ContentsVFLUIDMECHANICS ii13 Foundations of Fluid Dynamics 113.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 The Macroscopic Nature of a Fluid: Density, Pressure, Flow Velocity; Liquidsvs. Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.3.1 Archimedes’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.3.2 Nonrotating Stars and Planets . . . . . . . . . . . . . . . . . . . . . . 913.3.3 Hydrostatics of Rotating Fluids . . . . . . . . . . . . . . . . . . . . . 1113.4 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.5 The Dynamics of an Ideal Fluid . . . . . . . . . . . . . . . . . . . . . . . . . 2013.5.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2013.5.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 2113.5.3 Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2113.5.4 Bernoulli’s Theorem ........................... 2213.5.5 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.5.6 Joule-Kelvin Method of Cooling a Real Gas . . . . . . . . . . . . . . 3013.6 Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.7 Viscous Flows with Heat Conduction . ..................... 3413.7.1 Decomposition of the Velocity Gradient . . . . . . . . . . . . . . . . . 3413.7.2 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.7.3 Energy conservation and entropy production . . . . . . . . . . . . . . 3613.7.4 Molecular Origin of Viscosity . . . . . . . . . . . . . . . . . . . . . . 3813.7.5 Reynolds’ Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.7.6 Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.8T2 Relativistic Dynamics of an Ideal Fluid . . . . . . . . . . . . . . . . . . 4113.8.1T2 Stress-Energy Tensor and Equations of Relativistic Fluid Mechanics 4113.8.2T2 Relativistic Bernoulli Equation and Ultrarelativistic Astrophysi-cal Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4213.8.3T2 Nonrelativistic Limit of the Stress-Energy Tensor . . . . . . . . 44iPart VFLUID MECHANICSiiFluid MechanicsVersion 1113.2.K, 26 January 2012. [This supersedes version 1113.1.K, which was passedout to Yanbei Chen’s class]Please send comments, suggestions, and errata via email to [email protected] or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Having studied elasticity theory, we now turn to a second branch of continuum mechanics:fluid dynamics.Threeofthefourstatesofmatter(gases,liquidsandplasmas)canberegarded as fluids, so it is not surprising that interesting fluid phenomena surround us inour everyday lives. Fluid dynamics is an experimental discipline and much of our currentunderstanding has come in response to laboratory investigations. Fluid dynamics findsexperimental application in engineering, physics, biophysics, chemistry and many other fields.The observational sciences of oceanography, meteorology, astrophysics and geophysics, inwhich experiments are less frequently performed, are also heavily reliant on 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 applica-tions. The key assumption which underlies the governing equations that describe the motionof a fluid is that the length and time scales associated with the flow are long compared withthe corresponding microscopic scales, so thecontinuumapproximationcanbeinvoked.The fundamental equations of fluid dynamics are, in some respects, simpler than thecorresponding laws of elastodynamics. How ever, as with particle dynamics, simplicity ofthe equations does not imply that the solutions are simple, and indeed they are not! Onereason is that there is no restriction that fluid displacements be small (by constrast withelastodynamics where the elastic limit keeps them small), so most fluid phenomena areimmediately 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, com-puter simulations have begun to complement laboratory experiments and measurements.Fluid dynamics is a subject where considerable insight accrues from being able to visualizethe flow . This is true of fluid experiments, where muc h technical skill is devoted to markingthe fluid so it can be photographed; it is also true of numerical simulations, where frequentlymore time is devoted to computer graphics than to solving the underlying partial differentialiiiivequations. 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 present the fundamental concepts of fluid dynamics in Chap. 13, focusing partic-ularly on the underlying physical principles and the conservation laws for mass, momentumand energy. We shall explain why, when flow velocities are very subsonic, a fluid’s densitychanges very little, i.e. it is effectively incompressible;andweshallspecializethefundamentalprinciples and …
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