7708d_c06_298-383 - Flow past an inclined plate: The...

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Flow past an inclined plate: The streamlines of a viscous fluid flowing slowly past a two-dimensional object placed between two closely spaced plates 1 a Hele-Shaw cell 2 approximate inviscid, irrotational 1 potential 2 flow. 1 Dye in water between glass plates spaced 1 mm apart. 2 1 Photography courtesy of D. H. Peregrine. 2
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In the previous chapter attention is focused on the use of finite control volumes for the so- lution of a variety of fluid mechanics problems. This approach is very practical and useful, since it does not generally require a detailed knowledge of the pressure and velocity varia- tions within the control volume. Typically, we found that only conditions on the surface of the control volume entered the problem, and thus problems could be solved without a de- tailed knowledge of the flow field. Unfortunately, there are many situations that arise in which the details of the flow are important and the finite control volume approach will not yield the desired information. For example, we may need to know how the velocity varies over the cross section of a pipe, or how the pressure and shear stress vary along the surface of an air- plane wing. In these circumstances we need to develop relationships that apply at a point, or at least in a very small region 1 infinitesimal volume 2 within a given flow field. This approach, which involves an infinitesimal control volume , as distinguished from a finite control vol- ume, is commonly referred to as differential analysis , since 1 as we will soon discover 2 the governing equations are differential equations. In this chapter we will provide an introduction to the differential equations that de- scribe 1 in detail 2 the motion of fluids. Unfortunately, we will also find that these equations are rather complicated, partial differential equations that cannot be solved exactly except in a few cases, at least without making some simplifying assumptions. Thus, although differ- ential analysis has the potential for supplying very detailed information about flow fields, this information is not easily extracted. Nevertheless, this approach provides a fundamental basis for the study of fluid mechanics. We do not want to be too discouraging at this point, since there are some exact solutions for laminar flow that can be obtained, and these have proved to be very useful. A few of these are included in this chapter. In addition, by making some simplifying assumptions many other analytical solutions can be obtained. For exam- ple, in some circumstances it may be reasonable to assume that the effect of viscosity is small and can be neglected. This rather drastic assumption greatly simplifies the analysis and pro- vides the opportunity to obtain detailed solutions to a variety of complex flow problems.
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This note was uploaded on 09/06/2009 for the course CEE cv2601 taught by Professor Kellas,j during the Spring '09 term at Nanyang Technological University.

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7708d_c06_298-383 - Flow past an inclined plate: The...

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