Finitely Extensible Nonlinear Elastic Model Research Articles

Analytical model of friction at low shear rates for soft materials in 3D printing.

The biological properties of silicone elastomers such as polydimethylsiloxane (PDMS) have widespread use in biomedicine for soft tissue implants, contact lenses, soft robots, and many other small medical devices, due to its exceptional biocompatibility. Additive manufacturing of soft materials still has significant challenges even with major advancements that have occurred in development of these technologies for customized medical devices and tissue engineering. The aim of this study was to develop a mathematical model of tangential stress in relation to shear stress, shear rate, 3D printing pressure and velocity, for non-Newtonian gels and fluids that are used as materials for 3D printing. This study used FENE (finitely extensible nonlinear elastic model) model, for non-Newtonian gels and fluids to define the dependences between tangential stress, velocity, and pressure, considering viscosity, shear stress and shear rates as governing factors in soft materials friction and adhesion. Experimental samples were fabricated as showcases, by SLA and FDM 3D printing technologies: elastic polymer samples with properties resembling elastic properties of PDMS and thermoplastic polyurethane (TPU) samples. Experimental 3D printing parameters were used in the developed analytical solution to analyse the relationships between governing influential factors (tangential stress, printing pressure, printing speed, shear rate and friction coefficient). Maple software was used for numerical modelling. Analytical model applied on a printed elastic polymer, at low shear rates, exhibited numerical values of tangential stress of 0.208-0.216 N m - 2 at printing velocities of 0.9 to 1.2 mm s - 1, while the coefficient of friction was as low as 0.09-0.16. These values were in accordance with experimental data in literature. Printing pressure did not significantly influence tangential stress, whereas it was slightly influenced by shear rate changes. Friction coefficient linearly increased with tangential stress. Simple analytical model of friction for elastic polymer in SLA 3D printing showed good correspondence with experimental literature data for low shear rates, thus indicating possibility to use it for prediction of printing parameters towards desired dimensional accuracy of printed objects. Further development of this analytical model should enable other shear rate regimes, as well as additional soft materials and printing parameters.

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.

Numerical study of viscoelastic upstream instability

In this work, we report numerical results on the flow instability and bifurcation of a viscoelastic fluid in the upstream region of a cylinder in a confined narrow channel. Two-dimensional direct numerical simulations based on the FENE-P model (the finite-extensible nonlinear elastic model with the Peterlin closure) are conducted with numerical stabilization techniques. Our results show that the macroscopic viscoelastic constitutive relation can capture the viscoelastic upstream instability reported in previous experiments for low-Reynolds-number flows. The numerical simulations reveal that the non-dimensional recirculation length (LD) is affected by the cylinder blockage ratio (BR), the Weissenberg number (Wi), the viscosity ratio (β) and the maximum polymer extension (L). Close to the onset of upstream recirculation, LD with Wi satisfy Landau-type quartic potential under certain parameter space. The bifurcation may exhibit subcritical behaviour depending on the values of L2 and β. The parameters β and L2 have nonlinear influence on the upstream recirculation length. This work contributes to our theoretical understanding of this new instability mechanism in viscoelastic wake flows.

On the similarities of the sPTT and FENE-P models for polymeric fluids

<h2>Abstract</h2> For many commonly used single mode viscoelastic constitutive equations of differential type, it is well known that they share many features. For example, in certain parameter limits the models due to Giesekus, Phan-Thien Tanner and FENE-type models approach the Oldroyd-B model. In this talk, I'll compare the response of the linear form of the simplified Phan-Thien Tanner model [due to Phan-Thien and Tanner, Journal of Non-Newtonian Fluid Mechanics, 2 (1977) 353–365] (the "sPTT") and the Finitely Extensible Nonlinear Elastic model that follows the Peterlin approximation [due to Bird et al., Journal of Non-Newtonian Fluid Mechanics, 7 (1980) 213–235] (the "FENE-P"). I'll show that for steady homogeneous flows such as steady simple shear flow or pure extension the response of both models is identical under certain conditions. For more general flows we show analytically that the results from the two models only formally approach each other when both the polymer concentration and Weissenberg number is small. We then use a numerical approach to investigate the response of the two models when the flow is "complex" under two different definitions: firstly, when the applied deformation field is homogenous in space but transient in time (so called "start-up" shear and planar extensional flow) and then for "complex" flows (through a range of geometries) which, although Eulerian steady, are unsteady in a Lagrangian sense. Under the limit that the flows remain Eulerian steady (so the Weissenberg number is typically small), we see once again a very close agreement between the FENE-P and sPTT models.

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.

Viscoelastic fluid flow simulations in the e-VROCTM geometry

Microfluidic contraction devices have been proposed for extensional rheometry measurements, in particular as a useful method for determining the extensional viscosity of low elasticity solutions. The first commercially available “Extensional Viscometer-Rheometer-On-a-Chip” (e-VROCTM), developed by Rheosense, is a hyperbolically-shaped contraction/expansion geometry which incorporates pressure-drop measurement capabilities. To better understand the underlying flow kinematics within this geometry we have conducted a numerical study performing three-dimensional numerical simulations for both Newtonian and viscoelastic fluids. For the viscoelastic fluids the simplified Phan-Thien and Tanner (sPTT) and the Finitely Extensible Nonlinear Elastic models (FENE-P) are employed, in order to investigate the efficiency of this configuration in terms of increasing Weissenberg numbers and to understand the effects of various model parameters on the flow field. Our Newtonian fluid results suggest that the e-VROCTM geometry produces only a small region of extensional flow and is mainly shear-dominated, potentially suggesting any pressure-drop measurements from this device may be related to viscoelastic first normal-stress differences developed via a combination of shear and extension, rather than solely pure extension. By a careful selection of the sPTT and FENE-P model parameters, such that steady-state viscometric properties in homogeneous flows are matched, we are able to show that a small enhanced pressure-drop is seen for both models, which is larger for the FENE-P model.

The L2 decay for the 2D co-rotation FENE dumbbell model of polymeric flows

In this paper we mainly study the long time behavior of solutions to the finite extensible nonlinear elastic (FENE) dumbbell model with dimension two in the co-rotation case. Firstly, we obtain the L2 decay rate of the velocity of the 2D co-rotation FENE model is (1+t)−12 with small data. Then, by virtue of the Littlewood–Paley theory, we can remove the small condition. Our obtained sharp result improves considerably the recent results in [8,14].

Effects of viscous dissipation on heat convection of viscoelastic flow inside isothermal channels and tubes

In the present paper, analytical solutions for thermal convection of the Finitely Extensible Nonlinear Elastic model with Peterlin’s closure (FENE-P) in isothermal slits and tubes are presented by considering the viscous dissipation effects for the first time. Temperature distributions are derived in closed form, namely the HeunT function, and Frobenius series solutions for the slit and the tube flows, respectively. The effects of fluid elasticity, the Brinkman number and the extensibility parameter on the thermal convection characteristics of FENE-P fluid flows are investigated in detail. Generally, the Brinkman number causes a fall in the Nusselt number, but a rise in the centerline temperature. The results reveal that during the cooling process the Nusselt number experiences a decrease with the Brinkman number, while the centerline temperature rises. However, during the heating process the former increases and the latter decreases. As the main innovation of this study, the results show a strong relation between the Nusselt number and the Brinkman number and also between the centerline temperature and the Brinkman number.

Multiple relaxation modes in suspensions of colloidal particles bridged by telechelic polymers

We build a Brownian dynamics simulation model to study the linear rheology of latex particles interacting with telechelic polymers (for example, so-called “HEUR” polymers), with the latter modeled as Finitely Extensible Nonlinear Elastic (FENE) dumbbells with ends that stick to Brownian colloidal spheres. We identify from this model four relaxation modes, including (1) a single chain relaxation time, τ′FENE, which is closely related to τFENE, the relaxation time of a free chain predicted using the FENE model, but is around 1.4 times longer because of surface constraints on polymer configuration; (2) a loop translational-rotational relaxation time, τloop, produced by migration of the chain over the particle surface, allowing it to further relax its orientational conformation while still trapped on the particle surface; (3) a bridge formation/breakage relaxation time, τbridge; and (4) one or more colloidal particle cluster relaxation times. We show that τloop scales as the square of particle radius. τbridge...

Elliptical, parabolic, and hyperbolic exchanges of energy in drag reducing plane Couette flows

In the present paper, we investigate the polymer–turbulence interaction by discriminating between the mechanical responses of this system to three different subdomains: elliptical, parabolic, and hyperbolic, corresponding to regions where the magnitude of vorticity is greater than, equal to, or less than the magnitude of the rate of strain, respectively, in accordance with the Q-criterion. Recently, it was recognized that hyperbolic structures play a crucial role in the drag reduction phenomenon of viscoelastic turbulent flows, thanks to the observation that hyperbolic structures, as well as vortical ones, are weakened by the action of polymers in turbulent flows in a process that can be referred to as flow parabolization. We employ direct numerical simulations of a viscoelastic finite extensible nonlinear elastic model with the Peterlin approximation to examine the transient evolution and statistically steady regimes of a plane Couette flow that has been perturbed from a laminar flow at an initial time and developed a turbulent regime as a result of this perturbation. We have found that even more activity is located within the confines of the hyperbolic structures than in the elliptical ones, which highlights the importance of considering the role of hyperbolic structures in the drag reduction mechanism.

Study on the molecular models commonly used in the simulation of dilute polymeric solutions in flow

Three molecular models, commonly used in the simulation of polymeric solutions and melts were employed to describe the rheological behaviour of dilute, elastic and constant viscosity solution formed by bead-and-bond chain molecules immersed into a soft-sphere solvent. The intermolecular interactions for the three models were calculated by the Lennard-Jones potential. The differences amongst the models proceeded from the intramolecular restrictions: the simplest one is a Freely-Joined-Chain (FJC) model with harmonic bond potentials, in the second model bonds are restricted by a finite extensible non-linear elastic (FENE) potential plus a repulsive WCA potential, and the third model is conformed by the United Atom (UA) approach which includes bond, flexion and torsion potentials. Both Couette and Poiseuille flows were simulated using Non-Equilibrium Molecular Dynamics. Deformation displayed by the three chain models; defined in terms of their radius of gyration was calculated and according to results it was found that for Couette flow, the three chains exhibit similar response to deformation. In Poiseulle flow, the FJC and FENE models behave similarly but the UA model presents a larger resistance to deformation. For both flow regimes, the forces involved to deform the chains were estimated in terms of the first normal stress differences. From these estimations it was found that the UA model depicted the stiffest chain, followed by the FENE model, and last the FJC model.

Global existence and well-posedness for the FENE dumbbell model of polymeric flows

In this paper we mainly investigate the Cauchy problem of the finite extensible nonlinear elastic (FENE) dumbbell model with dimension d≥2. We first proved the local well-posedness for the FENE model in Besov spaces by using the Littlewood–Paley theory. Then by an accurate estimate we get a blow-up criterion. Moreover, if the initial data is a small perturbation around equilibrium, we obtain a global existence result. Our obtained results generalize and cover recent results in Masmoudi (2008).

The Liouville Theorem and the L 2 Decay for the FENE Dumbbell Model of Polymeric Flows

In this paper we mainly investigate the finite extensible nonlinear elastic (FENE) dumbbell model with dimension $d\geq2$ in the whole space. We first proved that there is only the trivial solution for the steady-state FENE model under some integrable condition. Our obtained results generalize and cover the classical results to the stationary Navier-Stokes equations. Then, we study about the $L^2$ decay of the co-rotation FENE model. Concretely, the $L^2$ decay rate of the velocity is $(1+t)^{-\frac{d}{4}}$ when $d\geq3$, and $\ln^{-k}{(e+t)}, k\in \mathds{N}^{+}$ when $d=2$. This result improves considerably the recent result of \cite{Schonbek2} by Schonbek. Moreover, the decay of general FENE model has been considered.

Influence Study of the Viscoelastic Fluids Features in Drag Reduction in Laminar Regime Flow in Pipeline

The drag reduction by polymer addition is wide interest in several areas. It has been shown that the polymer addition cushions the dissipative effects in turbulent flows. The main objective of this work is to establish a methodology for the numerical simulation of viscoelastic fluid through internal subroutines implemented in the Fluent code, via UDF. The validation of this methodology is made for the laminar flow regime case in pipeline. To describe the viscoelastic effect, it was used the Finitely Extensible Nonlinear Elastic model closing with Peterlin model. To taking in account the viscous effects 50≤Re≤2000 values were used. In addition, for the polymer concentration analysis it was used values which depend on the polymers molecular weight and the solution concentration that ranged from 0≤Cw≤20. The molecular elasticity and extensibility were maintained at constant values. The results showed that the addition of polymers regardless of their molecular weight in laminar flow regime causes no change in power dissipation. This result, which is consistent with the literature, is a significant advance in defining a credible and appropriate methodology to viscoelastic fluid flow study by UDF implementation of constituent models that characterize these fluids.

Impact of the Peterlin approximation on polymer dynamics in turbulent flows.

We study the impact of the Peterlin approximation on the statistics of the end-to-end separation of polymers in a turbulent flow. The finitely extensible nonlinear elastic (FENE) model and the FENE model with the Peterlin approximation (FENE-P) are numerically integrated along a large number of Lagrangian trajectories resulting from a direct numerical simulation of three-dimensional homogeneous isotropic turbulence. Although the FENE-P model yields results in qualitative agreement with those of the FENE model, quantitative differences emerge. The steady-state probability of large extensions is overestimated by the FENE-P model. The alignment of polymers with the eigenvectors of the rate-of-strain tensor and with the direction of vorticity is weaker when the Peterlin approximation is used. At large Weissenberg numbers, the correlation times of both the extension and of the orientation of polymers are underestimated by the FENE-P model.

Doubly self-consistent field theory of grafted polymers under simple shear in steady state

We present a generalization of the numerical self-consistent mean-field theory of polymers to the case of grafted polymers under simple shear. The general theoretical framework is presented, and then applied to three different chain models: rods, Gaussian chains, and finitely extensible nonlinear elastic (FENE) chains. The approach is self-consistent at two levels. First, for any flow field, the polymer density profile and effective potential are calculated self-consistently in a manner similar to the usual self-consistent field theory of polymers, except that the calculation is inherently two-dimensional even for a laterally homogeneous system. Second, through the use of a modified Brinkman equation, the flow field and the polymer profile are made self-consistent with respect to each other. For all chain models, we find that reasonable levels of shear cause the chains to tilt, but it has very little effect on the overall thickness of the polymer layer, causing a small decrease for rods, and an increase of no more than a few percent for the Gaussian and FENE chains. Using the FENE model, we also probe the individual bond lengths, bond correlations, and bond angles along the chains, the effects of the shear on them, and the solvent and bonded stress profiles. We find that the approximations needed within the theory for the Brinkman equation affect the bonded stress, but none of the other quantities.

Hysteretic behavior of a moment-closure approximation for FENE model

We discuss hysteretic behaviors of dilute viscoelastic polymeric fluidswith moment-closure approximation approach in extensional/enlongational flows.Polymeric fluids are modeled by the finite extensible nonlinear elastic (FENE) spring dumbbell model.Hysteresis is one of key features to describe FENE model.We here investigate the hysteretic behavior of FENE-D model introduced in [Y. Hyon et al., Multiscale Model. Simul., 7(2008), pp.978--1002].The FENE-D model is established from a special equilibrium solution of the Fokker-Planck equationto catch extreme behavior of FENE model in large extensional flow rates.Since the hysteresis of FENE model can be seen during a relaxation in simple extensional flowemploying the normal stress/the elongational viscosity versus the mean-square extension,we simulate FENE-D in simple extensional flows to investigate its hysteretic behavior comparingto FENE-P, FENE-L [G. Lielens et al., J. Non-Newtonian Fluid Mech., 76(1999), pp.249--279]. The FENE-P is a well-known pre-averaged approximated model,and it shows a good agreement to macroscopic induced stresses.However, FENE-P does not catch any hysteretic phenomenon.In contrast, the FENE-L shows a better hysteretic behavior than the other models to FENE,but it has a limitation for macroscopic induced stresses in large shear rates.On the other hand, FENE-D presents a good agreement to macroscopic induced stresses even in large shear rates,and moreover, it shows a hysteretic phenomenon in certain large flow rates.

Newtonian Limit for Weakly Viscoelastic Fluid Flows

This article addresses the low Weissenberg asymptotic analysis (Newtonian limit) of some macroscopic models of viscoelastic fluid flows in the framework of global weak solutions. We investigate the convergence of the corotational Johnson--Segalman, the FENE-P (closure proposed by Peterlin of the Finitely Extensible Nonlinear Elastic model), and the Giesekus and Phan-Thien and Tannes models. Relying on a priori bounds coming from energy or free energy estimates, we first study the weak convergence toward the Navier--Stokes system. We then turn to the main focus of our paper, i.e., the strong convergence. The novelty of our work is to address these issues by relative entropy estimates, which require the introduction of some corrector terms. We also take into account the presence of defect measures in the initial data, uniform with respect to the Weissenberg number, and prove that they do not perturb the Newtonian limit of the corotational system.

Time-dependent non-linear dynamics of polymer solutions in microfluidic contraction flow—a numerical study on the role of elongational viscosity

A generalised form of the finitely extensible non-linear elastic (FENE) model for modelling non-linear flow of semi-dilute polymer solutions is proposed. It accounts for conformation-dependent polymer elasticity and predicts shear-thinning shear viscosity, non-linear elongational viscosity and first and second normal stress differences. The rheometric material functions predicted by the model are critically compared with the results of the linear Phan–Thien–Tanner model. The predictabilities of these constitutive models under benchmark flow problems are evaluated by time-dependent simulations, using finite volume method based on a CFD simulation toolbox. The effects of the model parameters, the inertia and the contraction ratio are numerically studied. The modified FENE model qualitatively captures the non-linear flow phenomena of polymer solution in the high elasticity number (\(\mathrm {El}\)) flow regime observed in experiments. The results show that an accurate growth function of the elongational viscosity is the key to the prediction of the time-dependent highly asymmetric flow patterns.

Analysis of Polymeric Flow Models and Related Compactness Theorems in Weighted Spaces

We studied coupled systems of the Fokker--Planck equation and the Navier--Stokes equation modeling the Hookean and the finitely extensible nonlinear elastic (FENE)-type polymeric flows. We proved the continuous embedding and compact embedding theorems in weighted spaces that naturally arise from related entropy estimates. These embedding estimates are shown to be sharp. For the Hookean polymeric system with a center-of-mass diffusion and a superlinear spring potential, we proved the existence of a global weak solution. Moreover, we were able to tackle the FENE model with $L^2$ initial data for the polymer density instead of the $L^\infty$ counterpart in the literature.