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.     

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.     

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.     

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.     

Effects of viscosity ratio on deformation of a viscoelastic drop in a Newtonian matrix under steady shear

Deformation of an Oldroyd B drop in a Newtonian matrix under steady shear is simulated using a front tracking finite difference method for varying viscosity ratio. For drop viscosity lower than that of the matrix, the long-time steady deformation behavior is similar to that of the viscosity matched system—the drop shows reduced deformation with increasing Deborah number due to the increased inhibiting viscoelastic normal stress inside the drop. However for higher viscosity ratio systems, the drop response is non-monotonic—the steady drop deformation first decreases with increasing Deborah number but above a critical Deborah number, it increases with further increase in Deborah number, reaching higher than the viscous case value for some viscosity ratios. We explain the increase in deformation with Deborah number by noting that at higher viscosity ratios, strain rate inside the drop is reduced, thereby reducing the inhibiting viscoelastic stress. Furthermore, similar to the viscosity matched system, the drop inclination angle increases with increasing Deborah number. A drop aligned more with the maximum stretching axis at 45 degree of the imposed shear, experiences increased viscous stretching. With increased ratio of polymeric viscosity to total drop viscosity, the drop deformation decreases and the inclination angle increases. Our simulation results compare favorably with a number of experimental and computational results from other researchers.     

Response of viscoelastic fluids in extensional flow generated by a pulsating sphere

A simple analysis of the periodic extensional flow generated by a pulsating sphere in an infinite sea of viscoelastic fluid has been carried out. The general procedure is illustrated by two specific constitutive equations: the corotational Jeffreys fluid and the Oldroyd fluid model B. The response of these fluids is reflected in the temporal variation of the pressure on the surface of the sphere, with Reynolds and Deborah numbers and parameters of the constitutive equations as independent variables. For the case of pulsation with infinitesimal amplitude the fluid response is summarised in the form of pressure amplitude and phase lag versus Deborah number plots. The role of the pulsating flow in the characterisation of viscoelastic fluids and the extension of the procedure to other constitutive equations are briefly discussed.     

Perturbation theory for viscoelastic fluids between eccentric rotating cylinders

The circumferential and radial profiles of velocity, pressure and stress are derived for the flow of model viscoelastic liquids between two slightly eccentric cylinders with the inner one rotating. Singular perturbation methods are used to derive expansions valid for small gaps between the cylinders, but for all Deborah numbers. Results for Newtonian, second-order, Criminale-Ericksen-Filbey, upper-convected Maxwell, and White-Metzner constitutive equation separate the effects of elasticity, memory, and shear thinning on the development of the large stress gradients that hinder numerical solutions with these models in more complicated geometries. The effect of the constitutive equation on the critical Deborah number for flow separation is presented.     

An experimental and numerical investigation of the dynamics of microconfined droplets in systems with one viscoelastic phase

The dynamics of single droplets in a bounded shear flow is experimentally and numerically investigated for blends that contain one viscoelastic component. Results are presented for systems with a viscosity ratio of 1.5 and a Deborah number for the viscoelastic phase of 1. The numerical algorithm is a volume-of-fluid method for tracking the placement of the two liquids. First, we demonstrate the validation of the code with an existing boundary integral method and with experimental data for confined systems containing Newtonian components. This is followed by numerical simulations and experimental data for the combined effect of geometrical confinement and component viscoelasticity on the droplet dynamics after startup of shear flow at a moderate capillary number. The viscoelastic liquids are Boger fluids, which are modeled with the Oldroyd-B constitutive model and the Giesekus model. Confinement substantially increases the viscoelastic stresses and the elongation rates in and around the droplet. We show that the latter can be dramatic for the use of the Oldroyd-B model in confined systems with viscoelastic components. A sensitivity analysis for the choice of the model parameters in the Giesekus constitutive equation is presented.     

Numerical simulation of viscoelastic flows with Oldroyd-B constitutive equations and novel algebraic stress models

The purpose of the present study is to compare numerical simulations of viscoelastic flows using the differential Oldroyd-B constitutive equations and two newly devised simplified algebraic explicit stress models (AES-models). The flows of a viscoelastic fluid in a 180° bent planar channel and in a 4:1 planar contraction are considered to illustrate and support the underlying theory. The flow in the bent channel is used to illustrate the frame-invariant property of the new models in a pure shear flow exhibiting strong streamline curvature. The flow in the 4:1 contraction serves as a benchmark test in a situation where strong elongation occurs. For both geometries, it is found that the predictions of the new AES-models are in good agreement with Oldroyd-B up to Deborah numbers of order 0.5, with a significant reduction in computational effort.     

A 1-D theory for extensional deformation of a viscoelastic filament under exponential stretching

We consider a viscoelastic filament placed between two coaxial discs, with the bottom plate fixed and the top plate pulled at an exponential rate. Using a slender rod approximation, we derive a one-dimensional (1-D) model which describes the deformation of a viscoelastic filament governed by the Oldroyd-B constitutive model. It is assumed that the flow is axisymmetric and that inertia and gravity are negligible. One solution of the model equations corresponds to ideal uniaxial elongation. A linear stability analysis shows that this solution is unstable for a Newtonian fluid and for viscoelastic filaments with small Deborah number (De ≤ 0.5). For Deborah number greater than 0.5, ideal uniaxial elongation is linearly stable. Numerical solution of the nonlinear equations confirms the result of the linear stability analysis. For initial conditions close to ideal uniaxial flow, our results show that if De > 0.5, the central portion of the filament undergoes considerable strain hardening. As a result, the sample remains almost cylindrical and the deformation approaches pure uniaxial extension as the Hencky strain increases. For De ≤ 0.5, the Trouton ratio based on the effective extension rate at the mid-plane radius gives a much better approximation to the true extensional viscosity than that based on the imposed stretch rate.     

Migration of a sphere suspended in viscoelastic liquids in Couette flow: experiments and simulations

The intrinsically coupled effects of the curvature of the flow-field and of the viscoelastic nature of suspending medium on the cross-stream lateral migration of a single non-Brownian sphere in wide-gap Couette flow are studied. Quantitative videomicroscopy experiments using a counterrotating device are compared to the results of 3D finite element simulations. To evaluate the effects of differences in rheological properties of the suspending media, fluids have been selected which highlight specific constitutive features, including a reference Newtonian fluid, a single relaxation time wormlike micellar surfactant solution, a broad spectrum shear-thinning elastic polymer solution and a constant viscosity, highly elastic Boger fluid. As expected for conditions corresponding to Stokes flow, migration is absent in the Newtonian fluid. In the wormlike micellar solution and the shear-thinning polymer solution, spheres are observed to migrate in the direction of decreasing shear rate gradient, i.e. the outer cylinder, except when the sphere is initially released close to the inner cylinder, in which case the migration is towards it. The migration is enhanced by faster relative angular velocities of the cylinders. Shear-thinning reduces the migration velocity, showing an opposite behavior as compared to previous results in planar shear flow. In the Boger fluid, within experimental error no migration could be observed, likely due to the large solvent contribution to the overall viscosity. For small Deborah numbers the migration results are well described by an heuristic argument based on a local stress balance.     

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.     

Viscoelastic fluid behavior in annulus using Giesekus model

An analytical solution is derived for the steady state, laminar, axial, fully developed flow of a viscoelastic fluid obeying the Giesekus model without any retardation time in a concentric annulus.An approximation is used for the estimation of radial normal stress. The influence of Deborah number (De) and the mobility factor (α) on the velocity profile, axial pressure gradient are investigated and results show strong effects of mobility factor and Deborah number on above parameters.     

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).     

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     

Uniform flow of viscoelastic fluids past a confined falling cylinder

Uniform steady flow of viscoelastic fluids past a cylinder placed between two moving parallel plates is investigated numerically with a finite-volume method. This configuration is equivalent to the steady settling of a cylinder in a viscoelastic fluid, and here, a 50% blockage ratio is considered. Five constitutive models are employed (UCM, Oldroyd-B, FENE-CR, PTT and Giesekus) to assess the effect of rheological properties on the flow kinematics and wake patterns. Simulations were carried out under creeping flow conditions, using very fine meshes, especially in the wake of the cylinder where large normal stresses are observed at high Deborah numbers. Some of the results are compared with numerical data from the literature, mainly in terms of a drag coefficient, and significant discrepancies are found, especially for the constant-viscosity constitutive models. Accurate solutions could be obtained up to maximum Deborah numbers clearly in excess of those reported in the literature, especially with the PTT and FENE-CR models. The existence or not of a negative wake is identified for each set of model parameters.     

Rheology of an emulsion of viscoelastic drops in steady shear

Steady shear rheology of a dilute emulsion with viscoelastic inclusions is numerically investigated using direct numerical simulations. Batchelor's formulation for rheology of a viscous emulsion is extended for a viscoelastic system. Viscoelasticity is modeled using the Oldroyd-B constitutive equation. A front-tracking finite difference code is used to numerically determine the drop shape, and solve for the velocity and stress fields. The effective stress of the viscoelastic emulsion has three different components due to interfacial tension, viscosity difference (not considered here) and the drop phase viscoelasticity. The interfacial contributions – first and second normal stress differences and shear stresses – vary with Capillary number in a manner similar to those of a Newtonian system. However the shear viscosity decreases with viscoelasticity at low Capillary numbers, and increases at high Capillary numbers. The first normal stress difference due to interfacial contribution decreases with increasing drop phase viscoelasticity. The first normal stress difference due to the drop phase viscoelasticity is found to have a complex dependence on Capillary and Deborah numbers, in contrast with the linear mixing rule. Drop phase viscoelasticity does not contribute significantly to effective shear viscosity of the emulsion. The total first normal stress difference shows an increase with drop phase viscoelasticity at high Capillary numbers. However at low Capillary numbers, a non-monotonic behavior is observed. The results are explained by examining the stress field and the drop shape.     

Spherical bubble collapse in viscoelastic fluids

The collapse of a spherical bubble in an infinite expanse of viscoelastic fluid is considered. For a range of viscoelastic models, the problem is formulated in terms of a generalized Bernoulli equation for a velocity potential, under the assumptions of incompressibility and irrotationality. The boundary element method is used to determine the velocity potential and viscoelastic effects are incorporated into the model through the normal stress balance across the surface of the bubble. In the case of the Maxwell constitutive equation, the model predicts phenomena such as the damped oscillation of the bubble radius in time, the almost elastic oscillations in the large Deborah number limit and the rebound limit at large values of the Deborah number. A rebound condition in terms of ReDe is derived theoretically for the Maxwell model by solving the Rayleigh–Plesset equation. A range of other viscoelastic models such as the Jeffreys model, the Rouse model and the Doi-Edwards model are amenable to solution using the same technique. Increasing the solvent viscosity in the Jeffreys model is shown to lead to increasingly damped oscillations of the bubble radius.     

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.     

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.