Finitely Extensible Nonlinear Elastic Research Articles

Polymers in turbulence: any better than dumbbells?

Polymer chains in turbulent flows are generally modelled as dumbbells, i.e. two beads joined by a nonlinear spring. The dumbbell only maps a single spatial configuration, described by the polymer end-to-end vector, thus a multi-bead FENE (finitely extensible nonlinear elastic) chain seems a natural improvement for a more accurate characterisation of the polymer spatial conformation. At a large Weissenberg number, a comparison with the more accurate Kuhn chain reveals that the multi-bead FENE chain drastically overestimates the probability of folded configurations. Surprisingly, the dumbbell turns out to be the only meaningful bead-spring model to coarse-grain a polymer macromolecule in turbulent pipe flows.

Multiscale simulations of viscoelastic fluids in complex geometries using a finitely extensible nonlinear elastic transient network model

This paper presents a novel implementation of a numerical scheme for predicting complex flows of viscoelastic fluids using a finitely extensible nonlinear elastic (FENE) transient network model. This model extends the FENE model by incorporating chain interactions and accounting for the way in which the maximum chain length, drag, and relaxation time are influenced by entanglement and disentanglement processes. Three different initial networks are considered (disentanglement, entanglement, and aleatory), and the influence of variables such as the kinetic rate constants, elasticity, and chain length on the microstate concentration, stresses, and drag force is investigated. It is shown that although the concentrations of the microstates are independent of the Weissenberg number and the maximum extension length, the stresses and hence the drag are influenced by them.

Existence of large-data global weak solutions to kinetic models of nonhomogeneous dilute polymeric fluids

We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $ \mathbb{R}^d $, $ d = 2 $ or $ 3 $. The class of models under consideration involves the Navier–Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker–Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.

Macro–Micro-Coupled Simulations of Dilute Viscoelastic Fluids

Modeling the flow of polymer solutions requires knowledge at various length and time scales. The macroscopic behavior is described by the overall velocity, pressure, and stress. The polymeric contribution to the stress requires knowledge of the evolution of polymer chains. In this work, we use a microscopic model, the finitely extensible nonlinear elastic (FENE) model, to capture the polymer’s behavior. The benefit of using microscopic models is that they remain faithful to the polymer dynamics without information loss via averaging. Their downside is the computational cost incurred in solving the thousands to millions of differential equations describing the microstructure. Here, we describe a multiscale flow solver that utilizes GPUs for massively parallel, efficient simulations. We compare and contrast the microscopic model with its macroscopic counterpart under various flow conditions. In particular, significant differences are observed under nonlinear flow conditions, where the polymers become highly stretched and oriented.

A new class of Finitely Extensible Nonlinear Elastic (FENE-P) models obtained with a thermodynamical approach and the use of compressible natural configurations. Part II: Decoupled thermo-mechanical deformations

In this paper which is the continuation of the isothermal viscoelastic model presented in Gomez-Constante and Palade (2023), we incorporate temperature changes into our model. To achieve this, we present a temperature dependent Helmholtz potential from where the model will be derived using the idea of evolving natural configurations. To simplify the analysis, we assume that the temperature and the invariants of deformation are decoupled so the Helmholtz potential can be expressed as the product of two independent functions. The model thus derived is consistent with fundamental thermodynamical postulates and constrains. To show its qualitative behavior we chose to compare the isothermal model in Gomez-Constante and Palade (2023) with this paper non-isothermal version of the model and show how they behave in the classical Couette flow between infinite parallel plates and analyze their differences. We also present a simple extensional flow simulation for the non-isothermal version of the model to complete the analysis. Ideas towards the following steps towards a generalization of the model are also presented and discussed.

Kinetic Energy Budget in Turbulent Flows of Dilute Polymer Solutions

Direct numerical simulation of a turbulent pipe flow of a realistic solution of 108\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^8$$\\end{document} polymers, modelled as finitely extensible nonlinear elastic (FENE) dumbbells, and directly momentum coupled with the incompressible Navier–Stokes equations, are performed by means of an Eulerian-Lagrangian approach. Besides the drag reduction, the polymers significantly modify mean and turbulent kinetic energy budgets. The polymer backreaction to the solvent reduces the Reynolds stress and thus decreases the turbulent production and, at large Weissenberg number, the polymers act as a source of turbulent kinetic energy for y+>40\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y^+> 40$$\\end{document}, leading to an increase in the dissipation. This effect is peculiar to large Weissenberg polymers and it is particularly apparent at a small Reynolds number. At a smaller Weissenberg number, the effect of the polymers remains confined in the buffer layer, with the kinetic energy budget not significantly altered elsewhere.

A new class of Finitely Extensible Nonlinear Elastic FENE-P model obtained with a thermodynamical approach and the use of compressible natural configurations. Part I: Isothermal deformations

In a recent contribution to the fundamental understanding of polymer fluid dynamics, Khambhampati and Rajagopal (2021) established a connection between the natural configuration theory of Rajagopal and Srinivasa (2000) and the FENE-P model of Bird et al. (1987 [6]). In this paper we capitalize on the result in Khambhampati and Rajagopal (2021) and present a new class of FENE-P models using a more general Helmholtz potential within the conceptual framework of evolving natural configurations. To show its qualitative behavior, we exemplify with a classical Couette flow between infinite parallel plates. The model is capable of reproducing key features experimentally observed such as stress relaxation and the overshoot of the shear stress at the beginning of typical shear stress growth experiments. Comparison against the FENE-P type model obtained in Khambhampati and Rajagopal (2021) is used for comparison.

Molecular dynamics simulations of the monomer density profiles of knotted ring polymer chains confined in a slit of two parallel walls with one attractive and another repulsive surface.

We have used Molecular Dynamics simulations to obtain the monomer density profiles for real linear and ring polymer chains of 360 monomers length with different topological structures such as simple knots: 31, 61, 91, 10124, complex knots 313151 and twisted knots with n = 10 and n = 20 in a slit geometry of two parallel walls with one attractive and another repulsive surface. We have used Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software to perform simulations with the Verlet integration algorithm. The interactions between monomers were simulated as Lennard-Jones 12-6 potential, for bonds we have used Finitely Extensible Nonlinear Elastic (FENE) potential and the interaction with the walls was taken into account via Lennard-Jones 9-3 potential. We observed that topologically complex polymers have lower monomer density profiles near the attractive wall, but at some distance in the direction to the repulsive wall this tendency changes to the opposite. We showed that most complex twisted knots have two maxima in narrow slits. In the wide slits we do not observe such relation for twisted knots at higher temperatures. These results are important for better understanding the nature of the depletion forces which arise in a slit geometry of two parallel walls with one attractive and one repulsive wall.

Drag Reduction in Polymer-Laden Turbulent Pipe Flow

The turbulence of a realistic dilute solution of DNA macromolecules is investigated through a hybrid Eulerian–Lagrangian approach that directly solves the incompressible Navier–Stokes equation alongside the evolution of 108 polymers, modelled as finitely extensible nonlinear elastic (FENE) dumbbells. At a friction Reynolds number of 320 and a Weissenberg number of 2×104, the drag reduction is equal to 26%, which is similar to the one obtained at the lower Reynolds number of 180. The polymers induce an increase in the flow rate and the turbulent kinetic energy, whose axial contribution is predominantly augmented. The stress balance is analysed to investigate the causes of the drag reduction and eventually the effect of the friction Reynolds number on the probability distribution of the polymer configuration. Near the wall, the majority of the polymers are fully stretched and aligned along the streamwise direction, inducing an increase in the turbulence anisotropy.

Molecular Insights into the Wall Slip Behavior of Pseudoplastic Polymer Melt in Nanochannels during Micro Injection Molding

Wall slip directly affects the molding quality of plastic parts by influencing the stability of the filling flow field during micro injection molding. The accurate modeling of wall slip in nanochannels has been a great challenge for pseudoplastic polymer melts. Here, an effective modeling method for polymer melt flow in nanochannels based on united-atom molecular dynamics simulations is presented. The effects of driving forces and wall–fluid interactions on the behavior of polyethylene melt under Poiseuille flow conditions were investigated by characterizing the slip velocity, dynamics information of the flow process, and spatial configuration parameters of molecular chains. The results indicated that the united-atom molecular dynamics model could better describe the pseudoplastic behavior in nanochannels than the commonly used finitely extensible nonlinear elastic (FENE) model. It was found that the slip velocity could be increased with increasing driving force and show completely opposite variation trends under different orders of magnitude of the wall–fluid interactions. The influence mechanism was interpreted by the density distribution and molecular chain structure parameters, including disentanglement and orientation, which also coincides with the change in the radius of gyration.

Effect of Weissenberg number on polymer-laden turbulence

By adopting the direct force formulation in a finitely extensible nonlinear elastic (FENE) model for the feedback force by laden polymers to stationary isotropic turbulence, we clearly identified the energy flow between turbulence and polymers. The stretching motion of turbulence is suppressed by the polymer springs with large Weissenberg number, resulting in suppression of turbulent kinetic energy and energy dissipation.The direct force model seems to be more efficient in providing physical insight than the polymer stress model based on the conformation tensor.

Large Scale Hydrodynamically Coupled Brownian Dynamics Simulations of Polymer Solutions Flowing through Porous Media.

Large scale simulations of polymer flow through porous media provide an important tool for solving problems in enhanced oil recovery, polymer processing and biological applications. In order to include the effects of a wide range of velocity and density fluctuations, we base our work on a coarse-grain particle-based model consisting of polymers following Brownian dynamics coupled to a background fluid flow through momentum conserving interactions. The polymers are represented as Finitely Extensible Non-Linear Elastic (FENE) dumbbells with interactions including slowly decaying transient forces to properly describe dynamic effects of the eliminated degrees of freedom. Model porous media are constructed from arrays of parallel solid beams with circular or square cross-sections, arranged periodically in the plane perpendicular to their axis. No-slip boundary conditions at the solid–fluid interfaces are imposed through interactions with artificial particles embedded within the solid part of the system. We compare the results of our simulations with those of standard Smoothed Particle Hydrodynamics simulations for Newtonian flow through the same porous media. We observe that in all cases the concentration of polymers at steady state is not uniform even though we start the simulations with a uniform polymer concentration, which is indicative of shear-induced cross-flow migration. Furthermore, we see the characteristic flattening of the velocity profile experimentally observed for shear-thinning polymer solutions flowing through channels as opposed to the parabolic Poiseuille flow profile for Newtonian fluids.

Vanishing viscosity limit to the FENE dumbbell model of polymeric flows

In this paper we mainly investigate the inviscid limit for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. By virtue of the Littlewood-Paley theory, we first obtain a uniform estimate for the solution to the FENE dumbbell model with viscosity in Besov spaces. Moreover, we show that the data-to-solution map is continuous. Finally, we prove that the strong solution of the FENE dumbbell model converges to a Euler system couple with a Fokker-Planck equation. Furthermore, convergence rates in Lebesgue spaces are obtained also.

A constitutive equation for Rouse model modified for variations of spring stiffness, bead friction, and Brownian force intensity under flow

We derived a constitutive equation for the Rouse model (the most frequently utilized bead-spring model) with its spring constant κ, bead friction coefficient ζ, and the (squared) Brownian force intensity B being allowed to change under flow. Specifically, we modified the Langevin equation of the original Rouse model by introducing time (t)-dependent κ, ζ, and B (of arbitrary t dependence), which corresponded to the decoupling and preaveraging approximations often made in bead-spring models. From this modified Langevin equation, we calculated time evolution of second-moment averages of the Rouse eigenmode amplitudes and further converted this evolution into a constitutive equation. It turned out that the equation has a functional form, σ(t)=∫−∞tdt′{κ(t)/κ(t′)}M(t,t′)C−1(t,t′), where σ(t) and C−1(t,t′) are the stress and Finger strain tensors, and M(t,t′) is the memory function depending on κ(t′), ζ(t′), and B(t′) defined under flow. This equation, serving as a basis for analysis of nonlinear rheological behavior of unentangled melts, reproduces previous theoretical results under specific conditions, the Lodge–Wu constitutive equation for the case of t-independent κ, ζ, and B [A. S. Lodge and Y. Wu, “Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm,” Rheol. Acta 10, 539 (1971)], the finite extensible nonlinear elastic (FENE)-Peterlin mean-Rouse formulation for the case of t-dependent changes of the only κ reported by Wedgewood and co-workers [L. E. Wedgewood et al., “A finitely extensible bead-spring chain model for dilute polymer solutions,” J. Non-Newtonian Fluid Mech. 40, 119 (1991)], and analytical expression of steady state properties for arbitrary κ(t), ζ(t), and B(t) reported by ourselves [H. Watanabe et al., “Revisiting nonlinear flow behavior of Rouse chain: Roles of FENE, friction reduction, and Brownian force intensity variation,” Macromolecules 54, 3700 (2021)]. It is to be added that a constitutive equation reported by Narimissa and Wagner [E. Narimissa and M. H. Wagner, “Modeling nonlinear rheology of unentangled polymer melts based on a single integral constitutive equation,” J. Rheol. 64, 129 (2020)] has a significantly different functional form and cannot be derived from the Rouse model with any simple modification of the Rouse–Langevin equation.

New Temperature Dependent Configurational Probability Diffusion Equation for Diluted FENE Polymer Fluids: Existence of Solution Results

The theory for the non-isothermal rheology of polymer fluids proposed in Curtiss and Bird (Adv Polym Sci 125:1–101, 1996) used several approximations including the so-called linear gradient approximations for the temperature field and Brownian forces. While it had the significant advantage of dealing with linear equations, the approximations involved may have led to several non-physical predictions. This work is a continuation of Curtiss and Bird (1996) in that it obtains the corresponding non-linear configurational probability density equation in dimensionless form without the linear gradient approximations for the temperature field and Brownian forces. It does so for incompressible diluted polymer solutions with polymer molecules being modeled as FENE (Finitely Extensible Nonlinear Elastic) chains. Next we prove the existence of temperature dependent, positive variational solutions for the probability density equation of the FENE model.

Deformation Model of Chains and Networks with Extendable Bonds

The nonlinear stress–strain response of synthetic and biological networks and gels manifested in a monotonic increase of an instantaneous modulus is a result of the nonlinear deformation of individual network strands. To describe this behavior, we develop a nonlinear network deformation model, which relates macroscopic stress–strain response with force-elongation behavior of polymer chains with bending rigidity and extendable bonds. In particular, network mechanical properties such as structural shear modulus, extensibility ratio, and bond deformation modulus are expressed in terms of chain bending constant, Kuhn length, bond elastic constant, and chain sizes in undeformed and fully extended states. The chain deformation model is tested by molecular dynamics simulations of deformation of coarse-grained chains with harmonic and finitely extensible nonlinear elastic (FENE) bonds and is applied to analyze the DNA deformation data. The network model predictions are confirmed by molecular dynamics simulations of diamond networks made of coarse-grained chains and are used to describe experimental data for biological networks of collagen, fibrin, and neurofilaments.

Global existence of 2-D FENE Dumbbell Model with concentrated center-of-mass

In this paper, we prove global existence of smooth solutions to the two dimensional FENE (Finite Extensible Nonlinear Elastic) systems with an extra concentrated center-of-mass diffusion.

Controlling the Chain Folding for the Synthesis of Single-Chain Polymer Nanoparticles Using Thermoresponsive Polymers

Synthetic polymer single-chain nanoparticles (SCNPs) are an emerging new class of nanomaterials that possess similar folded structures as natural proteins. However, most SCNPs reported so far are p...

Analysis of viscoelastic fluid flow in different geometries abrupt contraction

In this work, two-dimension incompressible laminar viscoelastic fluid flow in an expansion-contraction duct has been investigated numerically. The effect of expansion-contraction shape, Deborah number, Reynolds number, expansion ratio and contraction length on the flow field variables has been investigated. Four cases for the expansion-contraction duct configurations are tested, the first case is the expansion duct, the second case is the contraction-expansion duct, the third case is the triangular expansion-contraction duct and the fourth case is the two-side triangular expansion-contraction duct. The study investigated the effect of the parameters: Reynolds number (Re), Deborah number (De), contraction length and expansion ratio (ER) on the flow field variables (pressure, stresses and velocity). The FENE-P (Finitely Extensible Nonlinear Elastic in the Peterlin approximation) viscoelastic fluid model is chosen for the modeling of viscoelastic fluid flow. The code is written in Simply Fortran 2 computer language. Numerically the momentum equations and the FENE-P model equation are solved by using the finite volume method passed on the collocated grid arrangement by using the Rhie-Chow interpolation. The power low scheme is used for the convective terms in the momentum equations, the up-wind scheme is used in the convective terms in the FENE-P model equation and the TDMA algorithm is used to solve the algebraic equations. The results show that the expansion ratio when decreasing causes an increase in the stresses and the pressure for all the studied cases. Further, the contraction length increment playing a major effect on the increasing of the pressure. It is discovered that increasing contraction length by 50% causes an increase in the pressure by 50%. For Deborah number, the results show that Deborah number effects all variables in the flow field, where the stresses show increasing with the increasing of Deborah number at the expansion side of all the cases. Moreover, the pressure is also increased as Deborah number increase for all the cases. The expansion duct it shows that increasing Deborah number from 0 to 1 causes an increase of 1.2% of the pressure, while for the contraction-expansion duct this percentage is increased. It is shown that increasing Deborah from 0 to 0.5 causes an increasing by 5% of the pressure. Furthermore, the effect of Deborah number on the recirculation zones is to shrink their size for all the cases. A comparison between the studied cases are presented and shows that the contraction-expansion duct has the maximum stresses and maximum required pressure, while the triangle contraction-expansion has the smallest required pressure. The limit of Deborah number gets to 1.2 in the expansion duct, while it is limited to 0.5 for the other cases.

The incompressible limit of compressible finitely extensible nonlinear bead-spring chain models for dilute polymeric fluids

We explore the behaviour of global-in-time weak solutions to a class of bead-spring chain models, with finitely extensible nonlinear elastic (FENE) spring potentials, for dilute polymeric fluids. In the models under consideration the solvent is assumed to be a compressible, isentropic, viscous, isothermal Newtonian fluid, confined to a bounded open domain in R3, and the velocity field is assumed to satisfy a complete slip boundary condition. We show that as the Mach number tends to zero the system is driven to its incompressible counterpart.