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2015 | OriginalPaper | Buchkapitel

4. Macroscopic Balances

verfasst von : Roberto Mauri

Erschienen in: Transport Phenomena in Multiphase Flows

Verlag: Springer International Publishing

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Abstract

In fluid mechanics, conservation of mass, momentum and energy lead to the so-called continuity and Navier-Stokes equations. These equations can be written in either integral or differential form. In this chapter, we consider their integral formulation, which is applicable to a finite mass of fluid in motion. The more rigorous differential treatment will be the subject of Chap. 6. Here, setting up macroscopic balances and, at first, neglecting all diffusive effects, we derive the continuity equation in Sect. 4.1, the Bernoulli equation in Sect. 4.2, and the Euler equation in Sect. 4.3. Then, in Sects. 4.4, 4.5 and 4.6, the corrections to the Bernoulli equation are analyzed, discussing the pressure losses due to friction forces, in particular for the cases of flows within pipes and around submerged objects.

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Vorheriges Kapitel General Features of Fluid Mechanics

Nächstes Kapitel Laminar Flow Fields

Named after Daniel Bernoulli (1700–1782) a Swiss mathematician and physicist, who was born into a family of very distinguished mathematicians from Holland. In fact, his father, Johann Bernoulli, convinced Paul Euler (Leonhard’s father, a pastor of the Reformed Church) that his son was destined to become a great mathematician.

Named after the French hydraulic engineer Henri Pitot (1695–1771).

Named after Evangelista Torricelli (1608–1647), an Italian physicist and mathematician.

Here we are considering the momentum transport through a streamtube. Note that at large Reynolds numbers the diffusive contribution is indeed very small.

Luigi Crocco (1909–1986), an Italian aerospace engineer.

For an iso-entropic flow, entropy is constant along any streamline, while homo-entropic flows have the entropy homogeneous everywhere. Accordingly, the flow of any inviscid and non-conducting fluid is always iso-entropic (see Eq. 6.8.4), but not necessarily homo-entropic, as, for example, there might be a temperature gradient.

Named after John Thomas Fanning (1837–1911), an American hydraulic engineer.

Also referred to as the Darcy–Weisbach, the Blasius or the Moody friction factor.

A qualitative explanation of this fact is that in pipe flow we are never too distant from the boundaries, so that wall effects can never be neglected. That means that the instability vortices will move away from the walls, where they form, towards the center of the tube, until the fluid motion will be turbulent everywhere. On the other hand, in the case of the flow past a submerged object, we can always move far enough from the body, where the fluid flow is unperturbed and convective momentum fluxes are the dominant form of transport, thus explaining the quadratic dependence of the drag force on the velocity that we have seen in Eq. (2.2.3).

As seen in Sect. 17.5, Blasius obtained this expression assuming that the fluid velocity depends on the distance from the center of the tube to the 1/7 power. A similar approach was followed by von Karman, who considered the self-similar logarithmic velocity profile at the wall. Other researchers simply tried to correlate the experimental data, such as Moody’s diagram.

Johann Nikuradse (1894–1979), a Georgian-German engineer and physicist.

In 1932, in an address to the British Association for the Advancement of Science, Sir Horace Lamb, a famous British applied mathematician and physicist, reportedly said, “I am an old man now, and when I die and go to heaven [Sir Horace was a hopeless optimist] there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.”

See S.B. Pope, Turbulent Flows, Cambridge Univ. Press (2000), Par. 7.3.

In many textbooks, the friction factor for submerged objects is denoted by C _D.

George Gabriel Stokes (1819–1903) was a mathematician, physicist, politician and theologian. Born in Ireland, Stokes spent all of his career at the University of Cambridge, where he served as Lucasian Professor of Mathematics.

Titel: Macroscopic Balances
verfasst von: Roberto Mauri
Verlag: Springer International Publishing
Buch: Transport Phenomena in Multiphase Flows
Print ISBN: 978-3-319-15792-4

Electronic ISBN: 978-3-319-15793-1

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-15793-1_4

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