FENE Dumbbell Research Articles

A semi-Lagrangian micro–macro method for viscoelastic flow calculations

We present in this paper a semi-Lagrangian algorithm to calculate the viscoelastic flow in which a dilute polymer solution is modeled by the FENE dumbbell kinetic model. In this algorithm the material derivative operator of the Navier–Stokes equations (the macroscopic flow equations) is discretized in time by a semi-Lagrangian formulation of the second order backward difference formula (BDF2). This discretization leads to solving each time step a linear generalized Stokes problem. For the stochastic differential equations of the microscopic scale model, we use the second order predictor-corrector scheme proposed in [22] applied along the forward trajectories of the center of mass of the dumbbells. Important features of the algorithm are (1) the new semi-Lagrangian projection scheme; (2) the scheme to move and locate both the mesh-points and the dumbbells; and (3) the calculation and space discretization of the polymer stress. The algorithm has been tested on the 2d 10:1 contraction benchmark problem and has proved to be accurate and stable, being able to deal with flows at high Weissenberg ( W i ) numbers; specifically, by adjusting the size of the time step we obtain solutions at W i = 444 .

Regularity of the solutions for micro–macro models near equilibrium

In this paper, we study the regularity of solutions to two microscopic–macroscopic models: Hookean spring model and FENE dumbbell model. By making use of the structure of Fokker–Planck operator and establishing the general Hardy–Littlewood's inequality, we show that the classical solutions near the equilibrium obtained in [F. Lin, C. Liu, Ping Zhang, On a micro–macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math. 60 (6) (2007) 838–866] and [F. Lin, Ping Zhang, The FENE dumbbell model near equilibrium, Acta Math. Sinica (Chin. Ser.) 24 (2008) 529–538] become smooth with respect to the end-to-end vector variable q for any positive time.

Well‐posedness for the FENE dumbbell model of polymeric flows

AbstractWe prove local and global well‐posedness for the FENE dumbbell model for a very general class of potentials. Indeed, in prior local or global well‐posedness results, conditions on the strength of the singularity (or on the parameter b) were made. Here we give a proof in the general case. We also prove global existence results if the data is small or if we restrict to the co‐rotational model in dimension 2. © 2008 Wiley Periodicals, Inc.

The FENE dumbbell model near equilibrium

We study the global existence of smooth solutions near the equilibrium to a coupled microscopic-macroscopic FENE dumbbell model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteracting polymer chains.

Global well-posedness for 2D polymeric fluid models and growth estimate

Motivated by [P. Constantin, N. Masmoudi, Global well-posedness for a Smoluchowski equation coupled with Navier–Stokes equations in 2D, Comm. Math. Phys. 278 (2008) 179–191; F. Lin, Ping Zhang, Zhifei Zhang, On the global existence of smooth solution to the 2-D FENE dumbbell model, Comm. Math. Phys. 277 (2008) 531–553], we prove the global existence of smooth solutions to a coupled microscopic–macroscopic model for polymeric fluid in 2D under the co-rotational assumption. Furthermore, we provide an estimate on the time growth of these solutions. Our method is general and can be applied to the co-rotational FENE model and to the rod-like model without the co-rotational assumption.

On the Global Existence of Smooth Solution to the 2-D FENE Dumbbell Model

In two space dimension, we prove the global existence of smooth solutions to a coupled microscopic-macroscopic co-rotational FENE dumbbell model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteracting polymer chains.

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.

Global existence of weak solutions to some micro-macro models

We prove the global existence of weak solutions for the co-rotational FENE dumbbell model and the Doi model also called the Rod model. The proof is based on propagation of compactness, namely if we take a sequence of weak solutions which converges weakly and such that the initial data converges strongly then the weak limit is also a solution. To cite this article: P.-L. Lions, N. Masmoudi, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

603 FENEダンベルの可逆ネットワークから導出した構成方程式の一般的特性(1)(OS6-1 非ニュートン流体の流動現象,OS6 非ニュートン流体の流動現象,オーガナイズドセッション)

603 FENEダンベルの可逆ネットワークから導出した構成方程式の一般的特性(1)(OS6-1 非ニュートン流体の流動現象,OS6 非ニュートン流体の流動現象,オーガナイズドセッション)

1419 高分子流体のブラウン動力学シミュレーションによるレオロジー特性の解析(S30-4 複雑流体の流動現象(4),S30 複雑流体の流動現象)

Recently, complex fluid that forms structures inside of the fluid is paid attention. Polymer fluids, a representative example of the complex fluid, show different rheological properties from Newtonian fluid. Thus, In this research, the FENE dumbbell model was used to model a solution of polymers without side chain. The Brownian dynamics simulation of shear flows for a system of suspension of model polymers was performed to evaluate rheological properties of polymeric fuid and to analyze the configuration of polymers under shear flows. When the length of model polymer is long, both the shear viscosity and the first normal stress coefficient are large. Brownian force largely affects the rheological property and the configuration at low shear rate.

Prediction of Rheological Properties of Associative Polymeric Fluids Using a Reversible Network with Non-Interacting FENE Dumbbell Model

In our previous work, we have improved the reversible network model proposed by Cifre and co-workers, and performed a Brownian dynamics simulation to investigate the capability of the model in predicting the shear and elongational properties of non-associative polymeric fluids. In the present work, the capability of the improved model in predicting the rheological properties of associative polymeric fluids is presented. Similar to that for non-associative polymers, substitution of Hookean connection force with FENE connection force is required for better prediction in shear and elogational properties of associative polymer. In shear flow, the model with FENE connection force predicts shear thickening in the shear viscosity, and the peak of the shear thickening depends strongly on the maximum extension parameter. The responses in the first normal stress difference show that the model is able to predict relatively fast growth at low shear rates and relatively slow growth at high shear rates. In the elongational flow, the model is able to predict elongational viscosity with strain thickening at low strain rate followed by strain thinning or constant viscosity at moderate and high strain rates depending on various model parameters.

603 FENEダンベルの可逆ネットワークから導出した構成方程式の一般的特性(2)(OS6-1 非ニュートン流体の流動現象,OS6 非ニュートン流体の流動現象,オーガナイズドセッション)

603 FENEダンベルの可逆ネットワークから導出した構成方程式の一般的特性(2)(OS6-1 非ニュートン流体の流動現象,OS6 非ニュートン流体の流動現象,オーガナイズドセッション)

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, W i = λ γ ˙ (where λ is the characteristic relaxation time of polymer, and γ ˙ 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.

Asymptotic basis of the L closure for finitely extensible dumbbells in suddenly started uniaxial extension

This paper applies the recent L closure [G. Lielens, P. Halin, I. Jaumain, R. Keunings, V. Legat, J. Non-Newtonian Fluid Mech., 76 (1998), 249–279], which was originally developed for FENE dumbbells with fixed friction, to two alternative dumbbell models of polymers in a dilute solution undergoing uniaxial extension: (A) a linear-locked dumbbell with fixed friction and (B) a FENE dumbbell with variable friction. The simplified box-spike representation of the probability density function (PDF) – used for Model B to close conformational averages of nonlinear quantities in terms of a reduced set of state variables (moments of the PDF) – is justified through detailed asymptotic analysis (singular perturbations combined with multiple scales) of the Smoluchowski equation, in the limit of large extensibility parameter at fixed elongation rate. Both dumbbell models A and B are actually more amenable to the L closure than the FENE to which it had previously been applied. The resulting closure relations compare favorably with corresponding integrals of asymptotic or numerical PDF’s (the latter obtained via atomistic SPH simulations of the Smoluchowski equation). Example calculations show the L closure to yield reasonably accurate stress–extension curves, even for a moderate (dimensionless) limit of extension ( L = 4 ).

Nonlinear elastic polymers in random flow

Polymer stretching in random smooth flows is investigated within the framework of the FENE dumbbell model. The advecting flow is Gaussian and short-correlated in time. The stationary probability density function of polymer extension is derived exactly. The characteristic time needed for the system to attain the stationary regime is computed as a function of the Weissenberg number and the maximum length of polymers. The transient relaxation to the stationary regime is predicted to be exceptionally slow in the proximity of the coil–stretch transition.

New constitutive equations derived from a kinetic model for melts and concentrated solutions of linear polymers

In this paper, new constitutive equations for linear entangled polymer solutions and melts are derived from a recently proposed kinetic model (Fang et al. 2004) by using five closure approximations available in the literature. The simplest closure approximation considered is that due to Peterlin (1966). In this case, a mean-field-type Fokker-Planck equation underlying the evolution equation for an equilibrium averaged polymer segment orientation tensor is shown to be consistent with the fluctuation-dissipation theorem (Kubo et al. 1985). We compare the performance of the five new constitutive equations in their capacity to faithfully reproduce the predictions of the modified encapsulated FENE dumbbell model of Fang et al. (2004) for a number of shear and extensional flows. Comparisons are also made with the experimental data of Kahvand (1995) and Bhattacharjee et al. (2002, 2003). In the case of the Hinch-Leal and Bingham closures (Hinch and Leal 1976; Chaubal and Leal 1998) a combination with the quadratic closure of Doi (1981) is found to be necessary for stability in fast flows. The Hinch-Leal closure approximation, modified in this way, is found to outperform the other closures and its mathematical description is considerably simpler than that of the Bingham closure.

On the effects of viscoelasticity on two-dimensional vortex dynamics in the cylinder wake

The present paper represents, as far as we are aware, the first linear stability analysis for inertial flows of a viscoelastic fluid around a bluff body. Our particular interest is the effect of polymer additives on the linear stability of two-dimensional viscous flow past a confined cylinder and our investigation involves both direct numerical simulation and the solution of a generalized eigenvalue problem arising from the linearized perturbation equations. For a constant viscosity FENE-type model polymer additive is shown to have a stabilizing effect upon two-dimensional flow past a confined cylinder. In particular, our results reveal that the greater the maximum extensibility of the FENE dumbbells, the larger the value of the critical Reynolds number marking the onset of vortex shedding. Thus the stabilization brought to bear by the presence of the FENE-type dumbbells would appear to depend strongly upon the extensional properties of the polymer solution. This is in agreement with the results of the recent experimental paper of Cressman et al. [J.R. Cressman, Q. Bailey, W.I. Goldburg, Modification of a vortex street by a polymer additive, Phys. Fluids 13 (2001) 867–871] which highlight the importance of the extensional properties of high molecular weight polymers (even at low concentrations) in reducing the kinetic energy contained in the velocity fluctuations behind a cylinder. The impact of elasticity on Strouhal number, time-averaged recirculation length, drag and lift in the vortex-shedding regime are also discussed at some length.

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.

Towards more realistic kinetic models for concentrated solutions and melts

In this paper we consider modifications of two kinetic models for concentrated polymer solutions and melts: the simplified uniform (SU) version of a thermodynamically admissible single segment reptation model by Öttinger [A thermodynamically admissible reptation model for fast flows of entangled polymers. J. Rheol. 43 (1999) 1461–1493] and Fang et al. [A thermodynamically admissible reptation model for fast flows of entangled polymers. II. Model predictions for shear and extensional flows, J. Rheol. 44 (2000) 1293–1317] and the encapsulated FENE dumbbell (EFD) model [R.B. Bird, J.R. Deaguiar, An encapsulated dumbbell model for concentrated polymer solutions and melts. I. Theoretical developments and constitutive equation, J. Non-Newtonian Fluid Mech. 13 (1983) 149–160]. The two modified models incorporate double reptation, convective constraint release (CCR), and chain stretching. In the modified SU model, orientation and stretching are still modelled in a decoupled way, while in the case of the modified EFD model they are modelled in a coupled way. Quantitative comparisons are made for a number of shear and extensional flows between the predictions of the modified EFD model with those of the SU model, the modified SU model, the original EFD model and a recent simple constitutive model by Marrucci and Ianniruberto [Flow-induced orientation and stretching of entangled polymers, Phil. Trans. R. Soc. Lond. A 361 (2003) 677–687]. The models’ predictions are also tested against experimental data for entangled polystyrene solutions. For steady simple shear flow it is shown that good quantitative agreement with experimental data using physically realistic parameters requires that orientation and stretching be modelled in a coupled way. In particular, predictions with the modified SU model of the steady shear viscometric functions are only accurate with a maximum chain stretch ratio parameter an order of magnitude smaller than is required by the polymer chemistry. The use of the modified EFD model with realistic model parameters leads to better overall fits in all cases with the experimental data than either variant of the SU model or the Marrucci–Ianniruberto model.

Simulation of dilute polymer solutions using a Fokker–Planck equation

A numerical method based on the Fokker–Planck equation is applied to the simulation of a dilute solution of polymer modelled by two-dimensional FENE dumbbells. The proposed method is tested for both planar homogeneous flows and the flow around a cylinder in a channel.