Do general viscoelastic stresses for the flow of an upper convected Maxwell fluid satisfy the momentum equation?

Given a general velocity field consistent with the stagnation point flow, can the viscoelastic stresses arising in the flow of an upper convected Maxwell fluid found by solving the constitutive equation also satisfy the momentum equation? Consideration is given to the study of the stress tensor arising in the steady flow of an upper convected Maxwell (UCM) fluid with a velocity field consistent with the stagnation point flow. By the method of characteristics, exact solutions to the partial differential equations arising in the approximating model of the viscoelastic stresses in the flow of an upper convected Maxwell (UCM) fluid are obtained for the three components of the stress tensor, for reasonably general velocity fields. We are able to account for the effects of variable boundary data at the inflow by considering the viscoelastic stresses over two spatial variables. Furthermore, we assume a relatively general velocity field. As a special case, some results present in the recent literature are obtained; it is known that these special case solutions do not satisfy the momentum equation. In the general case we consider, we find that the general solution will not satisfy the momentum equation except in a limited restricted case. We discuss how this shortcoming might be rectified by use of a more general velocity field.     

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.     

A finite element analysis of laminar unsteady flow of viscoelastic fluids through channels with non-uniform cross-sections

The effects of non-Newtonian behaviour of a fluid and unsteadiness on flow in a channel with non-uniform cross-section have been investigated. The rheological behaviour of the fluid is assumed to be described by the constitutive equation of a viscoelastic fluid obeying the Oldroyd-B model. The finite element method is used to analyse the flow. The novel features of the present method are the adoption of the velocity correction technique for the momentum equations and of the two-step explicit scheme for the extra stress equations. This approach makes the computational scheme simple in algorithmic structure, which therefore implies that the present technique is capable of handling large-scale problems. The scheme is completed by the introduction of balancing tensor diffusivity (wherever necessary) in the momentum equations. It is important to mention that the proper boundary condition for pressure (at the outlet) has been developed to solve the pressure Poisson equation, and then the results for velocity, pressure and extra stress fields have been computed for different values of the Weissenberg number, viscosity due to elasticity, etc. Finally, it is pertinent to point out that the present numerical scheme, along with the proper boundary condition for pressure developed here, demonstrates its versatility and suitability for analysing the unsteady flow of viscoelastic fluid through a channel with non-uniform cross-section.     

Pure axial flow of viscoelastic fluids in rectangular microchannels under combined effects of electro-osmosis and hydrodynamics

This paper presents an analysis of the combined electro-osmotic and pressure-driven axial flows of viscoelastic fluids in a rectangular microchannel with arbitrary aspect ratios. The rheological behavior of the fluid is described by the complete form of Phan-Thien–Tanner (PTT) model with the Gordon–Schowalter convected derivative which covers the upper convected Maxwell, Johnson–Segalman and FENE-P models. Our numerical simulation is based on the computation of 2D Poisson–Boltzmann, Cauchy momentum and PTT constitutive equations. The solution of these governing nonlinear coupled set of equations is obtained by using the second-order central finite difference method in a non-uniform grid system and is verified against 1D analytical solution of the velocity profile with less than 0.06% relative error. Also, a parametric study is carried out to investigate the effect of channel aspect ratio (width to height), wall zeta potential and the Debye–Hückel parameter on 2D velocity profile, volumetric flow rate and the Poiseuille number in the mixed EO/PD flows of viscoelastic fluids with different Weissenberg numbers. Our results show that, for low channel aspect ratios, the previous 1D analytical models underestimate the velocity profile at the channel half-width centerline in the case of favorable pressure gradients and overestimate it in the case of adverse pressure gradients. The results reveal that the inapplicability of the Debye–Hückel approximation at high zeta potentials is more significant for higher Weissenberg number fluids. Also, it is found that, under the specified values of electrokinetic parameters, there is a threshold for velocity scale ratio in which the Poiseuille number is approximately independent of channel aspect ratio.     

A numerical analysis of calendering

An analysis of calendering of inelastic (power-law) and viscoelastic sheets of finite initial thickness has been carried out using (i) a perturbation method based on lubrication theory; (ii) an approximate treatment including normal stress effects; (ii) a full numerical analysis using the boundary element method. The Phan-Thien-Tanner (PTT) fluid model was used in the viscoelastic analyses. Attention is focused on the separation criterion at the roll exit plane. While it is usual to assume in the inelastic case that separation occurs when the pressure and pressure gradient vanish simultaneously, it is not clear that this is appropriate in the viscoelastic model. The main new results are (a) a method of determining the separation point numerically using the criterion of zero tangential traction; (b) a computation of welling (∼ 5%) after the sheet leaves the nip; (c) a demonstration that the roll force first decreases as Weissenberg number (roll speed) rises, and then increases.     

An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows

In this paper, an incompressible smoothed particle hydrodynamics (SPH) method is presented to solve unsteady free-surface flows. Both Newtonian and viscoelastic fluids are considered. In the case of viscoelastic fluids, both the Maxwell and Oldroyd-B models are investigated. The proposed SPH method uses a Poisson pressure equation to satisfy the incompressibility constraints. The solution algorithm is an explicit predictor-corrector scheme and employs an adaptive smoothing length based on density variations. To alleviate the numerical difficulties encountered when fluid is highly stretched, an artificial stress term is incorporated into the momentum equation which reduces the risk of unrealistic fractures in the material. Two challenging test cases, the impacting drop and the jet buckling problems, are solved to demonstrate the capability of the proposed scheme in handling viscoelastic flows with complex free surfaces. The jet buckling test case was solved for a wide range of Weissenberg numbers. It was shown that in all cases the method is stable and fairly accurate and agrees well with the available data.     

Purely tangential flow of a PTT-viscoelastic fluid within a concentric annulus

An analytical solution is presented for the steady state, purely tangential flow of a viscoelastic fluid obeying the Phan-Thien–Tanner (PTT) constitutive equation in a concentric annulus with relative rotation of the inner and outer cylinders. The influence on the velocity distribution within the annulus and on fRe of the Weissenberg number, aspect ratio and an elongational parameter are investigated. The results show that the differences between the radial location of the minimum velocity and of the critical angular velocity compared with their Newtonian counterparts increase as the fluid elasticity increases. The results also show that fRe decreases with increasing Weissenberg number, radius ratio and the elongational parameter in the case of inner-cylinder rotation. In contrast, fRe increases with increasing radius ratio when the outer cylinder is rotating while the inner cylinder is at rest.     

A new numerical algorithm for viscoelastic fluid flows : The grid-by-grid inversion method

We present a new algorithm for solving viscoelastic flows with a general constitutive equation. In our approach the hyperbolic constitutive equation is split such that the term for the convective transport of stress tensor is treated as a source. This allows the stress tensor at each grid point to be expressed mainly in terms of the velocity gradient tensor at the same point. Then, the set of six stress tensor components is found after inverting a six by six matrix at each grid point. Thus we call this algorithm the grid-by-grid inversion method. The convective transport of stress tensor in the constitutive equation, which has been treated as a source, is updated iteratively. The present algorithm can be combined with finite volume method, finite element method or the spectral methods. To corroborate the accuracy and robustness of the present algorithm we consider viscoelastic flow past a cylinder placed at the center between two plates, which has served as a benchmark problem. Also considered is the investigation of the pattern and strength of the secondary flows in the viscoelastic flows through a rectangular pipe. It is found that the present method yields accurate results even for large relaxation times.     

黏弹流体流动的数值模拟研究进展

综述了黏弹流体流动数值模拟的研究进展,突出介绍近十年来有限元法在黏弹流体流动数值模拟研究中取得的成果,通过动量方程的适当变形和本构方程离散权函数的合理选择,可以显著增强数值计算的稳定性。得到较高Weissenberg数下的解,同时文中对黏弹流体流动数值模拟中本构方程的应用、非等温情况和三维空间下的研究进行了介绍。     

A marker-and-cell approach to viscoelastic free surface flows using the PTT model

This work is concerned with the development of a numerical method capable of simulating two-dimensional viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien–Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a marker-and-cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. In order to validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated.     

Stress diffusion and high order viscoelastic effects in the 3D flow past a sedimenting sphere subject to orthogonal shear

The effect of polymer stress diffusion in the unbounded flow past a sedimenting, freely rotating, rigid sphere subject to shear in a plane perpendicular to the direction of sedimentation is investigated analytically. Steady state, creeping, incompressible, and isothermal flow is assumed. For viscoelastic fluids following the Oldroyd-B constitutive model, three-dimensional results for the velocity vector, pressure, and viscoelastic extra-stress tensor are derived by including an artificial diffusion term in the constitutive equation and using regular perturbation theory with the small parameter being the Deborah number. The analytical solution reveals that the influence of the stress diffusion term on the results may be significant (and sometimes unexpected) and strongly depends on the magnitude of the dimensionless diffusion coefficient. For instance, it is shown that the critical Deborah number, below which a physical solution arises, decreases with the increase in the diffusion coefficient. Also, comparison against simulation results from the literature shows excellent agreement up to shear Weissenberg number (defined as the product of the imposed shear rate with the single relaxation time of the fluid) approximately equal to unity.     

A stable unstructured finite volume method for parallel large-scale viscoelastic fluid flow calculations

A new stable unstructured finite volume method is presented for parallel large-scale simulation of viscoelastic fluid flows. The numerical method is based on the side-centered finite volume method where the velocity vector components are defined at the mid-point of each cell face, while the pressure term and the extra stress tensor are defined at element centroids. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance the pressure–velocity–stress coupling. The log-conformation representation proposed in [R. Fattal, R. Kupferman, Constitutive laws for the matrix–logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] has been implemented in order improve the limiting Weissenberg numbers in the proposed finite volume method. The time stepping algorithm used decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. The resulting algebraic linear systems are solved using the FGMRES(m) Krylov iterative method with the restricted additive Schwarz preconditioner for the extra stress tensor and the geometric non-nested multilevel preconditioner for the Stokes system. The implementation of the preconditioned iterative solvers is based on the PETSc library for improving the efficiency of the parallel code. The present numerical algorithm is validated for the Kovasznay flow, the flow of an Oldroyd-B fluid past a confined circular cylinder in a channel and the three-dimensional flow of an Oldroyd-B fluid around a rigid sphere falling in a cylindrical tube. Parallel large-scale calculations are presented up to 523,094 quadrilateral elements in two-dimension and 1,190,376 hexahedral elements in three-dimension.     

PTT黏弹性流体的光滑粒子动力学方法模拟

运用光滑粒子流体动力学(smoothed particle hydrodynamics, SPH)方法对基于PHan-Thien-Tanner (PTT)模型的黏弹性流动进行了数值模拟. 首先, 利用SPH方法模拟了基于PTT模型的平板 Poiseuille流, 通过与文献结果的比较, 验证了SPH方法模拟黏弹性流动的准确性和有效性; 随后, 基于PTT模型对黏弹性自由表面流-液滴碰撞问题进行了SPH模拟, 研 究了PTT模型中拉伸参数对碰撞过程的影响. 为了解决张力不稳定问题, 采用简化的 人工应力公式. 数值结果表明, SPH方法可有效而灵活地模拟黏弹自由表面流问题.     

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.     

Flow of an Oldroyd-fluid in a contracting channel with a moving boundary

The industrial process of coating flat surfaces with polymeric substances is numerically simulated by solving the full equations of motion for a flow through a contraction with a moving boundary. The four-constant Oldroyd constitutive equation is used to represent the viscoelastic fluid. Some adjustments to existing finite-difference methods are made in such a way as to avoid singular iterative matrices during the solution process. Results are presented for flow situations with Weissenberg numbers up to about three times larger than any previously published results for this problem.     

An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG

We report an adaptive viscoelastic stress splitting (AVSS) scheme, which was found to be extremely robust in the numerical simulation of viscoelastic flow involving steep stress boundary layers. The scheme is different from the elastic viscous split stress (EVSS) in that the local Newtonian component is allowed to depend adaptively on the magnitude of the local elastic stress. Two decoupled versions of the scheme were implemented for the Upper Convected Maxwell (UCM) model: the streamline integration (AVSS/SI), and the streamline upwind Petrov-Galerkin (AVSS/SUPG) methods of integrating the stress. The implementations were benchmarked against the known analytic Poiseuille solution, and no upper limiting Weissenberg number was found (the computation was stopped at Weissenberg number of O(10⁴)). The flow past a sphere in a tube was solved next, giving convergent results up to a Weissenberg number of 3.2 with the AVSS/SI and 1.55 with the AVSS/SUPG (both were decoupled schemes; without the adaptive scheme, the limiting Weissenberg number for the decoupled streamline integration method was about 0.3). These results show that the decoupled AVSS is more stable than the corresponding EVSS, and the SI is more robust than SUPG in solving the constitutive equation of hyperbolic type. In addition, we found a very long stress wake behind the sphere, and a weak vortex in the rear stagnation region at a Weissenberg number above W_i of about 1.6, which does not seem to increase in size or strength with increasing W_i.     

Wave propagation in 3-D poroelastic media including gradient effects

In the framework of the theory of mixtures, the governing equations of motion of a fluid-saturated poroelastic medium including microstructural (for both the solid and the fluid) and micro-inertia (for the solid) effects are derived. This is accomplished by appropriately combining the conservation of mass and linear momentum equations with the constitutive equations for both the solid and the fluid constituents. The solid is assumed to be gradient elastic, that is, its stress tensor depends on the strain and the second gradient of strain tensor. The fluid is assumed to have an analogous behavior, that is, its stress tensor depends on the pressure and the second gradient of pressure. A micro-inertia term in the form of the second gradient of the acceleration of the solid is also included in the equations of motion. The equations of motion in three dimensions are seven equations with seven unknowns, the six displacement components for the solid and the fluid and the pore-fluid pressure. Because of the microstructural effects, the order of these equations is two degrees higher than in the classical case. Application of the divergence and the rot operations on these equations enable one to study the propagation of plane harmonic waves in the infinitely extended medium separately in the form of dilatational and rotational dispersive waves. The effects of the microstructure and the micro-inertia on the dispersion curves are determined and discussed.     

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 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.     

Numerical simulation of the fountain flow instability in injection molding

For the first time, the viscoelastic flow front instability is studied in the full non-linear regime by numerical simulation. A two-component viscoelastic numerical model is developed which can predict fountain flow behavior in a two-dimensional cavity. The eXtended Pom-Pom (XPP) viscoelastic model is used. The levelset method is used for modeling the two-component flow of polymer and gas. The difficulties arising from the three-phase contact point modeling are addressed, and solved by treating the wall as an interface and the gas as a compressible fluid with a low viscosity. The resulting set of equations is solved in a decoupled way using a finite element formulation. Since the model for the polymer does not contain a solvent viscosity, the time discretized evolution equation for the conformation tensor is substituted into the momentum balance in order to obtain a Stokes like equation for computing the velocity and pressure at the new time level. Weissenberg numbers range from 0.1 to 10. The simulations reveal a symmetric fountain flow for Wi = 0.1–5. For Wi = 10 however, an oscillating motion of the fountain flow is found with a spatial period of three times the channel height, which corresponds to experimental observations.