Advances in Mechanics and Mathematics 40 Aleksandra A. Bozhko Sergey A. Suslov Convection in Ferro-Nanofuids: Experiments and Theory Physical Mechanisms, Flow Patterns, and Heat Transfer Advances in Mechanics and Mathematics Volume 40 Series Editors David Gao, Federation University Australia Tudor Ratiu, Shanghai Jiao Tong University Advisory Board Antony Bloch, University of Michigan John Gough, Aberystwyth University Darryl D. Holm, Imperial College London Peter Olver, University of Minnesota Juan-Pablo Ortega, University of St. Gallen Genevieve Raugel, CNRS and University Paris-Sud Jan Philip Solovej, University of Copenhagen Michael Zgurovsky, Igor Sikorsky Kyiv Polytechnic Institute Jun Zhang, University of Michigan Enrique Zuazua, Universidad Auto´noma de Madrid and DeustoTech Kenneth C. Land, Duke University More information about this series at http://www.springer.com/series/5613 Aleksandra A. Bozhko • Sergey A. Suslov Convection in Ferro-Nanofluids: Experiments and Theory Physical Mechanisms, Flow Patterns, and Heat Transfer 123 Aleksandra A. Bozhko Sergey A. Suslov Faculty of Physics Department of Mathematics Perm State University Swinburne University of Technology Perm, Russia Hawthorn, Victoria, Australia ISSN 1571-8689 ISSN 1876-9896 (electronic) Advances in Mechanics and Mathematics ISBN 978-3-319-94426-5 ISBN 978-3-319-94427-2 (eBook) https://doi.org/10.1007/978-3-319-94427-2 Library of Congress Control Number: 2018950062 Mathematics Subject Classification: 76E06, 76E25, 76E30, 76R10, 76B70, 80A20, 82D40 © Springer International Publishing AG, part of Springer Nature 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To the memory of Professor G. F. Putin, outstanding experimentalist, colleague and teacher Preface Scope This book is based on the results of experimental and theoretical studies of hydrodynamic stability and heat and mass transfer processes in ferrofluids that we have been involved with over the past several decades. The main motivation for such studies has been the growing interest in the use of mag- netically controllable media as heat carriers in various thermal management systems. Along with other non-gravitational mechanisms capable of inducing the motion of initially quiescent fluid such as vibrational and electrocon- vection thermomagnetic convection can be used to enhance heat transfer in conditions where natural convection is impossible, for example, in micro- gravitation conditions. The experimental investigation that is reported in this book was conducted at Perm State University, Russia, under the license from the Russian State Corporation for Space Research (RosCosmos) and formed the ground-based component of a larger programme involving experi- ments on board autonomous and piloted spacecrafts, the orbital station “Mir” and the International Space Station. From a fundamental point of view the book considers an intricate interaction of non-isothermal and electrically non- conducting magnetopolarisable fluid with gravitational and magnetic fields. In the absence of a magnetic field ferrofluids behave similarly to other non- magnetic nanofluids, studies of which have been growing exponentially over the past two decades due to their ever expanding applications in modern technology. vii viii Preface Audience We hope that this book will be of interest to researchers and practitioners working in the areas of fluid mechanics, hydrodynamic stability and heat and mass transfer with the view of perspective applications of ferrofluids in heat management systems, in particular, in microelectronics and space technolo- gies. The main emphasis of the book is on the influence of a uniform magnetic field on flows of non-isothermal ferrofluids and the associated heat transfer. However, we also discuss peculiar features of ferrofluid flows occurring in the absence of a magnetic field, which are shown to be drastically different from those of ordinary fluids and need to be taken into account by practitioners working with magnetic and non-magnetic nanofluids. Content Invention of ferrofluids, their industrial synthesis and numerous studies at micro and macro levels have been primarily motivated by their magnetic properties that are many orders of magnitude stronger than those of natural paramagnetic and diamagnetic fluids and gases. The composition of ferroflu- ids that defines their magnetic properties and the related mechanisms of heat and mass transfer in them are briefly reviewed in Chapter 1. The main equations describing motion of non-isothermal ferrofluids by treating them as magnetopolarisable continuous media are summarised in Chapter 2. While such a description has its limitations that become evident when the theoretically obtained results are compared with those of experi- mental observations, currently, such an approximation offers the most robust way of modelling ferrofluid flows. The reasons for this are outlined in the subsequent chapters of the book. The major governing non-dimensional pa- rameters are also defined and their physical meaning is discussed in Chapter 2. Results of a theoretical analysis of thermomagnetic convection in geomet- rically simple yet practically relevant domains are presented in Chapter 3. Such an analysis sheds light on physical processes taking place in the bulk of ferrofluid offering the insight that is successfully used to guide experi- mental observations and measurements. In particular, the existence of ther- momagnetic waves associated with the thermally induced non-uniformity of fluid magnetisation and of oscillatory regimes of convections caused by the nonlinear variation of magnetisation across a ferrofluid layer was discovered theoretically first and then was confirmed in specialised experiments. A com- prehensive analysis of magnetoconvection arising in the arbitrarily oriented magnetic field in gravity-free conditions is another example of a practically important situation considered in Chapter 3 that is out of reach for ground- based laboratory experiments. Preface ix Chapter 4 contains a detailed description of experimental setups specifi- cally designed for a comprehensive study of buoyancy and thermomagneti- cally driven ferrofluid flows. The details distinguishing experimental cham- bers and flow visualisation and heat flux measurement techniques used for working with magnetically active media from those used in experiments with non-magnetic fluids are emphasised. In particular, it is shown that the shape and size of the working chamber have a defining influence on the type of convection patterns arising in a magnetic field. The features of thermogravitational and thermomagnetic convection aris- ing in finite flat layers and spherical cavities filled with ferrofluids and placed in uniform gravitational and magnetic fields are detailed in Chapters 5 and 6, respectively. Notably, a strong influence of gravitational sedimentation of solid particles and their aggregates contained in ferrofluids is demonstrated experimentally. It changes qualitatively the character of convection compared to that observed in ordinary single-phase fluids. Specifically, it is shown in Chapter 5 that in the vicinity of convection threshold in ferrofluids flows be- come oscillatory and chaotic both in space and time. A hysteresis is observed when the onset of convection in the initially density-stratified ferrocolloid is delayed compared to that recorded for the same but pre-mixed fluid. Convec- tion is found to arise and decay spontaneously and irregularly and this found to be related to the concentration of solid phase in experimental fluids. The influence of magnetic fields of various orientations on ferrofluid con- vection and heat transfer is discussed next in Chapter 6. It is shown that such an influence is not monotonic. Depending on the values of the governing gravitational and magnetic parameters, the application of magnetic field can either enhance or suppress convection drastically changing the observed flows and offering a not-intrusive means of controlling them. The experimental ev- idence of the fact that conditions of a particular laboratory run, storage and past usage of a ferrofluid strongly affect its flows and performance as a heat carrier. These factors should be taken into account when interpreting physi- cal observations of a non-isothermal ferrofluid behaviour and when using it in practical applications. Overall, the book is intended to provide a guidance to a very rich and frequently ambiguous behaviour of non-uniformly heated fer- rocolloids caused by their complex composition and influenced by an external magnetic field. Acknowledgements This book would not be possible without the hard work of our colleagues and technical staff who were invaluable in building experimental equipment and maintaining it in working order over many months during which individual experimental runs were performed and over decades during which this re- search was conducted. We are especially grateful to our students T. Pilugina, D. Shupeinik, P. Bulychev, A. Sidorov, M. Krauzina, P. Krauzin, H. Rah- x Preface man, P. Dey and K. Pham for their effort and time preparing and running experiments and performing computations reported in this book. We extend our gratitude to Professor A. F. Pshenichnikov and Dr. A. S. Ivanov of the Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences for illuminating discussions of microstructure of ferro- colloids, Mr. A. N. Poludnitsyn for help with experimental photography and Dr. T. Tynja¨l¨a of Lappeenranta University of Technology, Finland, for fruitful collaboration on numerical modelling of ferrofluid flows. AAB is also grateful to the late Professors I. M. Kirko, Yu. K. Bratukhin and G. Z. Gershuni for their mentoring and help during the early years of this research. Perm, Russia Aleksandra A. Bozhko Melbourne, Australia Sergey A. Suslov April 2018 Contents 1 Ferrofluids: Composition and Physical Processes . . . . . . . . . 1 1.1 Brief History and Composition of Ferrofluids . . . . . . . . . . . . . . . 1 1.2 Physical Processes Taking Place in Ferrofluids . . . . . . . . . . . . . . 4 1.3 Physical Properties of Ferrofluids . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Simplifying Physical Assumptions and Basic Equations . . . . . . 11 2.2 Nondimensionalisation and Governing Parameters . . . . . . . . . . 15 3 Infinite Vertical Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Problem Definition and Basic Flow Solutions . . . . . . . . . . . . . . . 24 3.3 Flow Patterns in a Normal Magnetic Field . . . . . . . . . . . . . . . . . 30 3.3.1 Linearised Equations for Infinitesimal Perturbations . . 30 3.3.2 Stability Results for an Equivalent Two-Dimensional Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.3 Perturbation Energy Balance . . . . . . . . . . . . . . . . . . . . . . 42 3.3.4 Three-Dimensional Results . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.5 Symmetry-Breaking Effects of Nonuniform Fluid Magnetisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.6 Variation of Stability Characteristics and Summary of Results for Convection in Normal Field . . . . . . . . . . . 59 3.4 Flow Patterns in an Oblique Magnetic Field . . . . . . . . . . . . . . . 63 3.4.1 Linearised Perturbation Equations in Zero Gravity . . . 64 3.4.2 Flow Stability Characteristics in Zero Gravity . . . . . . . . 67 3.4.3 Perturbation Energy Balance in Zero Gravity . . . . . . . . 73 3.4.4 Perturbation Fields in Zero Gravity . . . . . . . . . . . . . . . . 74 3.4.5 Linearised Perturbation Equations in Non-zero Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 xi