Lecture Notes in Mathematics 2212 Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Zurich Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg More information about this series at http://www.springer.com/series/304
Angiolo Farina Lorenzo Fusi Andro Mikelić Giuseppe Saccomandi Adélia Sequeira Eleuterio F. Toro Non-Newtonian Fluid Mechanics and Complex Flows Levico Terme, Italy 2016 Angiolo Farina Andro Mikelić Fabio Rosso Editors In collaboration with 123
Authors Angiolo Farina Dipartimento di Matematica e Informatica Ulisse Dini Università degli Studi di Firenze Andro Mikelić Département de Mathématiques Institut Camille Jordan Université Claude Bernard Lyon 1 Villeurbanne, France Adélia Sequeira Department de Matemática Instituto Superior Técnico Lisboa, Portugal Editors Angiolo Farina Dipartimento di Matematica e Informatica Ulisse Dini Università degli Studi di Firenze Lorenzo Fusi Dipartimento di Matematica e Informatica Ulisse Dini Università degli Studi di Firenze Giuseppe Saccomandi Dipartimento di Ingegneria Università degli Studi di Perugia Perugia, Italy Eleuterio F. Toro Dipartimento di Matematica University of Trento Trento, Italy Andro Mikelić Département de Mathématiques Institut Camille Jordan Université Claude Bernard Lyon 1 Villeurbanne, France Fabio Rosso Dipartimento di Matematica e Informatica Ulisse Dini Università degli Studi di Firenze ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics C.I.M.E. Foundation Subseries ISBN 978-3-319-74795-8 ISBN 978-3-319-74796-5 (ebook) https://doi.org/10.1007/978-3-319-74796-5 Library of Congress Control Number: 2018938092 Mathematics Subject Classification (2010): 92-XX, 35-XX Springer International Publishing AG, part of Springer Nature 2018, corrected publication 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface CIME activity is carried out with the collaboration and financial support of INdAM (Istituto Nazionale di Alta Matematica), partially supported by the University of Firenze under the Fondi di Ateneo research grants. The CIME-CIRM Course on New Trends in Non-Newtonian Fluid Mechanics and Complex Flows was held in Levico Terme (Italy) from August 29 to September 2, 2016. The course was codirected by Andro Mikelić (Université Claude Bernard Lyon 1, France), Fabio Rosso, and Angiolo Farina (Università di ). The course was attended by 32 participants, about half of them not coming from Italy. All the lecturers tried to emphasize physical and mathematical methods that cross disciplinary boundaries, putting in evidence those aspects of flow problems relevant to biology and industrial processes. The following topics were treated: 1. Hemorheology: Non-Newtonian Constitutive Models for Blood Flow Simulations (Adélia Sequeira, Instituto Superior Técnico, Lisboa, Portugal) 2. Old problems revisited from new perspectives in implicit theories of fluids (Giuseppe Saccomandi and Luigi Vergori, Università di Perugia, Italy). 3. Lectures on Hyperbolic Equations and their Numerical Approximation (Eleuterio F. Toro, Università di Trento, Italy). 4. An Introduction to the Homogenization Modeling of Non-Newtonian and Electrokinetic Flows in Porous Media (Andro Mikelić, Université Lyon 1, France). 5. Viscoplastic Fluids: Mathematical Modeling and Applications (Angiolo Farina and Lorenzo Fusi, Università di ). The focus was largely on non-newtonian fluids and their applications on blood flow, with some digression to the numerical approximation of hyperbolic equations. Actually, the goal of the course was to present a series of challenging mathematical problems arising in non-newtonian fluid dynamics. Essentially the lectures focused on (i) mathematical and physical modeling of a variety of problems that address current work and (ii) the numerical and analytical treatment of some of these problems. v
vi Preface Although the course presented a remarkable review of quite advanced problems in non-newtonian flows and in the mathematics involved, one easily realizes that such a research area is extremely rich and calls for an impressively large variety of mathematical methods. This volume collects lecture notes of the courses and therefore focuses only on a limited number of problems which, however, are typical in the field of non-newtonian flows. Such volume can certainly be useful not only to applied mathematicians but also to physicists, biologists, and engineers, who can find in it an overview of the most advanced models and methods. The first chapter is devoted to the lectures given by Adélia Sequeira. She presented a fascinating and quite difficult lecture about the rheology of the blood and some recent advances in this field (which is very active because of the numerous applications to the circulatory system diseases). The audience was very excited by her colorful slides and videos showing blood flow in very complex geometries such as those seen in aneurysms. She posed stimulating questions about the impact of the most significant non-newtonian characteristics of blood on its flow behavior. We had exciting afternoon sessions discussing such problems. Giuseppe Saccomandi gave a series of lectures on implicit constitutive models for the Cauchy stress tensor which provided an excellent complementary view of the subjects treated by Farina and Fusi. He investigated some classical problems in fluid mechanics (such as the onset of Rayleigh Bénard convection, laminar flows, and flows over an inclined plane) by employing implicit constitutive relations for the stress tensor. The chapter by Eleuterio F. Toro illustrates some recent techniques for the numerical solution of mathematical equations arising in all areas of physics especially in the mechanics of non-newtonian fluids. Actually the issues of accurate solutions and efficiency are becoming increasingly important given the growing trend to use mathematical models (PDEs) and the physics they embody. Only very accurate solutions of the PDEs will achieve this and also reveal limitations of the mathematical models (the governing equations and their parameters). Then we have a chapter by A. Mikelić on the flow of complex fluids (quasi- Newtonian and electrokinetic flows) through porous media (a problem common to many engineering applications). The modeling of filtration was developed exploiting a homogenization technique. The last chapter, by A. Farina and L. Fusi, deals with the flow of Bingham fluids in geometry characterized by a small aspect ratio. The so-called lubrication paradox is analyzed and the way of circumventing this difficulty is explained. We must abstain from commenting the scientific level of this volume, since we are among the contributors, but at least we wish to express our deep gratitude to the authors for their valuable work. Finally, we wish to thank CIRM and CIME
Preface vii for having made this course possible. Particular thanks to the Director of CIME, Prof. Elvira Mascolo, to the Director of CIRM Prof. Marco Andreatta, and to the Secretary of CIRM, Mr. Augusto Micheletti who took care of so many details. Villeurbanne, France Angiolo Farina Andro Mikelić Fabio Rosso The original version of the book was revised. The correction to the book is available at https://doi. org/10.1007/978-3-319-74796-5_6
Contents Hemorheology: Non-Newtonian Constitutive Models for Blood Flow Simulations... 1 Adélia Sequeira Old Problems Revisited from New Perspectives in Implicit Theories of Fluids... 45 Giuseppe Saccomandi and Luigi Vergori Lectures on Hyperbolic Equations and Their Numerical Approximation... 91 Eleuterio F. Toro An Introduction to the Homogenization Modeling of Non-Newtonian and Electrokinetic Flows in Porous Media... 171 Andro Mikelić Viscoplastic Fluids: Mathematical Modeling and Applications... 229 Angiolo Farina and Lorenzo Fusi Correction to: Non-Newtonian Fluid Mechanics and Complex Flows... E1 ix