Roland Keunings

Nonlinear analysis of the surface tension driven breakup of viscoelastic filaments

Abstract The surface tension driven breakup of viscoelastic filaments into droplets is qualitatively different from that of Newtonian liquid filaments. Disturbances on filaments of dilute polymer solutions often grow to a configuration consisting of nascent droplets connected by a thin ligament; the breakup time for this configuration is much longer than that predicted by extensions of Rayleighs linear stability theory. We present here a nonlinear analysis of surface tension driven breakup of viscoelastic filaments using two complementary approaches that given equivalent results: a transient finite element solution and a one-dimensional thin filament approximation. We show that significant nonlinear effects lead to the experimentally-observed nascent droplet-ligament configuration, and we predict the entire evolution of the filament profile. Agreement with available experimental data for profile evolution and breakup of jets of Newtonian fluids and dilute polymer solutions is excellent.

On the high Weissenberg number problem

Abstract We address the outstanding problem affecting the numerical simulation of steady viscoelastic flows in complex geometries, namely the existence of a critical value of the Weissenberg number beyond which no discrete solutions can be obtained. The flow of Maxwell and Leonov-like fluids through a sudden contraction is selected as a test problem. Discrete solutions are obtained by means of a mixed Galerkin/Finite Element method. We find that limit points of the discrete solution families are responsible for the loss of convergence of the iterative scheme. Intensive mesh refinement shows, however, that these limit points are numerical artifacts.

Finite-element analysis of die swell of a highly elastic fluid

Abstract A mixed finite element method is applied to the die swell calculation of ana Oldroyd fluid B. The use of large entry lengths together with the presence of the retardation time in the constitutive equations allow us to reach values of the recoverable shear as high as four for the flow emerging from slit and circular dies, with swelling ratios of the order of 2. The numerical results are in good agreement with some available experimental data.

An algorithm for the simulation of transient viscoelastic flows with free surfaces

We propose a numerical procedure for solving a class of transient viscoelastic flows with free surfaces. It is based on a Galerkin/Finite Element technique on deforming elements combined with a predictor-corrector scheme. The method is applied to the analysis of jet breakup caused by capillary forces. Non-linear effects known to experimentalists are predicted and a detailed comparison with asymptotic results is carried out.

On the Peterlin approximation for finitely extensible dumbbells

For the simplest non-linear kinetic theory of dilute polymeric solutions (FENE dumbbells), the pre-averaging Peterlin approximation used to derive a macroscopic constitutive equation (FENE-P) is shown to have a significant impact on the statistical and rheological properties of the model. This is illustrated in simulations of transient elongational flows by means of standard and stochastic numerical techniques

Modelling of critical fibre length and interfacial debonding in the fragmentation testing of polymer composites

Micro-mechanical models based on a unidimensional load transfer approximation are used to predict the critical fibre length as a function of applied strain in the fragmentation testing of polymer matrix composites. Conditions of perfect adhesion, partial debonding, and total debonding are considered in turn. Situations are identified where the critical length cannot be viewed as a material constant, i.e. where it remains strain dependent as the applied strain increases. Numerical results based on the partial debonding model are given for the critical fibre length and the extent of the debonding zone as a function of applied strain. The prediction of the total debonding model is recovered asymptotically for large strains. We find, however, that the critical length predicted by the partial debonding model can be lower than the one predicted by the total debonding model if the interfacial bond strength is sufficiently larger than the frictional shear stress. These theoretical results show that both bond strength and frictional shear stress must be taken into account in the interpretation of the fragmentation test data.

The Lagrangian particle method for macroscopic and micro-macro viscoelastic flow computations

We propose a new numerical technique, referred to as the Lagrangian Particle Method (LPM), for computing time-dependent viscoelastic flows using either a differential constitutive equation (macroscopic approach) or a kinetic theory model (micro-macro approach). In LPM, 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 flow. 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. Results are given for the start-up flow between slightly eccentric rotating cylinders, using the FENE and FENE-P dumbbell models for dilute polymer solutions

Prediction of thermo-mechanical properties for compression moulded composites

We present a method to determine the thermo-mechanical properties of compression moulded composite parts. The flow-induced fibre orientation is first calculated by numerical simulation, and the resulting orientation state is used as input in a micromechanical model that predicts the thermo-mechanical properties of the part. A two-step homogenization scheme based on the grain model approach is followed. First, the properties of a reference composite with aligned fibres are estimated by means of a mixture rule between the upper and lower Hashin-Shtrikman bounds (derived by Willis). This method is in agreement with the Mori-Tanaka estimates for moderate concentrations, and gives better results for higher concentrations. Next, the properties of the composite are obtained by averaging several reference composites with different fibre directions. An example of a 3-D compression moulded composite part is analyzed and the results are discussed

Numerical-simulation of the Flow of a Viscoelastic Fluid Through An Abrupt Contraction

Abstract The flow of a Phan Thien-Tanner fluid through an abrupt 4/1 contraction is anaylzed by means of finite elements. It is found that the choice of such a fluid gives rise to an important corner vortex growth and to an increasing Couette correction when the flow-rate increases

The Proper Generalized Decomposition for Advanced Numerical Simulations

Many problems in scientific computing are intractable with classical numerical techniques. These fail, for example, in the solution of high-dimensional models due to the exponential increase of the number of degrees of freedom.Recently, the authors of this book and their collaborators have developed a novel technique, called Proper Generalized Decomposition (PGD) that has proven to be a significant step forward. The PGD builds by means of a successive enrichment strategy a numerical approximation of the unknown fields in a separated form. Although first introduced and successfully demonstrated in the context of high-dimensional problems, the PGD allows for a completely new approach for addressing more standard problems in science and engineering. Indeed, many challenging problems can be efficiently cast into a multi-dimensional framework, thus opening entirely new solution strategies in the PGD framework. For instance, the material parameters and boundary conditions appearing in a particular mathematical model can be regarded as extra-coordinates of the problem in addition to the usual coordinates such as space and time. In the PGD framework, this enriched model is solved only once to yield a parametric solution that includes all particular solutions for specific values of the parameters. The PGD has now attracted the attention of a large number of research groups worldwide. The present text is the first available book describing the PGD. It provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method. Throughout the book, the PGD is applied to problems of increasing complexity, and the methodology is illustrated by means of carefully selected numerical examples. Moreover, the reader has free access to the Matlab software used to generate these examples.