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0212.2 - Chapter 12 Foundations of Fluid Dynamics Version...

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Chapter 12 Foundations of Fluid Dynamics Version 0212.2 28 Jan 03 Please send comments, suggestions, and errata via email to [email protected] and [email protected], or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125 12.1 Overview Having 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 be regarded as fluids and so it is not surprising that interesting fluid phenomena surround us in our everyday lives. Fluid dynamics is an experimental discipline and much of what has been learned has come in response to laboratory investigations. Fluid dynamics finds experimental application in engineering, physics, biophysics, chemistry and many other fields. The observational sciences of oceanography, meteorology, astrophysics and geophysics, in which 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 flows under 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 describe the 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 approximation can be invoked. 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 with particle 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 displacements be small (by constrast with elastodynamics where the elastic limit keeps them small), so most fluid phenomena are immediately non-linear. 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 clever physical “models” and the use of judicious mathematical approximations. In more recent years numerical fluid dynamics has come of age and in many areas of fluid mechanics, finite difference simulations have begun to complement laboratory experiments and measurements. 1
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2 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 to marking the fluid so it can be photographed, and numerical simulations where frequently more time is devoted to computer graphics than to solving the underlying partial differential equations. We shall pay some attention to flow visualization. The reader should be warned that obtaining an analytic solution to the equations of fluid dynamics is not the same as understanding the flow; it is always a good idea to be able to sketch the flow pattern at the very least.
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