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Chapter 6 SOLUTION OF VISCOUS-FLOW PROBLEMS 6.1 Introduction T HE previous chapter contained derivations of the relationships for the con- servation of mass and momentum—the equations of motion —in rectangular, cylindrical, and spherical coordinates. All the experimental evidence indicates that these are indeed the most fundamental equations of fluid mechanics, and that in principle they govern any situation involving the flow of a Newtonian fluid. Unfortunately, because of their all-embracing quality, their solution in analytical terms is difficult or impossible except for relatively simple situations. However, it is important to be aware of these “Navier-Stokes equations,” for the following reasons: 1. They lead to the analytical and exact solution of some simple, yet important problems, as will be demonstrated by examples in this chapter. 2. They form the basis for further work in other areas of chemical engineering. 3. If a few realistic simplifying assumptions are made, they can often lead to approximate solutions that are eminently acceptable for many engineering purposes. Representative examples occur in the study of boundary layers, waves, lubrication, coating of substrates with films, and inviscid (irrotational) flow. 4. With the aid of more sophisticated techniques, such as those involving power series and asymptotic expansions, and particularly computer-implemented nu- merical methods, they can lead to the solution of moderately or highly ad- vanced problems, such as those involving injection-molding of polymers and even the incredibly difficult problem of weather prediction. The following sections present exact solutions of the equations of motion for several relatively simple problems in rectangular, cylindrical, and spherical coor- dinates. Throughout, unless otherwise stated, the flow is assumed to be steady , laminar and Newtonian, with constant density and viscosity. Although these as- sumptions are necessary in order to obtain solutions, they are nevertheless realistic in many cases. All of the examples in this chapter are characterized by low Reynolds numbers. That is, the viscous forces are much more important than the inertial forces, and 272
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6.1—Introduction 273 are usually counterbalanced by pressure or gravitational effects. Typical applica- tions occur at low flow rates and in the flow of high-viscosity polymers. Situations in which viscous effects are relatively unimportant will be discussed in Chapter 7. Solution procedure. The general procedure for solving each problem in- volves the following steps: 1. Make reasonable simplifying assumptions. Almost all of the cases treated here will involve steady incompressible flowo fa Newtonian fluid in a single coordinate direction. Further, gravity may or may not be important, and a certain amount of symmetry may be apparent.
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This note was uploaded on 10/01/2011 for the course ME 509 taught by Professor Wereley during the Spring '11 term at Purdue University-West Lafayette.

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