ContentsVFLUIDMECHANICS ii13 Foundations of Fluid Dynamics 113.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 The Macroscopic Nature of a Fluid: Density, Pressure, Flow velocity; Fluidsvs. Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.3.1 Archimedes’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.3.2 Stars and Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.3.3 Hydrostatics of Rotating Fluids . . . . . . . . . . . . . . . . . . . . . 1113.4 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.5 Conservation Laws for an Ideal Fluid . . . . . . . . . . . . . . . . . . . . . . 1913.5.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1913.5.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 2013.5.3 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2013.5.4 Bernoulli’s Theorem; Expansion, Vorticity and Shear ......... 2113.5.5 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 2313.6 Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.7 Viscous Flows - Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213.7.1 Decomposition of the Velocity Gradient . . . . . . . . . . . . . . . . . 3213.7.2 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3213.7.3 Energy conservation and entropy production . . . . . . . . . . . . . . 3413.7.4 Molecular Origin of Viscosity . . . . . . . . . . . . . . . . . . . . . . 3513.7.5 Reynolds’ Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.7.6 Blo od F low . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36iPart VFLUID MECHANICSiiChapter 13Foundations of Fluid DynamicsVersion 1013.1.K, 21 January 2009Please send comments, suggestions, and errata via email to [email protected] or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 13.1Reader’s Guide• This chapter relies heavily on the geometric view of Newtonian physics (includingvector and tensor analysis) laid out in the sections of Chap. 1 labeled “[N]”.• This chapter also relies on the concepts of strain and its irreducible tensorial parts(the expansion, shear and rotation) introduced in Chap. 11.• Chapters 13–18 (fluid mechanics and magnetoh ydrodynamics) are extensions ofthis chapter; to understand them, this chapter must be mastered.• Portions of Part V, Plasma Physics (especially Chap. 20 on the “two-fluid formal-ism”), rely heavily on this chapter.• Small portions of Part VI, General Relativity, will entail relativistic fluids, for whichconcepts in this chapter will be important.13.1 OverviewHaving studied elasticity theory, we now turn to a second branch of contin uum mechanics:fluid dynamics.Threeofthefourstatesofmatter(gases,liquidsandplasmas)canberegarded as fluids and so it is not surprising that interesting fluid phenomena surroundus in our everyday lives. Fluid dynamics is an experimental discipline and much of whathas been learned has come in response to laboratory investigations. Fluid dynamics findsexperimental application in engineering, physics, biophysics, chemistry and many other fields.12The observational sciences of oceanograph y, meteorology, astrophysics and geophysics, inwhich experiments are less frequently performed, are also heavily reliant upon fluid dynamics.Many of these fields have enhanced our appreciation of fluid dynamics by presenting flowsunder conditions that are inaccessible to laboratory study.Despite this rich diversity, the fundamental principles are common to all of these appli-cations. The fundamental assumption which underlies the governing equations that describethe motion of fluid is that the length and time scales associated with the flow are long com-pared with the corresponding microscopic scales, so the continuum appro ximation can beinvoked. In this chapter, we will derive and discuss these fundamental equations. They are,in some respects, simpler than the corresponding laws of elastodynamics. How ever, as withparticle dynamics, simplicity in the equations does not imply that the solutions are simple,and indeed they are not! One reason is that there is no restriction that fluid displacementsbe small (by constrast with elastodynamics where the elastic limit keeps them small), somost fluid phenomena are immediately nonlinear.Relatively few problems in fluid dynamics admit complete, closed-form, analytic solu-tions, so progress in describing fluid flo ws has usually come from the introduction of cleverphysical “models” and the use of judicious mathematical approximations. In more recentyears numerical fluid dynamics has come of age and in many areas of fluid mechanics, finitedifference simulations have begun to complement laboratory experiments and measurements.Fluid dynamics is a subject where considerable insight accrues from being able to vi-sualize the flow. This is true of fluid experiments where much technical skill is devoted tomarking the fluid so it can be photographed, andnumericalsimulationswherefrequentlymore time is devoted to computer graphics than to solving the underlying partial differentialequations. We shall pay some attention to flow visualization. The reader should be warnedthat obtaining an analytic solution to the equations of fluid dynamics is not the same asunderstanding the flow; it is usually a good idea to sketch the flow pattern at the very least,as a tool for understanding.We shall begin this chapter in Sec. 13.2 with a discussion of the physical nature of afluid: the possibility to describe it …
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