Chung Fang An Introduction to Fluid Mechanics Springer Textbooks in Earth Sciences, Geography and Environment TheSpringerTextbooksseriespublishesabroadportfoliooftextbooksonEarthSciences, Geography and Environmental Science. Springer textbooks provide comprehensive introductionsaswellasin-depthknowledgeforadvancedstudies.Aclear,reader-friendly layout and features such as end-of-chapter summaries, work examples, exercises, and glossaries help the reader to access the subject. Springer textbooks are essential for students,researchersandappliedscientists. More information about this series at http://www.springer.com/series/15201 Chung Fang An Introduction to Fluid Mechanics 123 ChungFang Department ofCivil Engineering National Cheng KungUniversity Tainan,Taiwan ISSN 2510-1307 ISSN 2510-1315 (electronic) SpringerTextbooks inEarth Sciences, GeographyandEnvironment ISBN978-3-319-91820-4 ISBN978-3-319-91821-1 (eBook) https://doi.org/10.1007/978-3-319-91821-1 LibraryofCongressControlNumber:2018948619 ©SpringerInternationalPublishingAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Coverillustration:PicturecreditforOlgaNikonova(Shutterstock) ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To Yen-I, Kolli and Meimei Preface In the past decade, I have been teaching fluid mechanics from the fundamental to advanced levels at Department of Civil Engineering at National Cheng Kung University in Taiwan, and at School of Aeronautics and Astronautics at Zhejiang UniversityinChina,andhaveanimpressionthatatextbookencompassingthetopics from the fundamental disciplines to more advanced treatments offluid mechanics withabalanceddiscussionbetweenthemathematicsandunderlyingphysicsoffluid motion isnot available. Thisbecame themotivation ofpresent work. The book comprises 12 chapters. Chapter 1 deals with the mathematical pre- requisites including tensor analysis, integral theorems, and theory of complex variables. Aclear understanding of themathematical knowledge provides notonly a better access to understand the underlying physics of fluid motion, but also is essentialtootherbranchesofscienceandtechnology.Thefundamentalconceptsof fluidmotionareintroducedinChap.2.Specifically,thedistinctionbetweensolids, liquidsandgases,methodofanalysis,continuumhypothesis,andNewton’slawof viscosityarethemaintopics.Thedisciplinesdevotingtothefluidbehaviorinstatic circumstance are discussed in Chap. 3, with the focus on the hydrostatic pressure distribution, hydrostatic forces on submerged surfaces, phenomena of surface ten- sion and buoyancy, and liquids in rigid-body motion. Flow kinematics without referring to the dynamics foundation such as flow lines and the concepts of cir- culation, vorticity, stream tube and stream filament, and vortex tube and vortex filament are introduced in Chap. 4, which are used intensively for flow visualiza- tion. The fundamental physical laws of classical physics, specifically the balances ofmass,linearmomentum,angularmomentum,energyandentropy,areformulated in the base of a general balance statement of an extensive variable in either the integral or differential form in Chap. 5, for which the basic concepts of continuum mechanics and a simplified introduction to the theory of material equations are given. These balance equations are important, because they are valid not only for fluids, but also for other deformable bodies within the continuum hypothesis, provided that thematerial equations can appropriately be formulated.The study of fluid motion uses model test intensively, and the complete similarity between a modelandaprototypeneedsaprioritobeestablished.Forthispurpose,thetheory ofdimensionalanalysisandmodelsimilitudeisdiscussedinChap.6.Theflowsof vii viii Preface ideal, incompressible viscous, and compressible inviscid fluids are discussed sep- arately in the forthcoming three chapters. For ideal-fluid flows, the discussions on the Euler and Bernoulli equations, Kelvin’s theorem, two- and three-dimensional potential flows, and surface liquid waves are given in Chap. 7. For incompressible viscous flows, the vorticity equation, exact and low-Reynolds-number solutions to theNavier-Stokesequation,boundary-layerandbuoyancy-drivenflows,andabrief discussiononturbulentflowswithapplicationstopipe-flowproblemsarepresented in Chap. 8. For compressible inviscid flows, the Crocco equation, propagations of sound and shock waves, and some selected topics in one- and multi-dimensional circumstancesarediscussedinChap.9.Chapter10dealswithopen-channelflows, which is provided particularly for students in civil and hydraulic engineering. The essentialknowledgeofclassicalthermodynamicsissummarizedinChap.11,which provides an energy perspective in parallel to the mechanics perspective in under- standing the physics offluid motion. The last chapter concerns with some features of granular flows, which is used to illustrate the applications of the mature disci- plines offluid mechanics and thermodynamics to complex problems. The chapter arrangement follows the sequence of statics, kinematics, and dynamics of deformable materials, which is the common lecture sequence used in manyuniversity-leveleducationfacilities. Thiswasdoneinordertoletstudentsto understand the disciplines offluid mechanics in a coherent manner. Although not explicitlyaccomplished,thebookcanbedividedintothreeparts.PartIcontainsthe first six chapters for the fundamental disciplines of fluid mechanics. Part II com- prisesthenextfourchaptersdevotingtoanadvancedtreatmentoffluidmechanics. Part III consists of the last two chapters, which may be used to show the appli- cations of fluid mechanics in various problems of interest. At the end of each chapter, some problems are given for exercises or testing materials of the intro- duced disciplines. The detailed solutions to selected problems are provided in Appendix B, while the orthogonal curvilinear coordinates introduced in the first chapter are represented in a more concise manner in Appendix A for reference. Associatedwitheachchapter,alistforfurtherreadingisprovidedforthosereaders whowanttoknowmoreabouttherelatedtopics.Thebookcanbeusedforone-or two-semester lectures to deliver a broad and deep discussion on fluid mechanics with balanced mathematical treatments and physical understanding. I would like to express my sincere gratitude to Prof. Kolumban Hutter for the constant encouragement of writing the book. Miss Annett Buettner, Miss Helen Rachner, Miss Raghavy Krishnan and Mr. Karthik Raj Selvaraj from Springer Verlagaregreatlyacknowledgedfortheirgreatcareofmanagingalladministrative andpublishingissuesofthebook.JohnWiley&SonsInc.,isgreatlyacknowledged forthekindpermissiontousethefiguresquotedfromitspublishedbooks.Alarge part of this book was carried out during a sabbatical semester at School of Aero- nautics and Astronautics, Zhejiang University, China, and I should like to thank Prof.WeiqiuChenandProf.ZhaoshengYufortheirhospitalitythroughoutthisstay. There will be errorsremaining in thebook, and for these Ialone amresponsible. Preface ix BeforeIfinishthisPreface,Iwouldliketosaythatwritingabookcanneverbe finished, and a finished book has to be abandoned. This is I am now going to do, wellknowingthatabookbearsintrinsicallyitsweaknesses,thatIwouldknownow how to do it better. While writing this book through all its stages needed isolation and separation from the beloved family members, who all deserve my deepest gratitude. Tainan, Taiwan Chung Fang March 2018 Contents 1 Mathematical Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Summation Convention, Dummy and Free Indices . . . 1 1.1.2 The Kronecker Delta and Permutation Symbol . . . . . . 2 1.2 Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Definition and Components of a Tensor . . . . . . . . . . . 4 1.2.2 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Orthogonal Tensor and Transformation Laws . . . . . . . 8 1.2.4 Eigenvalues and Eigenvectors of a Tensor . . . . . . . . . 10 1.2.5 Tensor Invariants and the Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.6 Isotropic Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.1 Time Rate of Change of a Tensor . . . . . . . . . . . . . . . 13 1.3.2 Gradient, Divergence, and Curl . . . . . . . . . . . . . . . . . 14 1.3.3 Nabla and the Laplacian Operators . . . . . . . . . . . . . . . 15 1.4 Orthogonal Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . 16 1.4.1 Rectangular Coordinates . . . . . . . . . . . . . . . . . . . . . . 17 1.4.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6.1 Complex Numbers, Complex and Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6.2 The Cauchy-Riemann Equations and Multi-valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6.3 The Cauchy-Goursat Theorem and Cauchy Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6.4 The Taylor, Maclaurin, and Laurent Series . . . . . . . . . 24 1.6.5 Residues and Residue Theorem . . . . . . . . . . . . . . . . . 25 1.6.6 Conformal Transformation . . . . . . . . . . . . . . . . . . . . . 26 1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 xi