We next employed the NS equations to derive Bernoullis equation one of the best

# We next employed the ns equations to derive

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We next employed the N.–S. equations to derive Bernoulli’s equation, one of the best-known and widely-used results of elementary fluid dynamics. We applied this in the analysis of the pitot tube used to measure airspeed, and to study flow in a simple gravity-driven fluid transport system. The next step up in difficulty involved derivation of two classical exact solutions to the N.–S. equations in planar geometry: Couette and plane Poiseuille flows. Following this we began the study of pipe flow by first considering the basic physics of boundary layers and their association with entrance length and fully-developed flow in a pipe with circular cross section. We then derived an exact solution to the N.–S. equations for this case, the Hagen–Poiseuille solution. Use of this led to a relationship between pressure changes over a length L of pipe, and a friction factor associated with viscous effects. Following this we made modifications to Bernoulli’s equation to permit its application to pipe flow. Specifically, we noted that this equation is actually an energy equation, and we related pressure losses (termed “head losses”) to changes in internal energy resulting from conversion of useful kinetic energy to unusable thermal energy due to entropy production during diffusion of momentum—which is mediated by viscosity. Hence, these head losses were associated with internal friction, and related to a friction factor. Further generalizations of Bernoulli’s equation permitted treatment of pumps and turbines and account of turbulence. Finally, we generalized treatment of head losses arising from internal friction to the case of so-called minor losses allowing empirical analysis of pressure losses in various practical flow devices such as contracting and expanding pipes, tees, bends, etc . This permits analysis of quite complex piping systems in an efficient, though only approximate, manner. But it must be emphasized that all of these practical techniques have their roots in the Navier–Stokes equations, once again underscoring the universality of these equations in the context of describing the motion of fluids.

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