Analytical and numerical study of the electro-osmotic annular flow of viscoelastic fluids

In this work we present semi-analytical solutions for the electro-osmotic annular flow of viscoelastic fluids modeled by the Linear and Exponential PTT models. The viscoelastic fluid flows in the axial direction between two concentric cylinders under the combined influences of electrokinetic and pressure forcings. The analysis invokes the Debye-Hückel approximation and includes the limit case of pure electro-osmotic flow. The solution is valid for both no slip and slip velocity at the walls and the chosen slip boundary condition is the linear Navier slip velocity model. The combined effects of fluid rheology, electro-osmotic and pressure gradient forcings on the fluid velocity distribution are also discussed.

Annular flow of viscoelastic fluids: Analytical and numerical solutions

This work provides analytical and numerical solutions for the linear, quadratic and exponential Phan–Thien–Tanner (PTT) viscoelastic models, for axial and helical annular fully-developed flows under no slip and slip boundary conditions, the latter given by the linear and nonlinear Navier slip laws. The rheology of the three PTT model functions is discussed together with the influence of the slip velocity upon the flow velocity and stress fields. For the linear PTT model, full analytical solutions for the inverse problem (unknown velocity) are devised for the linear Navier slip law and two different slip exponents. For the linear PTT model with other values of the slip exponent and for the quadratic PTT model, the polynomial equation for the radial location (β) of the null shear stress must be solved numerically. For both models, the solution of the direct problem is given by an iterative procedure involving three nonlinear equations, one for β, other for the pressure gradient and another for the torque per unit length. For the exponential PTT model we devise a numerical procedure that can easily compute the numerical solution of the pure axial flow problem

Slip flows of Newtonian and viscoelastic fluids in a 4:1 contraction

This work presents a numerical study of the 4:1 planar contraction flow of a viscoelastic fluid described by the simplified Phan-Thien–Tanner model under the influence of slip boundary conditions at the channel walls. The linear Navier slip law was considered with the dimensionless slip coefficient varying in the range ½0; 4500. The simulations were carried out for a small constant Reynolds number of 0.04 and Deborah numbers (De) varying between 0 and 5. Convergence could not be achieved for higher values of the Deborah number, especially for large values of the slip coefficient, due to the large stress gradients near the singularity of the reentrant corner. Increasing the slip coefficient leads to the formation of two vortices, a corner and a lip vortex. The lip vortex grows with increasing slip until it absorbs the corner vortex, creating a single large vortex that continues to increase in size and intensity. In the range De = 3–5 no lip vortex was formed. The flow is characterized in detail for De ¼ 1 as function of the slip coefficient, while for the remaining De only the main features are shown for specific values of the slip coefficient.

A Reynolds stress model for turbulent flows of viscoelastic fluids

A second-order closure is developed for predicting turbulent flows of viscoelastic fluids described by a modified generalised Newtonian fluid model incorporating a nonlinear viscosity that depends on a strain-hardening Trouton ratio as a means to handle some of the effects of viscoelasticity upon turbulent flows. Its performance is assessed by comparing its predictions for fully developed turbulent pipe flow with experimental data for four different dilute polymeric solutions and also with two sets of direct numerical simulation data for fluids theoretically described by the finitely extensible nonlinear elastic - Peterlin model. The model is based on a Newtonian Reynolds stress closure to predict Newtonian fluid flows, which incorporates low Reynolds number damping functions to properly deal with wall effects and to provide the capability to handle fluid viscoelasticity more effectively. This new turbulence model was able to capture well the drag reduction of various viscoelastic fluids over a wide range of Reynolds numbers and performed better than previously developed models for the same type of constitutive equation, even if the streamwise and wall-normal turbulence intensities were underpredicted.

Numerical simulation of some viscoelastic fluids

Numerical simulation of the Oldroyd-B type viscoelastic fluids is a very challenging problem. rnThe well-known High Weissenberg Number Problem" has haunted the mathematicians, computer scientists, and rnengineers for more than 40 years. rnWhen the Weissenberg number, which represents the ratio of elasticity to viscosity, rnexceeds some limits, simulations done by standard methods break down exponentially fast in time. rnHowever, some approaches, such as the logarithm transformation technique can significantly improve rnthe limits of the Weissenberg number until which the simulations stay stable. rnrnWe should point out that the global existence of weak solutions for the Oldroyd-B model is still open. rnLet us note that in the evolution equation of the elastic stress tensor the terms describing diffusive rneffects are typically neglected in the modelling due to their smallness. However, when keeping rnthese diffusive terms in the constitutive law the global existence of weak solutions in two-space dimension rncan been shown. rnrnThis main part of the thesis is devoted to the stability study of the Oldroyd-B viscoelastic model. rnFirstly, we show that the free energy of the diffusive Oldroyd-B model as well as its rnlogarithm transformation are dissipative in time. rnFurther, we have developed free energy dissipative schemes based on the characteristic finite element and finite difference framework. rnIn addition, the global linear stability analysis of the diffusive Oldroyd-B model has also be discussed. rnThe next part of the thesis deals with the error estimates of the combined finite element rnand finite volume discretization of a special Oldroyd-B model which covers the limiting rncase of Weissenberg number going to infinity. Theoretical results are confirmed by a series of numerical rnexperiments, which are presented in the thesis, too.

Modelling integral viscoelastic flows in OpenFOAM

[Excerpt] A large number of constitutive equations were developed for viscoelastic fluids, some empirical and other with strong physical foundations. The currently available macroscopic constitutive equations can be divided in two main types: differential and integral. Some of the constitutive equations, e.g. Maxwell are available both in differential and integral types. However, relevant in tegral models, like K - BKZ, just possesses the integral form. (...)

Dispersion of fillers in viscoelastic polymer matrices using Eulerian-Lagrangian approaches

Understanding the behavior of c omplex composite materials using mixing procedures is fundamental in several industrial processes. For instance, polymer composites are usually manufactured using dispersion of fillers in polymer melt matrices. The success of the filler dispersion depends both on the complex flow patterns generated and on the polymer melt rheological behavior. Consequently, the availability of a numerical tool that allow to model both fluid and particle would be very useful to increase the process insight. Nowadays there ar e computational tools that allow modeling the behavior of filled systems, taking into account both the behavior of the fluid (Computational Rheology) and the particles (Discrete Element Method). One example is the DPMFoam solver of the OpenFOAM ® framework where the averaged volume fraction momentum and mass conservation equations are used to describe the fluid (continuous phase) rheology, and the Newton’s second law of motion is used to compute the particles (discrete phase) movement. In this work the refer red solver is extended to take into account the elasticity of the polymer melts for the continuous phase. The solver capabilities will be illustrated by studying the effect of the fluid rheology on the filler dispersion, taking into account different fluid types (generalized Newtonian or viscoelastic) and particles volume fraction and size. The results obtained are used to evaluate the relevance of considering the fluid complex rheology for the prediction of the composites morphology

Measurements of the bulk and interfacial velocity profiles in oscillating Newtonian and Maxwellian fluids

We present the dynamic velocity profiles of a Newtonian fluid (glycerol) and a viscoelastic Maxwell fluid (CPyCl-NaSal in water) driven by an oscillating pressure gradient in a vertical cylindrical pipe. The frequency range explored has been chosen to include the first three resonance peaks of the dynamic permeability of the viscoelastic-fluid¿pipe system. Three different optical measurement techniques have been employed. Laser Doppler anemometry has been used to measure the magnitude of the velocity at the center of the liquid column. Particle image velocimetry and optical deflectometry are used to determine the velocity profiles at the bulk of the liquid column and at the liquid-air interface respectively. The velocity measurements in the bulk are in good agreement with the theoretical predictions of a linear theory. The results, however, show dramatic differences in the dynamic behavior of Newtonian and viscoelastic fluids, and demonstrate the importance of resonance phenomena in viscoelastic fluid flows, biofluids in particular, in confined geometries.

Numerical prediction of three-dimensional time-dependent viscoelastic extrudate swell using differential and algebraic models

This study investigates the numerical simulation of three-dimensional time-dependent viscoelastic free surface flows using the Upper-Convected Maxwell (UCM) constitutive equation and an algebraic explicit model. This investigation was carried out to develop a simplified approach that can be applied to the extrudate swell problem. The relevant physics of this flow phenomenon is discussed in the paper and an algebraic model to predict the extrudate swell problem is presented. It is based on an explicit algebraic representation of the non-Newtonian extra-stress through a kinematic tensor formed with the scaled dyadic product of the velocity field. The elasticity of the fluid is governed by a single transport equation for a scalar quantity which has dimension of strain rate. Mass and momentum conservations, and the constitutive equation (UCM and algebraic model) were solved by a three-dimensional time-dependent finite difference method. The free surface of the fluid was modeled using a marker-and-cell approach. The algebraic model was validated by comparing the numerical predictions with analytic solutions for pipe flow. In comparison with the classical UCM model, one advantage of this approach is that computational workload is substantially reduced: the UCM model employs six differential equations while the algebraic model uses only one. The results showed stable flows with very large extrudate growths beyond those usually obtained with standard differential viscoelastic models. (C) 2010 Elsevier Ltd. All rights reserved.

Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids. (C) 2010 Elsevier B.V. All rights reserved.

Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Numerical assessment of mass conservation on a MAC-type method for viscoelastic free surface flows

A numerical study of mass conservation of MAC-type methods is presented, for viscoelastic free-surface flows. We use an implicit formulation which allows for greater time steps, and therefore time marching schemes for advecting the free surface marker particles have to be accurate in order to preserve the good mass conservation properties of this methodology. We then present an improvement by using a Runge-Kutta scheme coupled with a local linear extrapolation on the free surface. A thorough study of the viscoelastic impacting drop problem, for both Oldroyd-B and XPP fluid models, is presented, investigating the influence of timestep, grid spacing and other model parameters to the overall mass conservation of the method. Furthermore, an unsteady fountain flow is also simulated to illustrate the low mass conservation error obtained.

Development of a low-Reynolds-number k-ω model for FENE-P fluids

A low-Reynolds-number k-ω model for Newtonian fluids has been developed to predict drag reduction of viscoelastic fluids described by the FENE-P model. The model is an extension to viscoelastic fluids of the model for Newtonian fluids developed by Bredberg et al. (Int J Heat Fluid Flow 23:731-743, 2002). The performance of the model was assessed using results from direct numerical simulations for fully developed turbulent channel flow of FENE-P fluids. It should only be used for drag reductions of up to 50 % (low and intermediate drag reductions), because of the limiting assumption of turbulence isotropy leading to an under-prediction of k, but compares favourably with results from k-ε models in the literature based on turbulence isotropy. © 2012 Springer Science+Business Media Dordrecht.

Drop formation and breakup of low viscosity elastic fluids: Effects of molecular weight and concentration

The dynamics of drop formation and pinch-off have been investigated for a series of low viscosity elastic fluids possessing similar shear viscosities, but differing substantially in elastic properties. On initial approach to the pinch region, the viscoelastic fluids all exhibit the same global necking behavior that is observed for a Newtonian fluid of equivalent shear viscosity. For these low viscosity dilute polymer solutions, inertial and capillary forces form the dominant balance in this potential flow regime, with the viscous force being negligible. The approach to the pinch point, which corresponds to the point of rupture for a Newtonian fluid, is extremely rapid in such solutions, with the sudden increase in curvature producing very large extension rates at this location. In this region the polymer molecules are significantly extended, causing a localized increase in the elastic stresses, which grow to balance the capillary pressure. This prevents the necked fluid from breaking off, as would occur in the equivalent Newtonian fluid. Alternatively, a cylindrical filament forms in which elastic stresses and capillary pressure balance, and the radius decreases exponentially with time. A (0+1)-dimensional finitely extensible nonlinear elastic dumbbell theory incorporating inertial, capillary, and elastic stresses is able to capture the basic features of the experimental observations. Before the critical "pinch time" t(p), an inertial-capillary balance leads to the expected 2/3-power scaling of the minimum radius with time: R-min similar to(t(p)-t)(2/3). However, the diverging deformation rate results in large molecular deformations and rapid crossover to an elastocapillary balance for times t>t(p). In this region, the filament radius decreases exponentially with time R-min similar to exp[(t(p)-t)/lambda(1)], where lambda(1) is the characteristic time constant of the polymer molecules. Measurements of the relaxation times of polyethylene oxide solutions of varying concentrations and molecular weights obtained from high speed imaging of the rate of change of filament radius are significantly higher than the relaxation times estimated from Rouse-Zimm theory, even though the solutions are within the dilute concentration region as determined using intrinsic viscosity measurements. The effective relaxation times exhibit the expected scaling with molecular weight but with an additional dependence on the concentration of the polymer in solution. This is consistent with the expectation that the polymer molecules are in fact highly extended during the approach to the pinch region (i.e., prior to the elastocapillary filament thinning regime) and subsequently as the filament is formed they are further extended by filament stretching at a constant rate until full extension of the polymer coil is achieved. In this highly extended state, intermolecular interactions become significant, producing relaxation times far above theoretical predictions for dilute polymer solutions under equilibrium conditions. (C) 2006 American Institute of Physics

On the evaluation of some differential formulations for the pom-pom constitutive model

In recent years, progress has been made in modelling long chain branched polymers by the introduction of the so-called pompom model. Initially developed by McLeish and Larson (1998), the model has undergone several improvements or alterations, leading to the development of new formulations. Some of these formulations however suffer from certain mathematical defects. The purpose of the present paper is to review some of the formulations of the pom-pom constitutive model, and to investigate their possible mathematical defects. Next, an alternative formulation is proposed, which does not appear to exhibit mathematical defects, and we explore its modelling performance by comparing the predictions with experiments in non-trivial rheometric flows of an LDPE melt. The selected rheometric flows are the double step strain, as well as the large amplitude oscillatory shear experiments. For LAOS experiments, the comparison involves the use of Fourier-transform analysis.