45 Pipe Flow Analysis of pipe flow is one of the most important practical

45 pipe flow analysis of pipe flow is one of the most

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4.5 Pipe Flow Analysis of pipe flow is one of the most important practical problems in fluid engineering, and it provides yet another opportunity to obtain an exact solution to the Navier–Stokes equations, the Hagen–Poiseuille flow. We will derive this solution in the present section of these notes. We begin with a physical description of the problem being considered and by doing this introduce some important terminology and notation, among these some basic elements of boundary-layer theory. Following this we present the formal solution of the N.–S. equations that provides the Hagen–Poiseuille velocity profile for steady, fully-developed flow in a pipe of circular cross section, and then we use this to produce simple formulas useful in engineering calculations. In particular, we will see how to account for pressure losses due to skin-friction effects, thus providing a simple modification to Bernoulli’s equation that makes it applicable to viscous flow problems. Then we extend this to situations involving pipes with arbitrary cross-sectional shapes and other geometric irregularities including expansions and contractions, bends, tees, etc . 4.5.1 Some terminology and basic physics of pipe flow In this subsection we consider some basic features of pipe flow that allow us to solve the N.–S. equations for a rather special, but yet quite important, case corresponding to steady, fully-developed flow in a pipe of circular cross section. We have schematically depicted this in Fig. 4.11. What is apparent from the figure is a uniform velocity profile entering at the left end of the region of pipe under consideration and then gradually evolving to a velocity profile that is much smoother and, in fact, as we will later show is parabolic in the radial coordinate r . As indicated in the figure, the distance over which this takes place is called the entrance length , denoted L e , and this corresponds to the distance required for merging of regions starting at the pipe walls within which the originally uniform velocity adjusts from zero at the walls (imposed by the no-slip condition) to a free-stream velocity ultimately set by mass conservation.
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4.5. PIPE FLOW 127 z R r L e U boundary layers on pipe wall Figure 4.11: Steady, fully-developed flow in a pipe of circular cross section. Elementary Boundary-Layer Theory This region in which flow adjusts from zero velocity at the wall to a relatively high free-stream value is termed the boundary layer . The concept of a boundary layer is one of the most important in all of viscous fluid dynamics, so we will at least briefly describe it here although a complete treatment is well beyond the intended scope of the present lectures. We first do this in the simpler context of external flow over a flat plate but then argue that the same ideas also apply to the internal pipe flow under consideration here. Figure 4.12 presents the basic physical situation.
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