ContentsIV FLUID MECHANICS ii12 Foundations of Fluid Dynamics 112.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 The Macroscopic Nature of a Fluid: Density, Pressure, Flow velocity; Fluidsvs. Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.3.1 Archimedes’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.3.2 Stars and Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.3.3 Hydrostatics of Rotating Fluids . . . . . . . . . . . . . . . . . . . . . 1112.4 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512.5 Conservation Laws for an Ideal Fluid . . . . . . . . . . . . . . . . . . . . . . 1912.5.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1912.5.2 Momentum Conservat io n . . . . . . . . . . . . . . . . . . . . . . . . . 2 012.5.3 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2012.5.4 Bernoulli’s Theorem; Expansion, Vorticity and Shear . . . . . . . . . 2112.5.5 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.6 Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.7 Viscous Flows - Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.7.1 Decomposition of the Velocity Gradient . . . . . . . . . . . . . . . . . 3212.7.2 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.7.3 Energy conservation and entropy production . . . . . . . . . . . . . . 3412.7.4 Molecular Origin of Viscosity . . . . . . . . . . . . . . . . . . . . . . 3512.7.5 Reynolds’ Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.7.6 Blood Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36iPart IVFLUID MECHANICSiiChapter 12Foundations of Fluid DynamicsVersion 0812.1.K, 21 January 2009Please send comments, suggestions, and e rrata via email to [email protected] or on paperto Kip Thorne, 130-33 Caltech, Pasadena CA 91125Box 12.1Reader’s Guide• This chapter relies heavily on the geometric view of Newtonian physics (includingvector and tensor analysis) laid o ut 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 ro t ation) introduced in Chap. 11.• Chapters 13–18 (fluid mechanics and magnetohydrodynamics) 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 R elativity, will entail relativistic fluids, for whichconcepts in this chapter will be import ant.12.1 Ove rviewHaving studied elasticity theory, we now turn to a second branch of continuum mechanics:fluid dynamics. Three of the four states of matter (gases, liquids and plasmas) can beregarded 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 labor atory investigations. Fluid dynamics findsexperimental application in engineering, physics, biophysics, chemistry and many other fields.12The observational sciences of oceanography, meteorology, astrophysics and geophysics, inwhich experiments are less f r equently 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 describ ethe motion o f fluid is that the length and time scales associated with the flow are long com-pared with the corresponding microscopic scales, so the continuum approximation 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. However, 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 flows 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, and numerical simulations where frequentlymore time is devoted to computer graphics than to solving t he underlying partial differentia lequations. 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 …
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