The FENE-L and FENE-LS closure approximations to the kinetic theory of finitely extensible dumbbells

We address the closure problem for the Warner Finitely Extensible Non-Linear Elastic (FENE) dumbbell model of a dilute polymer solution. The FENE-L closure model, introduced recently for one-dimensional elongational flows [G. Lielens, P. Halin, I. Jaumain, R. Keunings, V. Legat, J. Non-Newtonian Fluid Mech. 76 (1998) 249-279], is extended to general Row kinematics. A simplified version of the theory, referred to as the FENE-LS model, is also proposed. Simulations of steady-state and transient rheometrical flows reveal the superiority of the FENE-L and FENE-LS constitutive equations with respect to the simpler FENE-P closure in describing the rheological response of the FENE kinetic theory. (C) 1999 Elsevier Science B.V. All rights reserved

An efficient method for evaluating diffuse field joint acceptance functions for cylindrical and truncated conical geometries

The evaluation of the response of elastic structures subjected to distributed random excitations is usually performed in the modal space. Random excitations (like acoustic diffuse fields) are usually modeled as weakly stationary random processes and are assumed to be homogeneous. Their characterization basically relies on the power spectral density (PSD) function of the pressure at a particular reference position and a suitable spatial correlation function. In the modal space, the distributed random excitation is characterized by a modal PSD matrix made from the joint acceptance functions related to the mode pairs. The joint acceptance function is a double surface integral involving the product of the considered mode shapes and the spatial correlation function. The paper shows how to evaluate efficiently this quadruple integral for cylindrical and truncated conical structures excited by an acoustic diffuse field. Basically, the procedure relies on the derivation of alternative expressions for the spatial correlation function. The related expressions prove to be more convenient for these geometries and are leading to a reduction of the double surface integral to a combination of simple integrals. A very substantial breakdown of the computational cost can be achieved using the resulting expressions. (c) 2005 Acoustical Society of America

On the hysteretic behaviour of dilute polymer solutions in relaxation following extensional flow

The hysteretic behaviour of dilute polymer solutions in relaxation following extensional now is studied in the framework of three distinct theoretical models. For ideal kinematics of uniaxial elongation, we show that the kinetic theory of FENE dumbbells and its FENE-L approximation present a hysteresis when plotting polymer stress versus average molecular extension. Similar behaviour is obtained for ideal extensional kinematics using a FENE-P constitutive equation with a spectrum of finite extensibility parameters. Finally, a numerical simulation of the filament-stretching device shows that spatial inhomogeneities of the stress and average conformation fields also lead to hysteretic behaviour with a single-mode FENE-CR constitutive equation. In all three cases, hysteretic behaviour results from the combined effect of dispersity and non-linearity. We also address the validity of the stress-optic law for FENE dumbbells in relaxation following start-up of uniaxial extension. The simulation results show that the stress-optic coefficient remains constant at low strains only. Plots of stress-optic coefficient versus birefringence show hysteresis as well. This rules out a modified stress-optic law for FENE dumbbells wherein the stress-optic coefficient would be a function of the second moment of the configuration distribution function alone. Finally, it is shown in the Appendix that a proper selection of the spectrum of finite extensibilities can be made so that the multi-mode FENE-P model gives essentially the same stress response as the kinetic theory of FENE dumbbells in transient uniaxial extension. (C) 1999 Elsevier Science B.V. All rights reserved

The adaptive Lagrangian particle method for macroscopic and micro-macro computations of time-dependent viscoelastic flows

We propose a new numerical technique, referred to as the Adaptive Lagrangian Particle Method (ALPM), for computing time-dependent viscoelastic flows using either a differential constitutive equation (macroscopic approach) or a kinetic theory model (micro-macro approach). In ALPM, the Eulerian finite element solution of the conservation equations is decoupled from the Lagrangian computation of the extra-stress at a number of discrete particles convected by the how. In the macroscopic approach, the extra-stress carried by the particles is obtained by integrating the constitutive equation along the particle trajectories. In the micro-macro approach, the extra-stress is computed by solving along the particle paths the stochastic differential equation associated with the kinetic theory model. At each time step, ALPM automatically enforces that all elements of the mesh have a number of Lagrangian particles ranging within a user-specified interval. Results are given for the start-up flow between highly eccentric rotating cylinders, using the FENE and FENE-P dumbbell models for dilute polymer solutions. (C) 1999 Elsevier Science S.A. All rights reserved

Simulation of randomly excited acoustic insulation systems using finite element approaches

Acoustic transmission properties of sound-insulating structures play an important role in the design of many transport systems (as, for instance, automotive, aircraft and railway products). Design procedures require efficient computational tools based on the selection of appropriate numerical models for both the insulating structure and the surrounding (bounded or unbounded) acoustic domains. Additionally the description of excitation mechanisms requires some caution and weak or strong coupling effects should be taken into account. The paper presents the main features of a refined finite element approach for addressing acoustic transmission problems in a random context. The model relies on the usual elasto-acoustic approximations. A particular attention is devoted to the description of turbulent boundary layer excitations. The handling of such spatially correlated random excitations is described both in direct (physical) and reduced (modal) contexts. A powerful asymptotic modal formulation is identified. Comparisons with reference analytical solutions are presented and demonstrate the validity of the proposed method. Further application to a reduced scale greenhouse model (from the EC-funded project SMILE) is presented.Anglai

New closure approximations for the kinetic theory of finitely extensible dumbbells

We address the closure problem for the most elementary non-linear kinetic model of a dilute polymeric solution, known as the Warner finitely extensible non-linear elastic (FENE) dumbbell model. In view of the closure problem, the FENE theory cannot be translated into an equivalent macroscopic constitutive equation for the polymer contribution to the stress tensor. We present a general framework for developing closure approximations, based on the concept of canonical distribution subspace first introduced by Verleye and Dupret (in: Developments in Non-Newtonian Flows, AMD-Vol. 175, ASME, New York, 1993, 139-163) in the context of fiber suspension modeling. The classical consistent pre-averaging approximation due to Peterlin (that yields the FENE-P constitutive equation) is obtained from the canonical approach as the simplest first-order closure model involving only the second moment of the configuration distribution function. A second-order closure model (referred to as FENE-P-2) is derived, which involves the second and fourth moments of the distribution function. We show that the FENE-P-2 model behaves like the FENE-P equation with a reduced extensibility parameter. In this respect, it is a close relative of the FENE-P* equation proposed by van Heel et al. (J. Non-Newton. Fluid Mech., 1998, in press). Inspired by stochastic simulation results for the FENE theory, we propose a more sophisticated second-order closure model (referred to as FENE-L). The rheological response of the FENE-P, FENE-P-2 and FENE-L closure models are compared to that of the FENE theory in various time-dependent, one-dimensional elongational flows. Overall, the FENE-L model is found to provide the best agreement with the FENE results. In particular, it is capable of reproducing the hysteretic behaviour of the FENE model, also observed in recent experiments involving polystyrene-based Boger fluids (Doyle et al., J. Non-Newton. Fluid Mech., submitted), in stress versus birefringence curves during startup of flow and subsequent relaxation. (C) 1998 Elsevier Science B.V. All rights reserved