Multiscale simulation of viscoelastic free surface flows

A micro–macro approach based on combining the Brownian configuration fields (BCF) method [M.A. Hulsen, A.P.G. van Heel, B.H.A.A. van den Brule, Simulation of viscoelastic flow using Brownian configuration fields, J. Non-Newtonian Fluid Mech. 70 (1997) 79–101] with an Arbitrary Lagrangian–Eulerian (ALE) Galerkin finite element method, using elliptic mesh generation equations coupled with time-dependent conservation equations, is applied to study slot coating flows of polymer solutions. The polymer molecules are represented by dumbbells with both linear and non-linear springs; hydrodynamic interactions between beads are incorporated. Calculations with infinitely extensible (Hookean) and pre-averaged finitely extensible (FENE-P) dumbbell models are performed and compared with equivalent closed-form macroscopic models in a conformation tensor based formulation [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on the conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409]. The BCF equation for linear dumbbell models is solved using a fully implicit time integration scheme which is found to be more stable than the explicit Euler scheme used previously to compute complex flows. We find excellent agreement between the results of the BCF based formulation and the macroscopic conformation tensor based formulation. The computations using the BCF approach are stable at much higher Weissenberg numbers, (where λ is the characteristic relaxation time of polymer, and is the characteristic rate of strain) compared to the purely macroscopic conformation tensor based approach, which fail beyond a maximum Wi. A novel computational algorithm is introduced to compute complex flows with non-linear microscopic constitutive models (i.e. non-linear FENE dumbbells and dumbbells with hydrodynamic interactions) for which no closed-form constitutive equations exist. This algorithm is fast and computationally efficient when compared to both an explicit scheme and a fully implicit scheme involving the solution of the non-linear equations with Newton’s method for each configuration field.     

Viscoelastic flow of polymer solutions around a periodic,linear array of cylinders: comparisons of predictions for microstructure and flow fields

Numerical simulation is used to investigate the flow of polymer solutions around a periodic, linear array of cylinders by using three constitutive equations derived from kinetic theory of dilute polymer solutions: the Giesekus model; the finitely extensible, nonlinear elastic dumbbell model with Peterlin's approximation (FENE-P); and the FENE dumbbell model of Chilcott–Rallison (CR). In the Giesekus model, intramolecular forces are described by a Hookean spring, whereas a finitely extensible spring whose modulus is given by the Warner approximation is used in both the FENE-P and CR models. Hydro dynamic drag on the beads is taken to be anisotropic for the Giesekus model and isotropic for the other two models. The CR and FENE-P models differ subtly in their approximate treatment of the nonlinear force law. The three models exhibit very similar rheological behavior in viscometric flow and steady elongational flow, with the notable exception that the viscosity for the CR model is shear-rate independent. Finite element simulations are performed by using two different formulations: the elastic-viscous split-stress gradient (EVSS-G) method and a new variant of this formulation, the discrete EVSS-G (DEVSS-G) formulation, in which the elliptic stabilization term is added only to the discrete version of the momentum equation, and the constitutive equation is solved directly in terms of the polymer contribution to the stress tensor. Calculations are performed for all models up to a Weissenberg number We, where the configuration tensor 〈QQ〉 loses positive definiteness. However, by locally refining the mesh in the gap region, the positive definiteness of 〈QQ〉 is recovered. The flow and stress fields predicted by the three constitutive equations are qualitatively similar. A `birefringent strand' of highly stretched polymer molecules, which appears to emanate from the rear stagnation point in the cylinder, strengthens as We is increased. Not surprisingly, the molecular extension computed for the Giesekus model is considerably larger than that of the two FENE spring models. The drag force on the cylinders differs for the FENE-P and CR models, because of the difference in the shear-thinning viscosity resulting from the different approximations used in these models.     

Viscoelastic flow past confined objects using a micro–macro approach

This paper is concerned with the numerical prediction of viscoelastic flow past a cylinder in a channel and a sphere in a cylinder using molecular-based models. The basis of the numerical method employed is a micro–macro model in which the polymer dynamics is described by the evolution of an ensemble of Brownian configuration fields. The spectral element method is used to discretize the equations in space. Comparisons are made between the macroscopic simulations based on the Oldroyd B constitutive model and microscopic simulations based on Hookean dumbbells, and excellent agreement is found. The micro–macro approach can be used to simulate models, such as the finitely extensible nonlinear elastic (FENE) dumbbell model, which do not possess a closed-form constitutive equation. Numerical simulations are performed for the FENE model. The influence of the model parameters on the flow is described and, in particular, the dependence of the drag as a function of the Weissenberg number.     

The FENE Dumbbell Polymer Model: Existence and Uniqueness of Solutions for the Momentum Balance Equation

We consider the FENE dumbbell polymer model which is the coupling of the incompressible Navier-Stokes equations with the corresponding Fokker–Planck–Smoluchowski diffusion equation. We show global well-posedness in the case of a 2D bounded domain. We assume in the general case that the initial velocity is sufficiently small and the initial probability density is sufficiently close to the equilibrium solution; moreover an additional condition on the coefficients is imposed. In the corotational case, we only assume that the initial probability density is sufficiently close to the equilibrium solution.     

Dilute solution rheology: experimental results and finitely extensible nonlinear elastic dumbbell theory

Experimental results for intrinsic viscosity and for intrinsic complex viscosity of polymer solutions were compared with the rheological predictions of the finitely extensible, nonlinear elastic (FENE) dumbbell theory of a dilute suspension. The FENE dumbbell adequately models the intrinsic viscosity of flexible polymers, but less successfully portrays the behavior in small amplitude oscillatory motion. Expressions for the high frequency asymptotic limit of the intrinsic complex viscosity of a FENE dumbbell suspension, and the mean-square end-to-end distance of FENE dumbbells in steady shear flow are given.     

Crucial properties of the moment closure model FENE-QE

We investigate four crucial properties for testing and evaluating a moment closure approximation of the FENE dumbbell model for dilute polymer solutions: non-negative configuration distribution function, energy dissipation, accuracy of approximation and computational expense. Through mathematical analysis, numerical experiments and comparisons with closure model FENE-P and FENE-YDL, we prove that the FENE-QE approximation has non-negative configuration distribution function, approximates the energy dissipation behavior of original kinetic theory and provides good accuracy. To improve the efficiency of this closure approximation, we introduce a piecewise linear approximation technique that greatly reduces the computational cost. This extension of FENE-QE, FENE-QE-PLA, is the closure model we recommend for simulating dilute polymer solutions.     

An extended FENE dumbbell model theory for concentration dependent shear-induced anisotropy in dilute polymer solutions: addenda

Schneggenburger et al. [C. Schneggenburger, M. Kröger, S. Hess, An extended FENE dumbbell theory for concentration dependent shear-induced anisotropy in dilute polymer solutions, J. Non-Newtonian Fluid Mech. 62 (1996) 235] extended the original FENE dumbbell kinetic theory to describe concentration dependent shear-induced anisotropy in dilute polymer solutions by a mean-field approach. Besides providing an erratum to the above-mentioned paper and two revised figures we present related analytic results for steady shear and uniaxial elongational flow. Within the same framework we further consider a modified FENE potential and briefly discuss its implications.     

The flow of dilute polymer solutions in confined geometries: a consistent numerical approach

Starting from rigorous expressions derived from phase space kinetic theory for dumbbell models of polymer solutions, a new numerical approach is presented. It enables one to solve the Langevin equations governing the motion of the dumbbells in a confined geometry consistently with the momentum balance equation. As an example, we discuss the flow of a polymer solution between two parallel shearing planes. For this purpose, we consider linear and nonlinear dumbbell models and investigate typical phenomena such as, for example, the slip effect.     

Dissipative stresses in dilute polymer solutions

Planar extensional flows of a dilute polymer solution are investigated using a free-draining bead-rod model. For steady flows, an analytic expression for the probability density of the polymer configuration is available. It is found that part of the associated steady polymer stress is unambiguously viscous at all time scales, in the sense that on cessation of flow it disappears instantaneously, but, except at very high flow rates, the elastic component is larger.A Brownian dynamics simulation of the chain is constructed for start-up flows for which no analytic expression is known. A stress that is apparently viscous is found to develop alongside the elastic stress, having comparable magnitude at moderate flow rates. An interpretation of this result for a system having a wide spectrum of relaxation times is given. This feature is not captured by conventional FENE constitutive equations, and a novel model is developed. The consequences for calculations of complex flows are briefly discussed.     

DNS of the Turbulent Channel Flow of a Dilute Polymer Solution

A direct numerical simulation of the turbulent channel flow of a dilute polymer solution has been performed in order to compare its turbulence statistics with those obtained in a Newtonian channel flow. The viscoelastic flow has been simulated by solving the whole set of continuity, momentum and constitutive equations for the six independent components of the extra-stress tensor induced by polymer addition. The Finitely Extensible Nonlinear Elastic dumbbell model was adopted in order to simulate a non-linear modulus of elasticity and a finite extendibility of the polymer macromolecules. Simulations were carried out under the narrow channel assumption at a Reynolds number of 169 based on the channel half height and on the friction velocity; they showed a significant reduction in drag, dependent on the influence of the elastic properties of the chains. A qualitative comparison with experiments at a higher Reynolds number has shown that the model here adopted is capable of reproducing all the main features of the polymer solution flow. Analysis of the turbulence statistics suggests that a dilute polymer solution can affect the intensity of the streamwise vortices, leading to an increase in the spacing between low speed streaks and eventually to a turbulent shear stress reduction.     

A study of the interacting FENE dumbbell model for semi-dilute polymer solutions in extensional flows

The effect of polymer concentration on the conformation of semidilute polymer solutions in extensional flows is studied via the interacting elastic dumbbell model proposed by Hess (1984), here modified to include a nonlinear Warner spring (FENE dumbbell) instead of the linear Hookean spring of the original model. The length of flow-induced conformation changes for the polymer is predicted to be a decreasing function of concentration. In particular, increasing concentration tends to inhibit large extension of the polymer due to polymer-polymer interaction. The specific birefringence is thus proportional to c ^–1 for semi-dilute solutions, in contrast to dilute solutions where it is known to be independent of concentration. However, the correlation between birefringence and the principle eigenvalue of the velocity gradient tensor, also found originally for dilute solutions, is predicted to occur in the semi-dilute regime. All of these predictions agree qualitatively with experimental observations.Some recent exceptions to the neglect of segmental stretch can be found in Marrucci and Grizzuti (1988), Pearson et al. (1991), Mead et al. (1992).     

A detailed comparison of various FENE dumbbell models

The rheological behaviour of dilute solutions of finitely extensible non-linear elastic (FENE) dumbbells in both steady state and transient shear and simple elongational flow is investigated. Three dumbbell models are compared: the original FENE model with the Warner spring force, which is treated by brownian dynamics simulations, and the FENE-P model based on the Peterlin approximation and the FENE-CR model as suggested by Chilcott and Rallison, which are treated by standard numerical techniques. It is shown that in the linear viscoelastic limit and in steady state flows the behaviour is similar, except for the FENE-CR dumbbell in shear flow, modelling a Boger fluid. In transient flows larger differences appear.     

A possible alternative to the FENE dumbbell model of dilute polymer solutions

The dumbbell model of dilute polymer solutions is simple and successful, and its FENE version has progressively become a paradigm. In some transient extensional flows however, an increased dissipation was observed which could not be understood within the FENE model. This prompted us to look for a new dumbbell-like model in which the finite extensibility of the polymer is taken more thoroughly into account, i.e. not only through the Warner potential. The main lines of this alternative model are presented.     

Local Existence for the FENE-Dumbbell Model of Polymeric Fluids

We study the well-posedness of a multi-scale model of polymeric fluids. The microscopic model is the kinetic theory of the finitely extensible nonlinear elastic (FENE) dumbbell model. The macroscopic model is the incompressible non-Newton fluids with polymer stress computed via the Kramers expression. The boundary condition of the FENE-type Fokker-Planck equation is proved to be unnecessary by the singularity on the boundary. Other main results are the local existence, uniqueness and regularity theorems for the FENE model in certain parameter range.     

A study of conformation-dependent friction in a dumbbell model for dilute solutions

We consider predictions of rheological behavior in a variety of shear and extensional flows for an elastic dumbbell model with a nonlinear spring, and conformation-dependent hydrodynamic properties. The latter include a conformation-dependent anisotropic bead friction coefficient, and a related conformation-dependent degree of inefficiency for rotation in straining flows. With these features, the dumbbell exhibits hydrodynamic behavior consistent with a particle of finite axis ratio over the complete set of possible polymer conformations, from random-coil to a fully extended thread-like configuration. The predicted rheological behavior in shear flow is improved, relative to data, by the inclusion of anisotropy and strain-inefficiency in the frictional properties of the model, while other desirable features such as the sudden onset of fully extended states at a critical value of the velocity gradient, the presence of a hysteresis-loop in end-to-end dimension as a function of the velocity gradient, and the correlation of end-to-end distance (or birefringence) with the eigenvalue of the velocity gradient tensor for a wide variety of two-dimensional flows, are maintained.     

A finite element analysis of a free surface drainage problem of two immiscible fluids

A sharp interface problem arising in the flow of two immiscible fluids, slag and molten metal in a blast furnace, is formulated using a two-dimensional model and solved numerically. This problem is a transient two-phase free or moving boundary problem, the slag surface and the slag–metal interface being the free boundaries. At each time step the hydraulic potential of each fluid satisfies the Laplace equation which is solved by the finite element method. The ordinary differential equations determining the motion of the free boundaries are treated using an implicit time-stepping scheme. The systems of linear equations obtained by discretization of the Laplace equations and the equations of motion of the free boundaries are incorporated into a large system of linear equations. At each time step the hydraulic potential in the interior domain and its derivatives on the free boundaries are obtained simultaneously by solving this linear system of equations. In addition, this solution directly gives the shape of the free boundaries at the next time step. The implicit scheme mentioned above enables us to get the solution without handling normal derivatives, which results in a good numerical solution of the present problem. A numerical example that simulates the flow in a blast furnace is given.     

Viscosity,first normal-stress coefficient,and molecular stretching in dilute polymer solutions

Warner's numerical method for finitely extensible nonlinear elastic (“FENE”) dumbbells in a dilute suspension undergoing steady-state shear flow has been improved by assuming a form of the distribution function that removes the singularity at R = 0 and improves the behavior of the weight function in the Galerkin expression for large b. The comparison of the results of the present method to those of Christiansen and Bird's extrapolation and Warner's numerical solution indicates the success of this method. The material functions, the dumbbell elongation, and the distribution function for steady-state shear flow are given. In addition the newly obtained results are used to assess the accuracy of two approximate methods referred to as the FENE-P model and the FENE-P-B model.     

An improved algorithm for simulating three-dimensional, viscoelastic turbulence

We present a new finite-difference formulation to update the conformation tensor in dumbbell models (e.g., Oldroyd-B, FENE-P, Giesekus) that guarantees positive eigenvalues of the tensor (i.e., the tensor remains positive definite) and prevents over-extension for finite-extensible models. The formulation is a generalization of the second-order, central difference scheme developed by Kurganov and Tadmor [A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160 (2000) 241–282] that guarantees a scalar field remains everywhere positive. We have extended the algorithm to guarantee a tensor field remains everywhere positive definite following an update. Extensive testing of the algorithm shows that the volume average of the conformation tensor is conserved. Furthermore, volume averages of the conformation tensor in homogeneous turbulent shear flow made over the Eulerian grid are in quantitative agreement with Lagrangian averages made over fluid particles moving throughout the domain, highlighting the accuracy of the treatment of the convective terms.