Practical comparison of differential viscoelastic constitutive equations in finite element analysis of planar 4:1 contraction flow

Finite element modeling of planar 4:1 contraction flow (isothermal incompressible and creeping) around a sharp entrance corner is performed for favored differential constitutive equations such as the Maxwell, Leonov, Giesekus, FENE-P, Larson, White-Metzner models and the Phan Thien-Tanner model of exponential and linear types. We have implemented the discrete elastic viscous stress splitting and streamline upwinding algorithms in the basic computational scheme in order to augment stability at high flow rate. For each constitutive model, we have obtained the upper limit of the Deborah number under which numerical convergence is guaranteed. All the computational results are analyzed according to consequences of mathematical analyses for constitutive equations from the viewpoint of stability. It is verified that in general the constitutive equations proven globally stable yield convergent numerical solutions for higher Deborah number flows. Therefore one can get solutions for relatively high Deborah number flows when the Leonov, the Phan Thien-Tanner, or the Giesekus constitutive equation is employed as the viscoelastic field equation. The close relationship of numerical convergence with mathematical stability of the model equations is also clearly demonstrated.     

Stress gradient-induced migration effects in the Taylor–Couette flow of a dilute polymer solution

We present an investigation of the phenomenon of stress-induced polymer migration for dilute polymer solutions in the Taylor–Couette device, consisting of two infinitely long, concentric cylinders rotating at constant angular velocities. The underlying physical model is represented by the dilute limit of a two-fluid Hamiltonian system involving two components: one (the polymer) is viscoelastic and obeys the Oldroyd-B constitutive equation, and the other (the solvent) is viscous Newtonian. The two components are considered to be in thermal, but not mechanical equilibrium, interacting with each other through an isotropic drag coefficient tensor. This allows for stress-induced diffusion of polymer chains. The governing equations consist of the continuity and the momentum equations for the bulk velocity, the constitutive model for the polymer chain conformation tensor and the diffusion equation for the polymer concentration. The diffusion equation contains an extra source term, which is proportional to gradients in the polymer stress, so that polymer concentration gradients can develop even in the absence of externally imposed fluxes in the presence of stress inhomogeneities. The solution to the steady-state purely azimuthal flow is obtained first using a spectral collocation method and an adaptive mesh formulation to track the steep changes of the concentration in the flow domain. The calculations show the development of strong polymer migration towards the inner cylinder with increasing Deborah number (De) in agreement with experimental observations. The migration is enhanced for increasing values of the gap thickness resulting in concentration changes by several orders of magnitude in the area between the inner and outer cylinder walls. The extent of the migration also depends strongly on the ratio of the solvent to the polymer viscosity. In addition to a strongly inhomogeneous polymer concentration, significant deviations from the homogenous flow are also observed in the velocity profile. Next, results are reported from a linear stability analysis around the steady-state solution against axisymmetric disturbances corresponding to various wavenumbers in the axial direction. The calculations show that the steady-state solution remains stable up to moderate values of the Deborah number, explaining why some of the predicted stress-induced migration effects should be experimentally observable. The role of the Peclet number (Pe) on the stability of the system is elucidated.     

On the fully developed tube flow of a class of non-linear viscoelastic fluids

The fully developed pipe flow of a class of non-linear viscoelastic fluids is investigated. Analytical expressions are derived for the stress components, the friction factor and the velocity field. The friction factor which depends on the Deborah and Reynolds numbers is substantially smaller than the corresponding value for the Newtonian flow field with implications concerning the volume flow rate. We show that non-affine models in the class of constitutive equations considered such as Johnson-Segalman and some versions of the Phan-Thien-Tanner models are not representative of physically realistic flow fields for all Deborah numbers. For a fixed value of the slippage factor they predict physically admissible flow fields only for a limited range of Deborah numbers smaller than a critical Deborah number. The latter is a function of the slippage.     

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.     

PIV measurements of slow flow of a viscoelastic fluid within a porous medium

Particle image velocimetry (PIV) and pressure loss measurements were used to investigate slow flow through a square array of cylinders having a solid fraction of 10%. The test fluids were a Newtonian fluid and a Boger fluid, both of high viscosity such that the Reynolds number did not exceed 0.1. The pressure loss data reveal that the onset of elastic effects occurred at a Deborah number around 0.5 and that flow resistance was up to several times Newtonian values at Deborah numbers up to 3. PIV showed that the transverse velocity profiles for the Newtonian and non-Newtonian fluid were the same at Deborah numbers below onset. Above onset, the profiles became skewed, increasingly so as the Deborah number increased. In the wake regions between cylinders in a column, periodic flow structures formed in the spanwise direction. The structures were staggered from column to column, consistent with the skewing and were offset. These flow patterns are the result of an apparent elastic instability.     

Numerical breakdown at high Weissenberg number in non-Newtonian contraction flows

 Planar contraction flows of non-Newtonian fluids with integral constitutive models are studied to investigate the problem of numerical breakdown at high Weissenberg or Debrorah numbers. Spurious shear stress extrema are found on the wall downstream of the re-entrant corner for both sharp and rounded corners. Moreover, a non-monotonic relation between shear stress and strain rate is found when the Deborah number limit is approached, which correlates with these shear extrema. This strongly suggests that non-monotonicity between shear stress and strain rate may be responsible for the Deborah number limit problem in contraction flow simulations. This non-monotonicity is caused by the inaccuracy of the quadrature, using constitutive equations that do not have shear stress maxima when exactly evaluated. This conclusion agrees with recent analytical findings by others that inaccuracy of the integration along the streamlines – either by numerical integration or asymptotic approximation – makes the problem ill-conditioned, with spurious growth occurring on the wall downstream of the re-entrant corner. Received: 5 March 1999/Accepted: 1 September 1999     

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.     

The characteristic‐based split (CBS) scheme for viscoelastic flow past a circular cylinder

A fully explicit, characteristic‐based split (CBS) method for viscoelastic flow past a circular cylinder, placed in a rectangular channel, is presented. The pressure equation in its explicit form is employed via an artificial compressibility parameter. The constitutive equations used here are based on the Oldroyd‐B model. No loss of convergence to steady state was observed in any of the results presented in this paper. Comparison of the present results with other available numerical data shows that the CBS algorithm is in excellent agreement with them at lower Deborah numbers. However, at higher Deborah numbers, the present results differ from other numerical solutions. This is due to the fact that the positive definitiveness of the conformation matrix is lost between a Deborah number of 0.6 and 0.7. However, the positive definitiveness is retained when an artificial diffusion is added to the discrete constitutive equations at higher Deborah numbers. It appears that the fractional solution stages used in the CBS scheme and the higher‐order time step‐based convection stabilization clearly reduce the instability at higher Deborah numbers. The Deborah number limit reached in the present work is three without artificial dissipation and two with artificial dissipation. Copyright © 2007 John Wiley & Sons, Ltd.     

Influence of viscoelasticity on drop deformation and orientation in shear flow: Part 1. Stationary states

The influence of matrix and droplet viscoelasticity on the steady deformation and orientation of a single droplet subjected to simple shear is investigated microscopically. Experimental data are obtained in the velocity–vorticity and velocity–velocity gradient plane. A constant viscosity Boger fluid is used, as well as a shear-thinning viscoelastic fluid. These materials are described by means of an Oldroyd-B, Giesekus, Ellis, or multi-mode Giesekus constitutive equation. The drop-to-matrix viscosity ratio is 1.5. The numerical simulations in 3D are performed with a volume-of-fluid algorithm and focus on capillary numbers 0.15 and 0.35. In the case of a viscoelastic matrix, viscoelastic stress fields, computed at varying Deborah numbers, show maxima slightly above the drop tip at the back and below the tip at the front. At both capillary numbers, the simulations with the Oldroyd-B constitutive equation predict the experimentally observed phenomena that matrix viscoelasticity significantly suppresses droplet deformation and promotes droplet orientation. These two effects saturate experimentally at high Deborah numbers. Experimentally, the high Deborah numbers are achieved by decreasing the droplet radius with other parameters unchanged. At the higher capillary and Deborah numbers, the use of the Giesekus model with a small amount of shear-thinning dampens the stationary state deformation slightly and increases the angle of orientation. Droplet viscoelasticity on the other hand hardly affects the steady droplet deformation and orientation, both experimentally and numerically, even at moderate to high capillary and Deborah numbers.     

An approximate solution for the Couette–Poiseuille flow of the Giesekus model between parallel plates

An approximate analytical solution is derived for the Couette–Poiseuille flow of a nonlinear viscoelastic fluid obeying the Giesekus constitutive equation between parallel plates for the case where the upper plate moves at constant velocity, and the lower one is at rest. Validity of this approximation is examined by comparison to the exact solution during a parametric study. The influence of Deborah number (De) and Giesekus model parameter (α) on the velocity profile, normal stress, and friction factor are investigated. Results show strong effects of viscoelastic parameters on velocity profile and normal stress. In addition, five velocity profile types were obtained for different values of α, De, and the dimensionless pressure gradient (G).     

Stability of planar stagnation flow of a highly viscoelastic fluid

Linear stability analysis is used to predict the onset of instabilities in inertialess viscoelastic planar stagnation flow. Beyond a critical value of the dimensionless flow rate, or Deborah number, the creeping base flow of similarity type, which is valid in the limit of vanishingly small Reynolds numbers, becomes unstable to localized three-dimensional disturbances. Stability calculations of the local similarity type viscoelastic flow in a small region near the stagnation plane are reported for the quasi-linear Oldroyd-B constitutive equation. The stability results for a range of Deborah numbers and viscosity ratio are presented to explore systematically the effects of elasticity and other rheological properties. The onset of instability and the temporal and spatial characteristics of the secondary flow predicted here resemble other purely elastic instabilities measured and predicted for viscoelastic flows in other simple and complex geometries with curved streamlines.     

FLOW OF A VISCOELASTIC FLUID BETWEEN TWO ROTATING CIRCULAR CYLINDERS SUBJECT TO SUCTION OR INJECTION

The steady flow of an Oldroyd-B fluid between two porous concentric circular cylinders is studied. The equation of motion and the constitutive equations form a system of non-linear ODEs that is solved numerically, and in a few cases the numerical results are compared with a known analytical solution. We consider the effect of the non-Newtonian nature of the fluid on the drag and on the boundary layer structure near the walls. Numerical computations show the effect of the non-Newtonian quantities on the velocity and on the shear stress as the dimensionless parameters are varied. © by 1997 John Wiley & Sons, Ltd.     

A new mixed finite element for calculating viscoelastic flow

A new mixed finite element has allowed us to calculate flows of Maxwell-B and Oldroyd-B fluids at very high values of the Deborah number, De. The element is divided into several bilinear sub-elements for the stresses, while streamline-upwinding is used for discretizing the constitutive equation. The method is applied to the stick-slip problem, the flow through a tapered contraction and the flow through four-to-one abrupt plane and circular contractions. Important corner vortices develop at high values of De in the circular contraction. We have not encountered upper limits for the Deborah number in our calculations with Oldroyd-B fluids.     

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     

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

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.     

A numerical comparison of two decoupled methods for the simulation of viscoelastic fluid flows

Two decoupled methods for the finite element solution of the planar stick-slip transition problem with Oldroyd-B fluids, namely the method of characteristics and the Lesaint-Raviart technique, are presented and compared. These procedures are used for the local treatment of the stress transport equation imbedded in the constitutive law and allow the approximation of stresses with discontinuous shape functions. Computations are carried out up to a Deborah number of 4 and the methods are shown to yield fairly similar results.     

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

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_Lc≡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.     

Stress analysis for a cavity flow of a memory fluid

The paper deals with the plane creeping flow of a viscoelastic liquid within an idealized rotor–stator system consisting of a rotating cylindrical drum and a fixed internal plate. The rheological properties of the liquid are modelled by a single-integral constitutive equation which considers a memory but disregards shear-thinning features. It is pointed out that the flow kinematics of such a memory fluid deviates only marginally from the Newtonian kinematics whereas the stress field is much more complicated. Therefore, the Newtonian velocity field is applied which can be represented in a closed analytical form using elementary complex-valued functions. The formulation allows computing the strain history and the resulting stress state with little numerical effort. A careful asymptotic analysis close to the corners yields details of the singular stress behaviour which differs markedly from the Newtonian characteristic. Finally, a second-order approximation being valid under certain restrictions leads to explicit analytical expressions also concerning the viscoelastic stress components. Altogether, a well-founded insight as regards the complex stress distribution and the effect of Deborah number is attained.     

Lattice-Boltzmann method for yield-stress liquids

In the present study we propose a new version of the lattice-Boltzmann (LB) method for the simulation of flow of yield-stress liquids. Unlike traditional LB methods, collisions are treated implicitly, i.e., the collision term is chosen in such a way that the stress and strain rate tensors satisfy the constitutive equation after the collision. This approach requires the solution of a (one-dimensional) non-linear algebraic equation at each point and at each time step. In the practically important cases of a Bingham liquid this equation can be solved analytically. We calculated the flow of Bingham fluid through a channel and periodic mesh of cylinders.