Submitted by Ellen Silva Gervazoni null (emillen_@hotmail.com) on 2016-08-25T17:45:32Z No. of bitstreams: 1 Disserataçaofinal_EllenGervazoni.pdf: 1426674 bytes, checksum: 25ce3b189028e33485b050b9f7676a94 (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-08-29T19:17:40Z (GMT) No. of bitstreams: 1 gervazoni_es_me_prud.pdf: 1426674 bytes, checksum: 25ce3b189028e33485b050b9f7676a94 (MD5) / Made available in DSpace on 2016-08-29T19:17:40Z (GMT). No. of bitstreams: 1 gervazoni_es_me_prud.pdf: 1426674 bytes, checksum: 25ce3b189028e33485b050b9f7676a94 (MD5) Previous issue date: 2016-06-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Diversos escoamentos de interesse prático são de fluidos viscoelásticos e muitas vezes é desejável saber se estes escoamentos propagam-se no estado laminar ou no turbulento. Embora a hidrodinâmica de fluidos viscoelásticos sejam fortemente afetadas pelo balanço entre forças inerciais e elásticas no escoamento, o efeito da elasticidade sobre a estabilidade de escoamentos inerciais não foi completamente estabelecida. No presente trabalho, estuda-se o que ocorre entre estes dois estados, na transição laminar-turbulenta. Especi- ficamente, é investigada a convecção de ondas de Tollmien-Schlichting para o escoamento incompressível de Poiseuille para um fluido viscoelástico, utilizando a equação constitutiva Oldroyd-B. Para isto, utiliza-se a Simulação Numérica Direta para verificar a estabilidade dos escoamentos de fluidos viscoelásticos a perturbações não estacionárias. Os resultados numéricos obtidos para escoamentos de fluidos viscoelásticos são comparados com os resultados de escoamentos de fluidos Newtonianos, que já estão bem documentados na comunidade científica. Além disso, uma equação de Orr-Sommerfeld modificada é deduzida para um escoamento viscoelástico utilizando a Teoria de Estabilidade Linear. / Several flows of practical interest are of viscoelastic fluids and it is often desirable to know if these flows are in a laminar or turbulent state. Although the hydrodynamics of viscoelastic fluids are strongly affected by the balance between inertia and elastic forces in the flow, the effect of elasticity on the stability of inertial flows has not been completely established. In this work is studied what happens between these two states, the laminar-turbulent transition. Specifically, it will be investigated the convection of Tollmien-Schlichting waves to incompressible Poiseuille flow of viscoelastic fluid, using the constitutive equation Oldroyd-B. For this, the analysis is carried out by means of Direct Numerical Simulation to verify the stability of the non-stationary disturbances viscoelastic fluids flows. The numerical results obtained for viscoelastic fluids flows are compared with the results of Newtonian fluids flows, which are already well documented in scientific community. In addition, an Orr-Sommerfeld modified equation is deducted for a viscoelastic flow using Linear Stability Theory.

In this thesis, we aim to create a framework for the derivation of thermodynamically consistent anisotropic viscoelastic models. As an example we propose simple models extending the isotropic Oldroyd-B and Giesekus models to illustrate the models' behavior and the process of finding the correct equations. We show what behavior in sheer we can expect and continue with a 3D simulation inspired by the experiment on a real liquid crystal mixture. Finally, we compare the simulation and the experiment to find similarities and possible further research topics.

Cette thèse est consacrée à l’analyse mathématique de trois problèmes d’écoulements de fluides viscoélastiques de type Oldroyd. Tout d’abord, nous étudions des écoulements stationnaires faiblement compressibles dans un domaine borné avec des conditions au bord de type "rentrante-sortante". Nous étudions aussi le problème d’écoulements stationnaires faiblement compressibles dans un coin convexe. En utilisant une méthode de point fixe (premier et deuxième problèmes) et une décomposition de Helmoltz (deuxième problème), nous montrons des résultats d’existence et d’unicité des solutions. Nous étudions également le cas d’un écoulement non stationnaire. Nous montrons un résultat d’existence locale et un résultat d’existence globale, avec des conditions initiales suffisamment petites, pour des fluides compressibles. Nous démontrons aussi la convergence du modèle d’écoulement viscoélastique compressible à faible nombre de Mach vers le modèle incompressible lorsque les données initiales sont "bien préparées" / In this PHD thesis, we study three problems for viscoelastic flows of Oldroyd type. First, we study steady flows of slightly compressible in a bounded domain with non-zero velocities on the boundary ; the pressure and the extra-stress tensor are prescribed on the part of the boundary corresponding to entering velocity. This causes a weak singularity in the solution at the junction of incoming and outgoing flows. We also study the problem of steady flows of slightly compressible fluids with zero boundary conditions in a domain with an isolated corner point. Using a method of fixed point (first and second problems) and a Helmoltz decomposition (second problem), we show some results of existence and uniqueness of solutions. In the last part, we study the case of a non-steady flow : we show some results of local and of global existence, with sufficiently small initial data, for compressible flows. The zero-Mach number limit is also established

Στόχος της παρούσας διδακτορικής διατριβής είναι η μελέτη χρονικά μεταβαλλόμενων και ασταθών υπό συνθήκες ροών ιξωδοπλαστικών και ιξωδο-ελαστικών ρευστών λόγω της σημασίας που παρουσιάζουν τα υλικά αυτά σε διάφορες βιομηχανικές διεργασίες. Στα πλαίσια της εργασίας αυτής μελετήθηκε αρχικά το πρόβλημα της συμπίεσης ιξωδοπλαστικού ρευστού μεταξύ δύο δίσκων το οποίο είναι ένα βασικό πείραμα ρεολογικού χαρακτηρισμού μη Νευτωνικών υλικών. Η χρονικά μεταβαλλόμενη προσομοίωση επέτρεψε τον προσδιορισμό των σημαντικών διαφορών μεταξύ των δύο εκδοχών του συγκεκριμένου ρεολογικού πειράματος (δίσκοι κινούμενοι υπό σταθερή ταχύτητα ή υπό σταθερή δύναμη) που ήταν αδύνατος με τα προηγούμενα μοντέλα, ενώ παρουσιάζεται και πλήρης παραμετρική μελέτη της διεργασίας αυτής. Το υπόλοιπο μέρος της παρούσας εργασίας αφιερώθηκε στη μελέτη της διεργασίας εκβολής ιξωδοελαστικών υλικών η οποία συναντάται ευρέως στη βιομηχανία μορφοποίησης πολυμερών. Παρουσιάζεται μια πλήρης παραμετρική ανάλυση για την εκβολή ενός ιξωδοελαστικού ρευστού Phan-Tien & Tanner (PTT) από ένα δακτυλιοειδή αγωγό προκειμένου να εξετάσουμε την επίδραση της γεωμετρίας του αγωγού, και των ρεολογικών ιδιοτήτων του ρευστού. Επιπλέον ένα σημαντικό πρόβλημα που εμφανίζεται κατά την εκβολή από ένα κυλινδρικό, επίπεδο ή δακτυλιοειδή αγωγό είναι η εμφάνιση ασταθειών οι οποίες επηρεάζουν σημαντικά την ποιότητα του τελικού προϊόντος. Προκειμένου να διερευνηθεί ο μηχανισμός που προκαλεί την εμφάνιση αυτών των ασταθειών πραγματοποιήθηκε γραμμική ανάλυση ευστάθειας για τη αξονοσυμμετρική και καρτεσιανή δισδιάστατη ροή επικόλλησης-ολίσθησης (stick-slip flow) ενός ρευστού PTT. Γύρω από τη λύση μόνιμης κατάστασης πραγματοποιείται γραμμική ανάλυση ευστάθειας, και υπολογίζονται οι ιδιοτιμές του γενικευμένου προβλήματος με τη βοήθεια της μεθόδου Arnoldi. Έτσι προσδιορίζεται η επίδραση των ιδιοτήτων του υλικού στην ευστάθεια ή μη της ροής. / The purpose of this dissertation is to perform a study on time-dependent and conditionally unstable flows of viscoplastic and viscoelastic fluids because of their importance in various industrial processes. A typical experiment which is used widely for the rheological characterization of non-Newtonian fluids is squeeze flow which is the subject of the first chapter of this dissertation. The time-dependent simulation permitted the determination of the distinct differences between the two versions of this rheological experiment (disks moving with constant velocity or under constant force) which was impossible with the models used up to now, while a complete parametric analysis of this process is presented. The rest of this work focuses in the important problem of the extrusion process of a viscoelastic fluid, which is frequently encountered in the polymer industry. A complete parametric analysis is presented for the extrusion of a Phan-Thien Tanner (PTT) fluid from an annular die in order to examine the effect of the die geometry and the rheological properties of the fluid on this process. Moreover, it is widely known that during the extrusion from a cylindrical, planar or an annular die various flow instabilities may arise affecting significantly the quality of the final product. In order to investigate the mechanism, which causes the appearance of these instabilities, a linear stability analysis is performed for the cylindrical or planar stick-slip flow of a PTT fluid. The stability analysis is performed around the steady state solution and the eigenvalues of the generalized problem are calculated using the Arnoldi algorithm. With this method the effect of the various rheological properties of the fluid on the stability of the flow is determined.

In comparison to the flow in a rigid channel, there is a multi-fold reduction in the transition Reynolds number for the flow in a micro channel when one of the walls is made sufficiently soft, due to a dynamical instability induced by the fluid-wall coupling. The flow after transition is characterized using Particle Image Velocimetry (PIV) in the x − y plane where x is the stream-wise direction and y is the cross-stream co-ordinate along the small dimension of the channel of height 0.2 − 0.3mm. For the two different soft walls of shear modulus 18 kPa and 2.19 kPaused here, the transition Reynolds number is about 250 and 330 respectively. The deformation of the microchannel due to the applied pressure gradient is measured in the experiments, and is used to predict the laminar mean velocity profiles for comparison with the experimental results. The mean velocity profiles in the microchannel are in quantitative agreement with those predicted for the laminar flow before transition, but are flatter near the centerline and have higher gradients at the wall after transition. The flow after transition is characterized by a mean velocity profile that is flatter at the center and steeper at the walls in comparison to that for a laminar flow. The root mean square of the stream-wise fluctuating velocity shows the characteristic sharp increase from the wall and a maximum close to the wall, as observed in turbulent flows in rigid-walled channels. However, the profile is asymmetric with a significantly higher maximum close to the soft wall in comparison to that close to the hard wall, and the Reynolds stress is found to be non-zero at the soft wall, indicating that there is a stress exerted by fluid velocity fluctuations on the wall. The turbulent energy production profile has a maximum at the soft wall, in contrast to the flow at a rigid surface where the turbulent energy production is zero at the wall (due to the zero Reynolds stress). The maximum of the root mean square of the velocity fluctuations and the Reynolds stress (divided by the fluid density) in the soft-walled microchannel for Reynolds numbers in the range 250-400, when scaled by suitable powers of the maximum velocity, are comparable to those in a rigid channel at Reynolds numbers in the range 5000-20000. The near-wall velocity profile shows no evidence of a viscous sub-layer for (yv∗/ν) as low as 2, but there is a logarithmic layer for (yv∗/ν) up to about 30, where the von Karman constants are very deferent from those for a rigid-walled channel. Here, v∗ is the friction velocity, ν is the kinematic viscosity and y is the distance from the soft surface. . The surface of the soft wall in contact with the fluid is marked with dye spots to monitor the deformation and motion along the fluid-wall interface. The measured displacement of the surface in the stream-wise direction, which is of the order of 5 − 12µm, is consistent with that calculated on the basis of linear elasticity. Low-frequency oscillations in the displacement of the surface are observed after transition in both the stream-wise and span-wise directions, indicating that the turbulent velocity fluctuations are dynamically coupled to motion in the solid. Modification of soft-wall turbulence in a micro channel due to the addition of small amounts of polymer The modification of soft-wall turbulence in a microchannel due to the addition of small amounts of polymer is experimentally studied using Particle Image Velocimetry (PIV) to measure the mean and the fluctuating velocities. The micro channels are of rectangular cross-section with height about 160 µm, width about 1.5 mm and length about 3 cm, with three walls made of hard Poly-dimethylsiloxane (PDMS) gel, and one wall made of soft PDMS gel with an elasticity modulus of about 18 kPa. A dynamical instabilty of the laminar flow due to the fluid-wall coupling, and a transition to turbulence, is observed at a Reynolds number of about 290 for the flow of pure water in the soft-walled microchannel (Verma and Kumaran, J. Fluid Mech., 727, 407-455, 2013). Solutions of polyacrylamide of molecular weight 5 × 106 and mass fraction up to 50 ppm, and of molecular weight 4 × 104 and mass fraction up to 1500 ppm, are used in the experiments. In all cases, the solutions are in the dilute limit be-low the critical concentration where the interactions between polymer molecules become important. The modification of the fluid viscosity due to addition of polymer molecules is small; the viscosity of the solutions with the highest polymer concentration exceed those for pure water by about 10% for the polymer with molecular weight 5 × 106, and by about 5% for the polymer with molecular weight 4 × 104. Two distinct types of flow modifications below and above a threshold mass fraction for the polymer, cTHRESHOLD , which is about 1 ppm for the polyacrylamide with molecular weight 5 × 106, and about 500 ppm for the polyacrylamide with molecular weight 4 × 104. As the polymer mass fraction increases up to the threshold value, there is no change in the transition Reynolds number, but there is significant turbulence attenuation the root mean square velocities in the stream wise and cross-stream directions decrease by a factor of 2, and the Reynolds stress decreases by a factor of 4 in comparison to that for pure water. When the polymer concentration increases beyond the threshold value, there is a decrease in the decrease in the transition Reynolds number by nearly one order of magnitude, and a further decrease in the intensity of the turbulent fluctuations. The lowest transition Reynolds number of about 35 for the solution of polyacrylamide with molecular weight 5 × 106 and mass fraction 50 ppm. For the polymer solutions with the highest concentrations, the fluctuating velocities in the stream wise and cross-stream direction are lower by a factor of 5, and the Reynolds stress is lower by a factor of 10, in comparison to pure water. Despite the significant turbulence attenuation, a sharp increase in the intensity of the fluctuating velocities is evident at transition for all polymer concentrations. Transitions to deferent kinds of turbulence in a channel with soft walls The flow in a rectangular channel with walls made of soft polyacrylamide gel is studied to examine the effect of soft walls on transition and turbulence. The width of the channel is much larger than the height, so that the flow can be considered approximately two-dimensional, the wall thickness is much larger than the channel height (smallest dimension), the bottom wall is fixed to a substrate and the top wall is unrestrained. The fluid velocity is measured using Particle Image Velocimetry, while the wall motion is studied by embedding beads in the soft wall, and measuring the time-variation of the displacement both parallel and perpendicular to the surface. As the Reynolds number increases, two different flow regimes are observed in sequence. The first is the ‘soft-wall turbulence’ resulting from a dynamical instability of the base flow due to the fluid-wall coupling. The flow in this case exhibits many of the features of the turbulent flow in a rigid channel, including the departure of the velocity profile from the parabolic profile, and the near-wall maxima in the stream-wise root mean square fluctuating velocity. However, there are also significant differences. The turbulence intensities, when scaled by suitable powers of the mean velocity, are much larger than those after the hard-wall laminar-turbulent transition at a Reynolds number of about 1000. The Reynolds stress profiles do not decrease to zero at the walls, indicating that the wall motion plays a role in the generation of turbulent fluctuations. There is no evidence of a viscous sub-layer close to the wall to within the experimental resolution. The mean velocity profile does satisfy a logarithmic law close to the surface within a region between 2-30 wall units from the surface, but the von Karman constants are very different from those for the hard-wall turbulence. The wall displacement measurements indicate that there is no observable motion perpendicular to the surface, but displacement fluctuations parallel to the surface are observed after transition, coinciding with the onset of velocity fluctuations in the fluid. The fluid velocity fluctuations are symmetric about the center line of the channel, and they show relatively little downstream variation after a flow development length of about 5 cm. As the Reynolds number is further increased, there is a second ‘wall flutter’ transition, which involves visible downstream traveling waves in the top (unrestrained) wall alone. Wall displacement fluctuations of low frequency (less than about 500 rad/s) are observed both parallel and perpendicular to the wall. The mean velocity profiles and turbulence intensities are asymmetric, with much larger turbulence intensities near the top wall. There is no evident logarithmic profile close to either the top or bottom wall. Fluctuations are initiated at the entrance of the test section, and the fluctuation intensities decrease with downstream distance, the fluctuation intensities first rapidly increase and then decrease as the Reynolds number is increased. For a channel with relatively small height (0.6 mm), the transition Reynolds number for the soft-wall instability is lower the hard-wall transition Reynolds number of about 1000, and the laminar flow becomes unstable to the soft-wall instability leading to soft-wall turbulence and then to wall flutter as the Reynolds number is increased. For a channel with relatively large height (1.8 mm), the transition Reynolds number for the soft-wall instability is higher than 1000, the flow first undergoes the hard-wall laminar-turbulent transition at a Reynolds number of about 1000, the turbulent flow undergoes the soft-wall transition leading to soft-wall turbulence, and then to wall flutter.

Dans une première partie, nous étudions des schémas numériques utilisant la méthode des éléments finis pour discrétiser le système d'équations Oldroyd-B modélisant un fluide viscolélastique avec conditions de collement dans un domaine borné, en dimension deux ou trois. Le but est d'obtenir des schémas stables au sens où ils dissipent une énergie libre, imitant ainsi des propriétés thermodynamiques de dissipation similaires à celles identifiées pour des solutions régulières du modèle continu. Cette étude s'ajoute a de nombreux travaux antérieurs sur les instabilités observées dans les simulations numériques d'équations viscoélastiques (dont celles connues comme étant des Problèmes à Grand Nombre de Weissenberg). A notre connaissance, c'est la première étude qui considère rigoureusement la stabilité numérique au sens de la dissipation d'une énergie pour des discrétisations de type Galerkin. Dans une seconde partie, nous adaptons et utilisons les idées d'une méthode numérique initialement développée dans des travaux de Y. Maday, A. T. Patera et al., la méthode des bases réduites, pour simuler efficacement divers modèles multi-échelles. Le principe est d'approcher numériquement chaque élément d'une collection paramétrée d'objets complexes dans un espace de Hilbert par la plus proche combinaison linéaire dans le meilleur sous-espace vectoriel engendré par quelques éléments bien choisis au sein de la même collection paramétrée. Nous appliquons ce principe pour des problèmes numériques liés : à l'homogénéisation numérique d'équations elliptiques scalaires du second-ordre, avec coefficients de diffusion oscillant à deux échelles, puis ; à la propagation d'incertitudes (calculs de moyenne et de variance) dans un problème elliptique avec coefficients stochastiques (un champ aléatoire borné dans une condition de bord du troisième type), enfin ; au calcul Monte-Carlo de l'espérance de nombreuses variables aléatoires paramétrées, en particulier des fonctionnelles de processus stochastiques d'Itô paramétrés proches de ce qu'on rencontre dans les modèles micro-macro de fluides polymériques, avec une variable de contrôle pour en réduire la variance. Dans chaque application, le but de l'approche bases-réduites est d'accélérer les calculs sans perte de précision / In a first part, we study numerical schemes using the finite-element method to discretize the Oldroyd-B system of equations, modelling a viscoelastic fluid under no flow boundary condition in a 2- or 3- dimensional bounded domain. The goal is to get schemes which are stable in the sense that they dissipate a free-energy, mimicking that way thermodynamical properties of dissipation similar to those actually identified for smooth solutions of the continuous model. This study adds to numerous previous ones about the instabilities observed in the numerical simulations of viscoelastic fluids (in particular those known as High Weissenberg Number Problems). To our knowledge, this is the first study that rigorously considers the numerical stability in the sense of an energy dissipation for Galerkin discretizations. In a second part, we adapt and use ideas of a numerical method initially developped in the works of Y. Maday, A.T. Patera et al., the reduced-basis method, in order to efficiently simulate some multiscale models. The principle is to numerically approximate each element of a parametrized family of complicate objects in a Hilbert space through the closest linear combination within the best linear subspace spanned by a few elementswell chosen inside the same parametrized family. We apply this principle to numerical problems linked : to the numerical homogenization of second-order elliptic equations, with two-scale oscillating diffusion coefficients, then ; to the propagation of uncertainty (computations of the mean and the variance) in an elliptic problem with stochastic coefficients (a bounded stochastic field in a boundary condition of third type), last ; to the Monte-Carlo computation of the expectations of numerous parametrized random variables, in particular functionals of parametrized Itô stochastic processes close to what is encountered in micro-macro models of polymeric fluids, with a control variate to reduce its variance. In each application, the goal of the reduced-basis approach is to speed up the computations without any loss of precision