Constitutive equation for dilute polymer solutions in strong elongational flow

A constitutive equation is proposed, which is constructed using both phenomenological and structural ideas. In this formulation, the kinematics of the fluid is characterized by the deformation rate and a structural vector. The vector follows an evolutionary law which is inspired by known molecular models. The expression for the stress is given by introducing a dissipative term related to the strong hydrodynamic interaction of the distorted molecules and a deformation term for the molecules, as well as by using the second law of thermodynamics.A study of the general properties of the evolutionary equation and its response in a homogeneous two-dimensional flow provides evidence of the performance of the proposed model.     

Branched viscoelastic surfactant solutions and their response to elongational flow

Several years ago, Münstedt and Laun reported on the influence of branching on the elongational flow properties of polymer chains (Münstedt and Laun, 1981). They concluded that, in addition to the molecular weight distribution, the degree of branching strongly affects the degree of strain thickening of the elongational viscosity in such a way that the maximum in this material function increases with branching. In a recent paper by Lin, a ternary system of dodecyldimethylamine oxide-sodium laureth sulphate-sodium chloride surfactant solutions was investigated by CryoTEM and rheology (Lin, 1996). He reported a linear relation between the added sodium chloride and the branching of the wormlike micelles. In this paper we present an investigation of these surfactant solutions in elongational flow. Our results indicate that for branched micellar systems the presence of branching enhances the maximum of the elongational viscosity in the same manner as in the case of polymer melts.     

The Giesekus fluid in ω–D form for steady two-dimensional flows

Using the fact that for simple fluids the most general constitutive equation in constant stretch history flows for the extra stress tensor τ is known in an explicit form, the Giesekus fluid model is cast into this (ω–D) form for two-dimensional flows. The three material functions needed to characterize τ are listed. The explicit results for simple shear and planar elongation reveal that the parameter α should be restricted to values less than 0.5. It is demonstrated that in this explicit form the constitutive equation is free from thermodynamic objections and can thus be used as a starting point for numerical calculations of general, but steady, two-dimensional flows. Received: 9 November 1998 Accepted: 20 May 1999     

Efficient simulation of nonlinear viscoelastic fluid flows

This paper presents a new and efficient method for computing the flow of a non-Newtonian fluid. The approach is based on two independent concepts:Time-dependent simulation of viscoelastic flow: A new decoupled algorithm, presented in P. Saramito, Simulation numérique d'ecoulements de fluides viscoélastiques par éléments finis incompressibles et une méthode de directions alternées; applications, Thèse de l'Institut National Polytechnique de Grenoble, 1990 and P. Saramito, Numerical simulation of viscoelastic fluid flows using incompressible finite element method and a θ-method, Math. Modelling Num. Anal., 35 (1994) 1–35, enables us to split the major difficulties of this problem, and to solve it more efficiently. Moreover, this scheme is of order two in time, and can be used to obtain stationary flows in an efficient way.Conservative finite element method: this method combines the incompressible Raviart Thomas element, the discontinuous Lesaint-Raviart element, and a finite volume element method. It satisfies exactly the mass conservation law, and leads to an optimal size for the nonlinear system in terms of the total degree of freedom versus the mesh size.We apply our numerical procedure to the Phan-Thien-Tanner model with a classical benchmark: the four to one abrupt contraction. The numerical solutions exhibit good behavior, especially near the singularity, in the vicinity of the re-entrant corner. The numerical experiments present the main features of such flows: vortex development and overshooting of the velocity profile along the axis of symmetry in the entry region.     

Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows

This study examines the effect of discretization schemes for the convection term in the constitutive equation on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a fully developed turbulent channel flow are selected as test cases, and eight different discretization schemes are considered. Among them, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much by these schemes and the corresponding flow fields are quite different from those obtained by higher-order upwind difference schemes. Among higher-order upwind difference schemes investigated in this study, a third-order compact upwind difference scheme (CUD3) with locally added AD shows stable and most accurate solutions for highly extensional flows even at relatively high Weissenberg numbers.     

The role of elongational viscosity in the mechanism of drag reduction by polymer additives

The role of elongational viscosity in the mechanism of drag reduction by polymer additives is investigated qualitatively by means of direct numerical simulations of a turbulent pipe flow. For the polymer solution, a generalised Newtonian constitutive model is utilised in which the viscosity depends on the second and third invariant of the rate-of-strain tensor via an elongation parameter. This elongation parameter is capable of identifying elongational type of regions within the flow. The simulations show that complementary to stretching of the polymers, also compression must be incorporated to have drag reduction, contrary to many suggestions done in the literature on the mechanism which assume that stretching of the polymers is most important.     

Comparison of the predictions of viscoelastic and plastic-viscoelastic fluid model to the rheological behaviour of polystyrene and polystyrene-carbon black compounds

The rheological properties of a polystyrene melt and its compounds with carbon black are compared with a non-linear viscoelastic fluid theory and a plastic-viscoelastic fluid model. Shear viscosities, principal normal stresses and elongational viscosities are investigated for melt and compound. The relaxation spectra for the melt was determined with an eccentric disc rheometer. The Bogue-White model gave satisfactory agreement for the melt with an a parameter of 0.5. For the plastic-viscoelastic fluid theory, a Bogue-White model was used to represent theH functional. The _i and a parameters were taken to be the same as for the pure melt while theG _i were increased by a hydrodynamic factor. A yield value was obtained from the compound data. Generally good agreement was found but some discrepancy exists for the elongational flow experiments.     

A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.     

Morphology development in polystyrene/polyethylene blends during uniaxial elongational flow

The deformation of linear low-density and low-density polyethylene particles dispersed in a polystyrene matrix was studied during defined uniaxial elongational flow conditions for different capillarity numbers and different temperatures. The morphology of the elongated samples was analysed by quenching the specimens in liquid nitrogen directly after the deformation. Furthermore, morphology development after recovery was investigated. By measuring the transient elongational viscosity of the blend matrix the true hydrodynamic stress during the flow process was calculated. Using a modified critical capillarity number, the fibril formation of the dispersed phase could be described at all test conditions. Virtually no break-up processes were observed. This finding could be explained by calculating the characteristic time of fibril break-up due to Rayleigh instabilities. By annealing the elongated samples a spherical shape of the dispersed droplets was regained. Compared with the initial sample morphology a pronounced increase of the particle sizes due to coalescence processes during elongation was observed.     

Material functions generated by the complex viscosity

Based on the complex viscosity model various steady-state and transient material functions have been completed. The model is investigated in terms of a corotational frame reference. Also, BKZ-type integral constitutive equations have been studied. Some relations between material functions have been derived. C ^–1 Finger tensor - F[], (F ^–1[]) Fourier (inverse) transform - rate of deformation tensor in corotating frame - h(I, II) Wagner's damping function - J (x) Bessel function - m parameter inh (I, II) - m(s) memory function - m _k, n_k integers (powers in complex viscosity model) - P principal value of the integral - parameter in the complex viscosity model - rate of deformation tensor - shear rates - [], [] incomplete gamma function - (a) gamma function - steady-shear viscosity - ^* complex viscosity - , real and imaginary parts of ^* - ₀ zero shear viscosity - ⁺, ₁ ⁺ stress growth functions - ^–, ₁ ^- stress relaxation functions - (s) relaxation modulus - ₁(s) primary normal-stress coefficient - ø(a, b; z) degenerate hypergeometric function - ₁, ₂ time constants (parameters of ^*) - frequency - extra stress tensor     

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.     

On the composite Riemann problem for multi‐material fluid flows

We develop an Eulerian fixed grid numerical method for calculating multi‐material fluid flows. This approach relates to the class of interface capturing methods. The fluid is treated as a heterogeneous mixture of constituent materials, and the material interface is implicitly captured by a region of mixed cells that have arisen owing to numerical diffusion. To suppress this numerical diffusion, we propose a composite Riemann problem (CRP), which describes the decay of an initial discontinuity in the presence of a contact point between two different fluids, which is located off the initial discontinuity point. The solution to the CRP serves to calculate multi‐material no mixed numerical flux without introducing any material diffusion. We discuss the CRP solution and its implementation in the multi‐material fluid Godunov method. Numerical results show that a simple framework of the CRP greatly improves capturing material interfaces in the Godunov method and reproduces many of the advantages of more complicated interface tracking multi‐material treatments. Copyright © 2014 John Wiley & Sons, Ltd.     

Stagnation flows for the Oldroyd-B fluid

Exact solutions to the plane and axi-symmetric stagnation flows of an Oldroyd-B fluid are reported. It is found that a steady flow is possible if the Weissenberg numberWi, defined by the product of the Maxwellian relaxation time and the shear rate at infinity, satisfies – 1/2 <Wi < 1/m, wherem = 1 in an axisym-metric flow andm = 2 in a plane flow. Furthermore, the fluid elasticity always decreases the boundary-layer thickness. An Oldroyd-B fluid with the parameters matched those of a typical Boger fluid behaves essentially like a Newtonian fluid in a stagnation flow.     

Linear stability of homogeneous elongational flow of the upper convected Maxwell fluid

We prove that planar elongational flow of the upper convected Maxwell fluid is linearly stable and analyze the associated spectral problem.     

A mixed finite element scheme for viscoelastic flows with XPP model

A mixed finite element formulation for viscoelastic flows is derived in this paper, in which the FIC （finite incremental calculus） pressure stabilization process and the DEVSS （discrete elastic viscous stress splitting） method using the Crank-Nicolson-based split are introduced within a general framework of the iterative version of the fractional step algorithm. The SU （streamline-upwind） method is particularly chosen to tackle the convective terms in constitutive equations of viscoelastic flows. Thanks to the proposed scheme the finite elements with equal low-order interpolation approximations for stress-velocity-pressure variables can be successfully used even for viscoelastic flows with high Weissenberg numbers. The XPP （extended Pom-Pom） constitutive model for describing viscoelastic behaviors is particularly integrated into the proposed scheme. The numerical results for the 4：1 sudden contraction flow problem demonstrate prominent stability, accuracy and convergence rate of the proposed scheme in both pressure and stress distributions over the flow domain within a wide range of the Weissenberg number, particularly the capability in reproducing the results, which can be used to explain the ＂die swell＂ phenomenon observed in the polymer injection molding process.     

A new elongational rheometer for polymer melts and other highly viscoelastic liquids

In polymer melt elongational rheometry only by the rotary clamp technique large elongations can be obtained homogeneously. However, as described in this paper, there still remain disadvantages that led to the development of a new rheometer with the following main features: The dimensions of the required sample are small (60 × 7 × 2 nun³), the sample is supported by a cushion of inert gas and, after having reached the test temperature of up to well above 300°C, it can be extended by a new type of clamps that make use of metal conveyor belts. The resulting tensile force is measured with a resolution of better than 100 mgf (0.001 N). The strain rate range is 0.001-1 s^–1, and the maximum Hencky strain is 7, corresponding to a maximum stretch ratio of 1100. Within the sample, the temperature variation in time and space is less than 0.1°C. For the evaluation and documentation of the test performance, a video camera records the top and side views of the sample that carries a marking powder to permit the evaluation of the true strain rate. The operation of the instrument is easy, and so is the sample preparation, but care must be taken concerning the necessary isotropy and internal homogeneity. Examples of test results are given for several polymer melts at various temperatures: (1) Polystyrene up to a total Hencky strain larger than 7 at 170°C, (2) several types of polyethylene (LDPE, LLDPE, HDPE) at 150°C, (3) poly(amide) at 250°C, and (4) poly(ethersulfone) at 350°C. The wide applicability of the new rheometer is demonstrated by adding results obtained from samples of bread dough. The surface tension has no influence on the results if an error of 3% can be tolerated. From the results it follows that by means of the newly developed rheometer many problems in polymer melt elongation have been solved.Dedicated to Professor Dr. Hermann Janeschitz-Kriegl on the occasion of his 70th birthday.Extended version of a paper presented at the XIth International Congress on Rheology, Brussels, Belgium, August 17–21, 1992.     

Forward roll coating flows of viscoelastic liquids

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter and apply a liquid layer to a substrate. Except at low speed, the two-dimensional film splitting flow that occurs in forward roll coating is unstable; a three-dimensional steady flow sets in, resulting in more or less regular stripes in the machine direction. For Newtonian liquids, the stability of the two-dimensional flow is determined by the competition of capillary and viscous forces: the onset of meniscus nonuniformity is marked by a critical value of the capillary number. Although most of the liquids coated industrially are non-Newtonian polymeric solutions and dispersions, most of the theoretical analyses of film splitting flows relied on the Newtonian model. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case.Forward roll coating flow is analyzed here with two differential constitutive models, the Oldroyd-B and the FENE-P equations. The results show that the elastic stresses change the flow near the film splitting meniscus by reducing and eventually eliminating the recirculation present at low capillary number. When the recirculation disappears, the difference of the tangential and normal stresses (i.e., the hoop stress) at the free surface becomes positive and grows dramatically with fluid elasticity, which explains how viscoelasticity destabilizes the flow in terms of the analysis of Graham [M.D. Graham, Interfacial hoop stress and instability of viscoelastic free surface flows, Phys. Fluids 15 (2003) 1702–1710].