Differential Constitutive Models Research Articles

Harmonic balance for differential constitutive models under oscillatory shear

Harmonic balance (HB) is a popular Fourier–Galerkin method used in the analysis of nonlinear vibration problems where dynamical systems are subjected to periodic forcing. We adapt HB to find the periodic steady-state response of nonlinear differential constitutive models subjected to large-amplitude oscillatory shear flow. By incorporating the alternating-frequency-time scheme into HB, we develop a computer program called FLASH (acronym for Fast Large Amplitude Simulation using Harmonic balance), which makes it convenient to apply HB to any differential constitutive model. We validate FLASH by considering two representative constitutive models, viz., the exponential Phan-Thien–Tanner model and a nonlinear temporary network model. In terms of accuracy and speed, FLASH typically outperforms the conventional approach of solving initial value problems by numerical integration via time-stepping methods often by several orders of magnitude. Exceptions to this rule are sometimes encountered for materials that are strongly shear thinning or described by constitutive models with discontinuous derivatives. We discuss how FLASH can be conveniently extended for other nonlinear constitutive models, which opens up potential applications in model calibration and selection, and stability analysis.

The medium amplitude response of nonlinear Maxwell–Oldroyd type models in simple shear

A general framework for Maxwell-Oldroyd type differential constitutive models is examined, in which an unspecified nonlinear function of the stress and rate-of-deformation tensors is incorporated into the well-known corotational version of the Jeffreys model discussed by Oldroyd. For medium amplitude simple shear deformations, the recently developed mathematical framework of medium amplitude parallel superposition (MAPS) rheology reveals that this generalized nonlinear Maxwell model can produce only a limited number of distinct signatures, which combine linearly in a well-posed basis expansion for the third order complex viscosity. This basis expansion represents a library of MAPS signatures for distinct constitutive models that are contained within the generalized nonlinear Maxwell model. We describe a framework for quantitative model identification using this basis expansion, and discuss its limitations in distinguishing distinct nonlinear features of the underlying constitutive models from medium amplitude shear stress data. The leading order contributions to the normal stress differences are also considered, revealing that only the second normal stress difference provides distinct information about the weakly nonlinear response space of the model. After briefly considering the conditions for time-strain separability within the generalized nonlinear Maxwell model, we apply the basis expansion of the third order complex viscosity to derive the medium amplitude signatures of the model in specific shear deformation protocols. Finally, we use these signatures for estimation of model parameters from rheological data obtained by these different deformation protocols, revealing that three-tone oscillatory shear deformations produce data that is readily able to distinguish all features of the medium amplitude, simple shear response space of this generalized class of constitutive models.

A Comparison Between Fractional-Order and Integer-Order Differential Finite Deformation Viscoelastic Models: Effects of Filler Content and Loading Rate on Material Parameters

Elastomers or rubber-like materials exhibit nonlinear viscoelastic behavior such as creep and relaxation upon mechanical loading. Differential constitutive models and hereditary integrals are the main frameworks followed in the literature for modeling the viscoelastic behavior at finite deformations. Regular differential operators can be replaced by fractional-order derivatives in the standard models in order to make fractional viscoelastic models. In the present paper, the relaxation behavior of elastomers is formulated both in terms of ordinary (integer-order) and fractional differential viscoelastic models. The derived constitutive equations are fitted to several experimental data to compare their efficiency in modeling the stress relaxation phenomenon. Specifically, a fractional viscoelastic model with one fractional dashpot (FD) is compared with two ordinary models including respectively one and two ordinary dashpots (OD). The models are compared in fitting accuracy, number of required material parameters and also variation of parameters from one compound to another to clarify the effects of filler content and deformation rate. It is shown that, the results of the ordinary model with one OD is not good at all. The fractional model with one FD and the ordinary model with two ODs provide good fittings for all compounds whereas the former uses only three parameters and the latter uses five material parameters. For the fractional model, the order of the Maxwell element and the associated relaxation time approximately remain the same for different compounds of each material at certain loading rates, but it is not the case for the ordinary differential models.

Finite element simulation of three-dimensional viscoelastic flow at high Weissenberg number based on the log-conformation formulation

Viscoelasticity is an important characteristic of many complex fluids such as polymer solutions and melts. Understanding the viscoelastic behavior of such complex fluids presents mathematical, modeling and computational challenges, particularly in the case of fluids affected by elastic turbulence at high Weissenberg number. A numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three-dimensional flow of viscoelastic fluids. The discrete elastic viscous split stress (DEVSS) formulation in cooperating with log-conformation formulation transformation is employed to improve computational stability at high Weissenberg number. The momentum equation is calculated after introducing an ellipticity factor and the constitutive equation is calculated based on the logarithm of the conformation tensor. The finite element-finite difference formulations of governing equations are derived. The planar contraction as a representative benchmark problem is used to test the robustness of the numerical method to predict real flow patterns of viscoelastic fluids at different Weissenberg numbers. The simulation results predicted with differential constitutive models based on the logarithm of the conformation tensor agree well with Quinzani’s experimental results. Both the stability and the accuracy are improved compared with traditional calculation method. The numerical methodology proposed in the study can well predict complex flow patterns of viscoelastic fluids at high Weissenberg number.

Application of fractional time derivatives in modeling the finite deformation viscoelastic behavior of carbon-black filled NR and SBR

Mechanical behavior of elastomers includes nonlinear and time dependent phenomena such as creep and relaxation where due to low stiffness, these materials usually undergo large deformations. The classical approaches of viscoelasticity,i.e. differential constitutive models and hereditary integrals have been widely used to model the viscoelastic behavior of these materials even at finite deformations. However, from the viewpoint of computational analysis, the integral based methods are often time and memory consuming and the differential methods usually need a large number of material parameters whose identification is a real challenge. A potential solution is to introduce fractional order derivatives instead of regular differential operators in the standard spring-dashpot mechanical models which remarkably reduces the number of required material parameters. In the present paper, relaxation behavior of NR and SBR is investigated in finite strains. Due to the remarkable effects of filler particles and deformation rate on the mechanical properties of elastomers, compounds of each material with different amounts of filler content are investigated at two strain rates. A main interest is to examine efficiency of the fractional derivative models in modeling the finite deformation relaxation behavior of elastomers with different filler contents. The other goal is to investigate the effect of loading rate on the obtained model parameters. It is observed that, the fractional Maxwell chain is able to well fit the stress relaxation response of all compounds with only three parameters. Analyzing the obtained parameters and the experimental data, it is concluded that, order of the associated Maxwell element is not much affected with the filler content. Seemingly, it depends on the matrix material and more noticeably on the deformation rate. Moreover, as much as the deformation rate, the associated relaxation time is low. Decrease of the relaxation time with the deformation rate is more evident for compounds of SBR compared with NR compounds.

Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

We present an Eulerian free-surface flow solver for incompressible pseudoplastic and viscoelastic non-Newtonian fluids. The free-surface flow solver is based on the streamfunction flow formulation and the volume-of-fluid method. The streamfunction solver computes the vector potential of a solenoidal velocity field, which ensures by construction the mass conservation of the solution, and removes the pressure unknown. Pseudoplastic liquids are modelled with a Carreau model. The viscoelastic fluids are governed by differential constitutive models reformulated with the log-conformation approach, in order to preserve the positive-definiteness of the conformation tensor, and to circumvent the high Weissenberg number problem. The volume fraction of the fluid is advected with a geometric conservative unsplit scheme that preserves a sharp interface representation. For the sake of comparison, we also implemented an algebraic advection scheme for the liquid volume fraction. The proposed numerical method is tested by simulating the planar extrudate swell with the Carreau, Oldroyd-B and Giesekus constitutive models. The swell ratio of the extrudates are compared with the data available in the literature, as well as with numerical simulations performed with the open-source rheoTool toolbox in OpenFOAM®. While the simulations of the generalized Newtonian fluids achieved mesh independence for all the methods tested, the flow simulations of the viscoelastic fluids are more sensitive to mesh refinement and the choice of numerical scheme. Moreover, the simulations of Oldroyd-B fluid flows above a critical Weissenberg number are prone to artificial surface instabilities. These numerical artifacts are due to discretization errors within the Eulerian surface-capturing method. However, the numerical issues arise from the stress singularity at the die exit corner, and the unphysical predictions of the Oldroyd-B model in the skin layer of the extrudate after the die exit, where large extensional deformations occur.

Improved both sides diffusion (iBSD): A new and straightforward stabilization approach for viscoelastic fluid flows

This paper reports the developments made to improve the numerical stability of the open-source finite-volume computational library OpenFOAM® developed for the numerical computation of viscoelastic fluid flows described by differential constitutive models. The improvements are based on the modification of the both-sides diffusion technique, named improved both-sides diffusion (iBSD), which promotes the coupling between velocity and stress fields. Calculations for two benchmark 2D case studies of an upper-convected Maxwell (UCM) fluid are presented and compared with literature results, namely the 4:1 planar contraction flow and the flow around a confined cylinder. The results obtained for the first case are computed in five meshes with different refinement levels and are compared with literature results. In this case study it was possible to achieve steady-state converged solutions in the range of Deborah numbers tested, De={0,1,2,3,4,5}, for all meshes. The corner vortex size predictions agree well with the literature and a relative error below 0.6% is obtained for De ≤ 5. In the flow around a confined cylinder, steady-state converged solutions were obtained in the range of Deborah numbers tested, De={0,0.3,0.6,0.8}, in four consecutively refined meshes. The predictions of the drag coefficient on the cylinder are similar to reference data with a relative error below 0.08%. For both test cases the developed numerical method was shown to have a convergence order between 1 and 2, in general very close to the latter. Moreover, the results presented for both case studies clearly extend the previous ones available in the literature in terms of accuracy. This was a direct consequence of the capability of performing the calculation with more refined meshes, than the ones employed before.

Two Ways to Examine Differential Constitutive Equations: Initiated on Steady or Initiated on Unsteady (LAOS) Shear Characteristics.

The exponential Phan–Tien and Tanner (PTT), Giesekus, Leonov, and modified extended Pom–Pom (mXPP) differential constitutive models are evaluated in two ways: with regard to steady shear characteristics and with regard to large amplitude oscillatory shear characteristics of a solution of poly(ethylene oxide) in dimethyl sulfoxide. Efficiency of the models with nonlinear parameters optimized with respect to steady shear measurements is evaluated by their ability to describe large amplitude oscillatory shear (LAOS) characteristics. The reciprocal problem is also analyzed: The nonlinear parameters are optimized with respect to the LAOS measurements, and the models are confronted with the steady shear characteristics. In this case, optimization is based on the LAOS measurements and equal emphasis is placed on both real and imaginary parts of the stress amplitude. The results show that the chosen models are not adequately able to fit the LAOS characteristics if the optimization of nonlinear parameters is based on steady shear measurements. It follows that the optimization of nonlinear parameters is much more responsible if it is carried out with respect to the LAOS data. In this case, when the optimized parameters are used for a description of steady shear characteristics, efficiency of the individual models as documented differs.

Improved convergence based on linear and non-linear transformations at low and high Weissenberg asymptotic analysis

We investigate the accuracy and efficiency of linear and non-linear transformations on the truncated series solutions obtained by asymptotic methods for simple shear and uniaxial elongational viscoelastic flows. The Upper Convected Maxwell, Oldroyd-B, exponential and linear Phan-Thien and Tanner, Giesekus, and FENE-P differential constitutive models are used. The major dimensionless quantity of interest is the Weissenberg number (Wi) in terms of which the unknown dependent flow variables are expanded suitably in series form. The standard regular perturbation method is performed at the Newtonian limit (zero Weissenberg number), while regular or singular perturbation techniques are required to perform the analysis at the purely elastic limit (infinite Weissenberg number). It is shown that the asymptotic approximations at low Wi converge very slowly and have limited radius of convergence, while at high Wi the asymptotic approximations are very accurate even for Wi close to unity. Further improved convergence is achieved by applying a variety of techniques on the original asymptotic solutions, such as Euler's linear transform, Shanks’ non-linear transform, and Padé-type approximants. It is found that the accuracy and the radius of convergence of the transformed solutions increase substantially compared to the original ones. By combining the transformed solutions from both limits, excellent agreement with the exact solutions is demonstrated over the entire range of the Weissenberg number, from zero to infinity, except for a very narrow region close to Wi= 1/2 under uniaxial elongational flow. The results show that, contrary to intuition, viscoelastic phenomena at Wi=O(1) can be predicted accurately using non-linear analysis.

Flexibility of three differential constitutive models evaluated by large amplitude oscillatory shear and Fourier transform rheology

Abstract Using Fourier transform rheology the Giesekus, exponential Phan-Tien and Tanner, and modified eXtended Pom-Pom constitutive models were examined in the non-linear regime represented by large amplitude oscillatory shear (LAOS) measurements. For the experiments a 5 wt% solution of poly(ethylene oxide) (molecular weight 1000 kg/mol) solved in dimethyl sulfoxide was used. The minimal number of harmonics required for an adequate reconstruction of the original raw time-stress signal was shown. An emphasis was paid to simultaneous consideration of the normalised stress magnitudes and the phases of individual harmonics in the fitting procedure adjusting the non-linear parameters of the studied constitutive models.

Electro-osmotic and pressure-driven flow of viscoelastic fluids in microchannels: Analytical and semi-analytical solutions

In this work, we present a series of solutions for combined electro-osmotic and pressure-driven flows of viscoelastic fluids in microchannels. The solutions are semi-analytical, a feature made possible by the use of the Debye–Hückel approximation for the electrokinetic fields, thus restricted to cases with small electric double-layers, in which the distance between the microfluidic device walls is at least one order of magnitude larger than the electric double-layer thickness. To describe the complex fluid rheology, several viscoelastic differential constitutive models were used, namely, the simplified Phan-Thien–Tanner model with linear, quadratic or exponential kernel for the stress coefficient function, the Johnson-Segalman model, and the Giesekus model. The results obtained illustrate the effects of the Weissenberg number, the Johnson-Segalman slip parameter, the Giesekus mobility parameter, and the relative strengths of the electro-osmotic and pressure gradient-driven forcings on the dynamics of these viscoelastic flows.

The role of the Gordon–Schowalter derivative term in the constitutive models—improved flexibility of the modified XPP model

Constitutive models that complete the set of equations describing the flow of polymer melts should respect objective thermodynamics and stability conditions ensuring their validity in the whole range of possible deformation flow. However, in practice, a very good description of flow situations can be achieved with the models not complying with the physical assumptions in all respects. Analogously to the term characterizing yield stress in empirical viscoplastic models, the term represented by the Gordon–Schowalter (GS) derivative in the differential constitutive models contributes to better fitting the experimental data, especially shear thinning. Efficiency of the recently presented modified eXtended Pom-Pom model (just one non-linear parameter per mode) implementing the GS derivative term (one additional non-affine motion parameter per mode) is improved (documented on LDPE, HDPE, and polyvinyl butyral (PVB) materials), and a comparison with the exponential Phan-Tien–Tanner (PTT) and PTT-XPP models (a priori containing the GS derivative term) are presented.

A computational extensional rheology study of two biofluid systems

The main focus of the present computational modelling work is to determine the extensional rheological response of some model biofluids, with a view to ultimately aiding experimentally based analyses and clinical practice. This is accomplished in the present study through model extensional flows and rheological investigation, addressing filament stretching and contraction flows, and upon which significant advances are presented. As such, two biofluid flow systems within the human body are of current interest: (i) respiratory disorders and sputum in the lung airways (associated with filament stretching), where stretchiness of mucus sputum in situ is vital, with clinical focus on chronic obstructive pulmonary disease (COPD/sputum); and (ii) bile flow in the biliary system (contraction flow), with clinical focus on disorders of primary sclerosis cholangitis and common bile duct narrowing. Both sputum and bile biofluid systems are represented through kinetic theory rheological fluid modelling, with capability to represent material structure entanglement, branching and anisotropy. This is practically achieved by appealing to the class of pom-pom differential constitutive models, extracted from polymer melt physics and deployed here through a single extended pom-pom (SXPP) approximation. This class of models is sufficiently rich to enable description of both network structure and rheological properties, exhibiting viscoelastic response (memory), with strain-hardening/softening and shear-thinning properties.

Predictive/fitting capabilities of differential constitutive models for polymer melts—reduction of nonlinear parameters in the eXtended Pom-Pom model

A phenomenological modification of the eXtended Pom-Pom (XPP) model is proposed with the aim to reduce the number of free nonlinear parameters. The modified XPP model includes three parameters per mode in total (two linear viscoelastic parameters—linear relaxation time λ and shear modulus G, and one nonlinear parameter). The original XPP model contains five parameters (two linear viscoelastic parameters and three nonlinear ones, one nonlinear parameter participates in the second normal stress difference prediction). The predictive/fitting capabilities of the modified model are compared with the Giesekus, eXtended Pom-Pom, and modified Leonov models using various low-density PE materials in steady and transient shear and uniaxial elongational flows. It has been found that the modified model is capable of predicting/fitting the rheological properties, with the exception of the second normal stress difference, for studied LDPE materials with sufficient accuracy, including strain hardening in uniaxial elongational flow.

Rheological effects in the 3D creeping flow past a sedimenting sphere subject to orthogonal shear

The effects of the rheological parameters of nonlinear differential constitutive models in the isothermal, steady, creeping, flow past a sedimenting sphere, in an incompressible viscoelastic matrix fluid, subject to simple shear in a plane perpendicular to the direction of sedimentation are studied analytically. The viscoelasticity of the ambient fluid is modeled using the Upper Convected Maxwell, Oldroyd-B, exponential Phan-Thien-Tanner, Giesekus, and FENE-P constitutive equations. The solution of the governing equations is expanded as a regular perturbation series for small values of the Deborah number, and the resulting sequence of three-dimensional partial differential equations is solved analytically up to fourth order. Approximate analytical expressions for the angular velocity of the rigid sphere, as well as for the drag force on the sphere, are derived and discussed. The solutions reveal both the increase of the drag in case of Boger-type fluids (modeled with the FENE-P model) and the decrease of the drag in case of elastic fluids (modeled with the Giesekus model).

Modeling and simulation of three‐dimensional extrusion swelling of viscoelastic fluids with PTT, Giesekus and FENE‐P constitutive models

SUMMARYThe investigation of the extrusion swelling mechanism of viscoelastic fluids has both scientific and industrial interest. However, it has been traditionally difficult to afford theoretical and experimental researches to this problem. The numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three‐dimensional extrusion swelling of viscoelastic fluids flowing through out of a circular die. The rheological responses of viscoelastic fluids are described by using three kinds of differential constitutive models including the Phan‐Thien Tanner model, the Giesekus model, and the finite extensible nonlinear elastic dumbbell with a Peterlin closure approximation model. A streamface‐streamline method is introduced to adjust the swelling free surface. The calculation stability is improved by using the discrete elastic‐viscous split stress algorithm with the inconsistent streamline‐upwind scheme. The essential flow characteristics of viscoelastic fluids are predicted by using the proposed numerical method, and the mechanism of swelling phenomenon is further discussed.Copyright © 2013 John Wiley & Sons, Ltd.

Modeling and simulation of three-dimensional planar contraction flow of viscoelastic fluids with PTT, Giesekus and FENE-P constitutive models

The planar contraction flow is a representative benchmark problem that is often used as a stringent test for the robustness of the numerical method to predict real flow patterns of viscoelastic fluids. A numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three-dimensional flow of viscoelastic fluids through planar contraction. The viscoelastic rheological responses are described by using three kind differential constitutive models including the Phan-Thien–Tanner (PTT) model, the Giesekus model and the finite extensible nonlinear elastic dumbbell with a peterlin closure approximation (FENE-P) model. The discrete elastic viscous split stress (DEVSS) formulation in cooperating with the inconsistent streamline upwind (SU) scheme is employed to improve the computational stability. The simulation results of flow velocity and stress on different sections of the flow field are compared with Quinzani’s experimental results that detected by laser-doppler velocimetry and flow-induced birefringence technologies. It is found that the simulation results predicted with three kind differential constitutive models agree well with the experimental results. The numerical methodology proposed in the study can be used successfully to predict complex flow patterns of viscoelastic fluids.

The kernel-conformation constitutive laws

This work presents a generic kernel-conformation tensor transformation for a large class of differential constitutive models, in which several matrix kernel-transformation families can be applied to the conformation tensor evolution equation. The discretization of this generic analytical framework generates numerical results consistent with the standard conformation tensor discretization at low Weissenberg numbers ( Wi). The numerical efficiency of High Weissenberg Number Problems is ultimately related with both the specific kernel function used in the matrix transformation, but is also related to the existence of mathematical singularities introduced either by the physical description of the flow or by the characteristics of the adopted constitutive equation. We also show that the log-conformation [R. Fattal, R. Kupferman, J. Non-Newtonian Fluid Mech., 123(2–3) (2004) 281–285] and the square-root-conformation [N. Balci, B. Thomases, M. Renardy, C.R. Doering, J. Non-Newtonian Fluid Mech., 166(11) (2011) 546–553] approaches are two relevant particular cases of the kernel-conformation tensor transformation. Simulations for the benchmark confined cylinder flow of an Oldroyd-B fluid provide a preliminary assessment of the merits of some tested transformation functions.

Analysis of inhomogeneous anisotropic viscoelastic bodies described by multi-parameter fractional differential constitutive models

The response to static loads of plane inhomogeneous anisotropic bodies made of linear viscoelastic materials is investigated. Multi-parameter differential viscoelastic constitutive equations are employed, which are generalized using fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plane body element, are two coupled linear fractional evolution partial differential equations in terms of the displacement components. Using the Analog Equation Method (AEM) in conjunction with the Boundary Element Method (BEM) these equations are transformed into a system of multi-term ordinary fractional differential equations (FDEs), which are solved using a numerical method for FDEs developed recently by Katsikadelis. Numerical examples are presented, which not only demonstrate the efficiency of the solution procedure and validate its accuracy, but also permit a better understanding of the response of plane bodies described by different viscoelastic models.

Alternative derivation of differential constitutive equations of the Oldroyd-B type

Now that almost 60 years have passed since the pioneering works of J.G. Oldroyd it seems appropriate as an homage to consider here constitutive equations that can be viewed as generalisations of the by now classical Oldroyd-B model. In this short communication we shall address heuristically the theme of differential constitutive models and will provide an alternative way of deriving a “modified FENE” equation (FENE-M) and inter-relating the PTT and FENE-P-like models.