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*, 12

Viscous FIow
Th" of fluid viscosity was developed and viscosity defined in Chapter 7. "on""pt Clearly, all fluids are viscous, but in certain situations and under certain conditions, a fluid may be considered ideal or inviscid, making possible an analysis by the methods of Chapter 10. Our task in this chapter is to consider viscous fluids and the role of viscosity as it affectsthe flow. Of particular interest is the case of flow past solid surfaces and the interrelations between the surfaces and the flowing fluid.


The existence of two distinct types of viscous flow is a universally accepted phenomenon. The smoke emanating from a lighted cigarette is seen to flow smoothly and uniformly for a short distance from its source and thenchange abruptly into a very irregular, unstable pattern. Similar behavior may be observed for water flowing slowly from a faucet. Thewell-orderedtype of flow occurs when adjacentfluid layers slide smoothly overone another with mixing between layers or lamina occurring only on a molecular level. It was for this type of flow that Newton's viscosity relation was derived, and in order for us to measure theviscosit¡ p, this laminar flow must exist. The second flow regime, in which small packets of fluid particles are transferred between layers, giving it a fluctuating nature, is called the turbulent flow regime.

The existence of laminar and turbulent flow, although recognized earlier, was first
described quantitatively by Reynolds in 1883. His classic experiment is illustrated in Figure 12. 1 .Water was allowed to flow through a transparent pipe, as shown, at a rate controlled by a valve. A dye having the same specific gravity as water was introduced at the pipe opening and its pattern observed for progressively larger flow rates ofwater. At low rates offlow, the dye pattern was regular and formed a single line of color as shown in Figure 12.1(a). At high flow rates, however, the dyebecame dispersed throughout the pipe cross section because of

(a) Re < 2300

(b) Re > 2300

Figure 12.1 Reynolds's experiment.



Chapter 12

Viscous Flow

the very irregular fluid motion. The difference in the appearance of the dye streak was, of course, due to the orderly nature of laminar flow in the first case and to the fluctuating character of turbulent flow in thelatter case.

The transition from laminar to turbulent flow in pipes is thus a function of the fluid velocity. Actually, Reynolds found that fluid velocity was the only one variable determining the nature of pipe flow, the others being pipe diameter, fluid densit¡ and fluid viscosity. These four variables, combined into the single dimensionless puru-Lt",

R": DP' p


flow ihls is2300.

form the Reynolds number, symbolized Re, in honor of osborne Reynolds and his important contributions to fluid mechanics. For flow in circular pipes, it is found that below a value for Reynolds number of2300 the flow is laminar. Above this value the flow may be laminar as well, and indeed, laminar flow has been observed for Reynolds numbers as high as 40,000 in experiments wherein externaldisturbances were minimized. Above a Reynolds number of 2300, small disturbances will cause a transition to turbulent flow whereas below this value disturbances are damped out and laminar flow prevails . The critical Reynolds number pipe


Reynolds's experiment clearly demonstrated the two different regimes of flow: laminar and turbulent. Another manner of illustrating thesedifferent flow regimes and their dependence upon Reynolds number is through the consideration of drag. A particularly illusfátive case is that of external flow (i.e., flow around a body as opposed to flow inside a conduit). The drag force due to friction is caused by the shear stresses at the surface of a solid object moving through a viscous fluid. Frictional drag is evaluated by using the...
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