Response to "Comment on: `Thermodynamics of viscoelastic fluids: the temperature equation'"

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A modification of the convective constraint release mechanism in the molecular stress function model giving enhanced vortex growth

The molecular stress function model with convective constraint release (MSF with CCR) constitutive model [J. Rheol. 45 (2001), 1387] is capable of fitting all viscometric data for IUPAC LDPE, with only two adjustable parameters (with difference found only on reported ¿steady-state¿ elongational viscosities). The full MSF with CCR model is implemented in a backwards particle-tracking implementation, using an adaptive method for the computation of relative stretch that reduces simulation time many-fold, with insignificant loss of accuracy. The model is shown to give improved results over earlier versions of the MSF (without CCR) when compared to well-known experimental data from White and Kondo [J. non-Newt. Fluid Mech., 3 (1977), 41]; but still to under-predict contraction flow opening angles. The discrepancy is traced to the interaction between the rotational dissipative function and the large stretch levels caused by the contraction flow. A modified combination of dissipative functions in the constraint release mechanism is proposed, which aims to reduce this interaction to allow greater strain hardening in a mixed flow. The modified constraint release mechanism is shown to fit viscometric rheological data equally well, but to give opening angles in the complex contraction flow that are much closer to the experimental data from White and Kondo. It is shown (we believe for the first time) that a constitutive model demonstrates an accurate fit to all planar elongational, uniaxial elongational and shear viscometric data, with a simultaneous agreement with this well-known experimental opening angle data. The sensitivity of results to inaccuracies caused by representing the components of the deformation gradient tensor to finite precision is examined; results are found to be insensitive to even large reductions in the precision used for the representation of components. It is shown that two models that give identical response in elongational flow, and a very similar fit to available shear data, give significantly different results in flows containing a mix of deformation modes. The implication for constitutive models is that evaluation against mixed deformation mode flow data is desirable in addition to evaluation against viscometric measurements

A study of the quadratic molecular stress function constitutive model in simulation

Constitutive models that conform to separable KBKZ specification have been shown to fit steady-state strain hardening rheological data in planar and uniaxial elongational flows, but with inaccuracy in the rate of strain hardening. The single parameter Molecular Stress Function model of Wagner [Rheol. Acta, 39 (2000), 97-109] has been shown to accurately fit the rise-rate in experimental data for a number of strain hardening and strain softening materials. We study this models accuracy against the well characterised IUPAC LDPE data, and present a method for full implementation of this model for flow solution which is suitable for incorporating into existing separable KBKZ software. A new method for particle tracking in arbitrarily aligned meshes, which is efficient and robust, is given. The Quadratic Molecular Stress Function (QMSF) model is compared to existing separable KBKZ based models, including one which is capable of giving planar strain hardening; the QMSF is shown to fit experimental rheological and contraction flow data more convincingly. The issue of `negative correction pressures¿ notable in some Doi-Edwards based models is addressed. The cause is identified, and leads to a logical method of calculation which does not give these anomalous results

Hyperbolic contraction measuring systems for extensional flow

In this paper an experimental method for extensional measurements on medium viscosity fluids in contraction flow is evaluated through numerical simulations and experimental measurements. This measuring technique measures the pressure drop over a hyperbolic contraction, caused by fluid extension and fluid shear, where the extensional component is assumed to dominate. The present evaluative work advances our previous studies on this experimental method by introducing several contraction ratios and addressing different constitutive models of varying shear and extensional response. The constitutive models included are those of the constant viscosity Oldroyd-B and FENE-CR models, and the shear-thinning LPTT model. Examining the results, the impact of shear and first normal stress difference on the measured pressure drop are studied through numerical pressure drop predictions. In addition, stream function patterns are investigated to detect vortex development and influence of contraction ratio. The numerical predictions are further related to experimental measurements for the flow through a 15:1 contraction ratio with three different test fluids. The measured pressure drops are observed to exhibit the same trends as predicted in the numerical simulations, offering close correlation and tight predictive windows for experimental data capture. This result has demonstrated that the hyperbolic contraction flow is well able to detect such elastic fluid properties and that this is matched by numerical predictions in evaluation of their flow response. The hyperbolical contraction flow technique is commended for its distinct benefits: it is straightforward and simple to perform, the Hencky strain can be set by changing contraction ratio, non-homogeneous fluids can be tested, and one can directly determine the degree of elastic fluid behaviour. Based on matching of viscometric extensional viscosity response for FENE-CR and LPTT models, a decline is predicted in pressure drop for the shear-thinning LPTT model. This would indicate a modest impact of shear in the flow since such a pressure drop decline is relatively small. It is particularly noteworthy that the increase in pressure drop gathered from the experimental measurements is relatively high despite the low Deborah number range explored

Nonisothermal flows of vicsoelastic fluids. Thermodynamics, analysis and numerical simulation.

Mechanical Maritime and Materials Engineerin

Impact of decoupling approximation between stretch and orientation in rheometrical and complex flow of entangled linear polymers

We study the rheometrical and complex flow response of the coupled version of the double-convection-reptation model with chain stretch. This model for monodisperse entangled linear polymers has recently been proposed by Marrucci and Ianniruberto [G. Marrucci, G. Ianniruberto, Flow-induced orientation and stretching of entangled polymers, Philos. Trans. R. Soc. A 361 (2003) 677-688] to overcome the anomalous shear thickening that was present in an earlier version of the theory. It avoids the decoupling approximation between orientation and stretch. Except for the shear thickening, both coupled and decoupled models show very similar results that are in qualitative agreement with available rheometrical data for two nearly monodisperse polymer solutions. In contraction/expansion flow simulations, however, higher Weissenberg numbers can be attained with the coupled model. Simulations in the stretch-dominated regime predict a dramatic growth of the upstream vortex activity and an increase of the pressure correction. (C) 2004 Elsevier B.V. All rights reserved

Simulation of linear polymer melts in transient complex flow

Recently, much progress has been made in improving the modelling of linear polymer melts with the aid of reptation theory. In simple shear flows, this has resulted in a much better prediction of the shear viscosity and normal stress ratio. Here, we evaluate in complex flow the transient and steady-state behaviour of a recently proposed reptation model, the Marrucci-Greco-Ianniruberto model [G. Marrucci, F Greco, G. Ianniruberto, Rheol. Acta, 2000, submitted for publication], that includes convective constraint release and a force balance on the entanglement nodes. To incorporate integral type models into the numerical framework of Lagrangian particle methods, developed previously to simulate dilute polymer solutions, we have included the so-called deformation field method. For the contraction/expansion flow that we consider, we find that a correction of the convective constraint release contribution to the relaxation time is necessary to avoid the unphysical situation of negative relaxation times. With this correction, we could obtain mesh and time convergence for high Weissenberg numbers without adding any solvent viscosity. We find that in complex flow also, both the steady-state and transient response of the integral model can be very well approximated by a constitutive equation of differential type. Due to the dominance of the strong thinning in both shear and elongational flows for the model, however, the inelastic Carreau-Yasuda model reproduces the steady-state kinematics and pressure drop as well. (C) 2000 Elsevier Science B.V. All rights reserved

Thermodynamics of viscoelastic fluids: The temperature equation

From the thermodynamics with internal variables we will derive the temperature equation for viscoelastic fluids. We consider the type of storage of mechanical energy, the dissipation df mechanical energy, the compressibility of the fluid, the nonequilibrium heat capacity and thermal expansion, and deformation induced anisotropy of the heat conduction. The well-known stress differential models that fit into the thermodynamic theory will be treated as an example. Adapting a power-law scaling of the sheer moduli on temperature and density, as is usual in rubber elasticity, we will derive an approximation of the temperature equation in measurable quantities. This equation will be compared with experimental results. (C) 1998 The Society of Rheology

Numerical simulation of branched polymer melts in transient complex flow using pom-pom models

In recent years, a number of constitutive equations have been derived from reptation theory to describe the rheology of both linear and branched polymer melts. While their predictions in rheometrical flows have been discussed in detail, not much is known of their behaviour in complex flows. In the present paper, we study by way of numerical simulation the transient, start-up how of branched polymers through a planar contraction/expansion geometry. The constitutive equation is the so-called pom-pom model introduced by McLeish and Larson [J. Rheol. 42 (1998) 81], and later modified by Blackwell et al. [J. Rheol. 44 (2000) 121]. By combining the backward-tracking Lagrangian particle [J. Non-Newtonian Fluid Mech. 91 (2000) 273] and deformation field [J. Non-Newtonian Fluid Mech. 89 (2000) 209] methods, we obtain results for the original, integral pom-pom model which makes use of the Doi-Edwards orientation tensor. Two simplified versions of the pom-pom model are also considered, namely one based on the Currie approximation for the orientation tensor, and a differential constitutive equation proposed in [J. Rheol. 42 (1998) 81]. Finally, the simulation results are compared to those obtained with the so-called MGI model proposed recently by Marrucci et al. [Rheol. Acta, submitted for publication] for describing linear polymer melts. (C) 2001 Elsevier Science B.V. All rights reserved