test mold is investigated by comparing the numerical results and photoelastic photos for polystyrene, showing flow-induced stresses are closely related to melt temperatures. The filling of a two-cavity box is also studied to investigate the viscoelastic effects on real injection molding.

A simple method for simulating general viscoelastic fluid flows with an alternate log-conformation formulation

The log-conformation formulation has alleviated the long-standing high Weissenberg number problem associated with the viscoelastic fluid flows [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285]. This formulation ensures that solutions of viscoelastic flow problems are physically admissible, and it is able to capture sharp elastic stress layers. However, the implementations presented in literature thus far require changing the evolution equation for the conformation tensor into an equation for its logarithm, and are based on loosely coupled (partitioned) solution procedures [M.A. Hulsen, et al., Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]. A simple alternate form of the log-conformation formulation is presented in this article, and an implementation is demonstrated in the DEVSS-TG/SUPG finite element method. Besides its straightforward implementation, the new log-conformation formulation can be used to solve all the governing equations (continuity, conservation of momentum and constitutive equation) in a strongly coupled way by Newton’s method. The method can be applied to any conformation tensor model. The flows of Oldroyd-B and Larson-type fluids are tested in the benchmark problem of a flow past a cylinder in a channel. The accuracy of the method is assessed by comparing solutions with published results. The benefits of this new implementation and the pending issues are discussed.

Numerical simulation of three‐dimensional polymer extrusion flow with differential viscoelastic model

AbstractThe numerical simulation of viscoelastic flow problems is nowadays an effective way of investigating the complex flow mechanism related to practical engineering problems, such as plastic injection, blow molding and extrusion. The mathematical model of a three‐dimensional (3D) viscoelastic flow in a typical contraction die for polymer extrusion is established and a stable solving method is investigated. The penalty finite element method (FEM) is performed to simulate the viscoelastic melts flow in the channel with a differential constitutive model. The discrete elastic‐viscous split stress (DEVSS) formulation and the streamline‐upwind Petrov–Galerkin (SUPG) technology are employed to improve the computation stability. Both the implementation of the numerical scheme and its application in the practical process analysis are investigated. The effects of various calculation control parameters and different material parameters upon the numerical results are discussed. The 3D flow patterns in the extrusion die with different contraction angles are further investigated based on the above discussions. Copyright © 2007 John Wiley & Sons, Ltd.

Numerical simulations of shear dependent viscoelastic flows with a combined finite element–finite volume method

A hybrid combined finite element–finite volume method has been developed for the numerical simulation of shear-dependent viscoelastic flow problems governed by a generalized Oldroyd-B model with a non-constant viscosity function. The method is applied to the 4:1 planar contraction benchmark problem, to investigate the influence of the viscosity effects on the flow and results are compared with those found in the literature for creeping Oldroyd-B flows, for a range of Weissenberg numbers. The method is also applied to flow in a smooth stenosed channel. It is shown that the qualitative behavior of the flow is influenced by the rheological properties of the fluid, namely its viscoelastic and inertial effects, as well as the shear-thinning viscosity. These results appear in the framework of a preliminary study of the numerical simulation of steady and pulsatile blood flows in two-dimensional stenotic vessels, using this hybrid finite element–finite volume method.

An efficient algorithm for multiscale flow simulation of dilute polymeric solutions using bead-spring chains

In this study, a computationally efficient algorithm for multiscale flow simulation of dilute polymer solutions using a bead-spring chain description of polymer molecules is presented. The algorithm combines a computationally efficient extension of the earlier BCF-based semi-implicit method (i.e., approximately four-fold speed up) for multiscale flow simulations using a bead-spring dumbbell description [M. Somasi, B. Khomami, Linear stability and dynamics of viscoelastic flows using time-dependent stochastic simulation techniques, J. Non-Newtonian Fluid Mech. 93 (2000) 339–362] with a highly CPU efficient predictor–corrector scheme for BD simulation of bead-spring chains [M. Somasi, B. Khomami, N. Woo, J. Hur, E. Shaqfeh, Brownian dynamics simulations of bead-rod and bead-spring chains: numerical algorithms and coarse-graining issues, J. Non-Newtonian Fluid Mech. 108 (2002) 227–255]. The fidelity and computational efficiency of the parallel implementation of the algorithm are demonstrated via three benchmark flow problems, namely, plane Couette flow, Poiseuille flow and 4:1:4 axisymmetric contraction–expansion flow. The algorithm shows linear speed up with the number of processors and more importantly with the number of chain segments. In addition, the proposed algorithm is approximately 50 times faster in comparison to the only existing fully implicit method [J. Ramirez, M. Laso, Size reduction methods for the implicit time-dependent simulation of micro-macro viscoelastic flow problems, J. Non-Newtonian Fluid Mech. 127 (2005) 41–49].

Basic viscoelastic fluid flow problems using the Jeffreys model

Using Laplace transformation technique, semi-analytical solutions are obtained for three basic viscoelastic fluid flow problems under the effect of the Jeffreys model. These semi-analytical solutions are not available in the literature. The present work investigates the effect of two types of driving forces on the flow behavior. These two types are the velocity-type and shear-type driving forces. The effect of the relaxation and retardation times on the flow behavior for these two types of driving forces may be viewed well using the obtained semi-analytical solutions. The three fundamental problems are transient Couette flow, transient wind-driven flow over finite domains and the transient Poiseuille flow in parallel-plates channels. It is shown that as the dimensionless relaxation time ( λ 1 ) increases, the flow response to the imposed driving force becomes slower. This implies that the flow needs more time to feel the presence of the driving force and hence needs more time to attain steady-state behavior. On the other hand, the effect of the dimensionless retardation time ( λ 2 ) depends on the type of the driving force imposed on the system. For a velocity-type driving force, the flow response becomes faster as the dimensionless retardation time ( λ 2 ) increases and for a shear-type driving force the flow response becomes slower as the dimensionless retardation time ( λ 2 ) increases.