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Mechanics of Fluids

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The Mathematical Structure of Classical and Relativistic Physics

Abstract

In this chapter, we analyse the global variables of fluid dynamics to determine their association with space and time elements. We also present the two major balance equations, the mass and momentum balances, without concerning ourselves with the differential formulation. Lastly, we show how to obtain the traditional equations in a differential formulation.

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Notes

  1. 1.

    This chapter presupposes a reading of Chaps. 1–9.

  2. 2.

    Chadwick [39, p. 50].

  3. 3.

    Lai et al. [117, p. 176], McLeod [155, p. 3].

  4. 4.

    Aris [6, p. 124].

  5. 5.

    Chorin and Marsden [41, p. 5], Aris [6, p. 105], McLeod [155, p. 7]. We remark that Meyer [158, p. 64] distinguishes between perfect and ideal fluids: an ideal fluid is a perfect fluid in isochoric motion.

  6. 6.

    Batchelor [10, p. 146], Yih [255, p. 31].

  7. 7.

    Cebeci and Smith [38, p. 2].

  8. 8.

    Billington and Tate [15, p. 96].

  9. 9.

    For this physical interpretation of the stream function, which is usually presented by the purely mathematical relations \(q_{x} = \partial _{y}\psi,\hspace{2.84526pt} q_{y} = -\partial _{x}\psi\), see Milne and Thomson [160, p. 476].

  10. 10.

    Kundu et al. [116, p. 99].

  11. 11.

    Recall the peculiar role of velocity in fluid dynamics, discussed in Sect. 12.2.

  12. 12.

    Recall that a closed line is said to be reducible when it can be contracted to a point by a continuous deformation, without passing outside the fluid region. See Batchelor [10, p. 92].

  13. 13.

    See for example Milne and Thomson [160, p. 53], Lamb [119, p. 17]. In the first case, one can write \(\mathbf{v} =\, \nabla \,\phi\), whereas with the old convention \(\mathbf{v} = -\,\nabla \,\phi ^{\prime}\).

  14. 14.

    See p. 293.

  15. 15.

    Chadwick [39, p. 65], Billington and Tate [15, p. 44], Jaunzemis [106, p. 207], Lai et al. [117, p. 76].

  16. 16.

    Billington and Tate [15, p. 47], Prager [188, p. 64], Aris [6, p. 84], Hunter [98, p. 111], Chadwick [39, p. 74], Jaunzemis [106, p. 208].

  17. 17.

    Temple [223, p. 3]. The expression fluid body is also used by Meyer [158, p. 3].

  18. 18.

    O’Neil [171, p. 55], Chorin and Marsden [41, p. 32].

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Authors and Affiliations

  1. Department of Engineering and Architecture, University of Trieste, Trieste, Italy

    Enzo Tonti

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  1. Enzo Tonti
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Tonti, E. (2013). Mechanics of Fluids. In: The Mathematical Structure of Classical and Relativistic Physics. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7422-7_12

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