Abstract
Fluid mechanics is the mechanics of fluids embracing liquids and gases and is the discipline within a broad field of applied mechanics concerned with the behavior of liquids and gases at rest and in motion. Knowledge of ordinary and partial differential equations, linear algebra, vector calculus, and integral transforms is a fundamental prerequisite. However, to better access the underlying physical interpretations and mechanisms of fluid motions, additional mathematical knowledge is required, which is introduced in this chapter. First, the index notation with free and dummy indices is discussed, followed by the elementary theory of the Cartesian tensor, including tensor algebra and tensor calculus. Based on these, field quantities and mathematical operations which are essential to fluid mechanics in orthogonal curvilinear coordinate systems can be expressed in a coherent manner. Useful integral theorems in establishing the theory of fluid mechanics, such as Gauss’s divergence theorem, Green’s and Stokes’ theorems, are summarized as an outline. A review of complex analysis which is used intensively in discussing two-dimensional potential-flow theory of fluid mechanics is provided at the end. Detailed derivations and proofs of most equations and theorems are absent. They provide additional exercises for readers to become familiar with the topics introduced in this chapter.
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Notes
- 1.
Albert Einstein, 1879–1955, a German-born theoretical physicist. The summation convention was first introduced in 1916 in his “The Foundation of the General Theory of Relativity”.
- 2.
Leopold Kronecker, 1823–1891, a German mathematician. His viewpoint of mathematics is reflected by the famous motto, which reads: “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk” (God made the integers, all else is the work of man).
- 3.
Tullio Levi-Civitá, 1873–1941, an Italian mathematician. The permutation symbol is intensively used in linear algebra, tensor analysis, and differential geometry.
- 4.
Stretching tensor is the symmetric part of velocity gradient, which will be discussed in Sect. 5.1.5.
- 5.
Arthur Cayley, 1821–1895, a British mathematician. Sir William Rowan Hamilton, 1805–1865, an Irish physicist, astronomer, and mathematician. Cayley helped found the modern British school of pure mathematics. The main contributions of Hamilton are in the fields of classical mechanics, optics, and algebra.
- 6.
The name “Nabla” comes from the Hellenistic Greek word for a Phoenician harp based on the symbol’s shape. Pierre-Simon Laplace, 1749–1827, a French scholar whose main contributions are the development of mathematics, statistics, physics, and astronomy.
- 7.
Joseph-Louis Lagrange, 1736–1813, an Italian Enlightenment Era mathematician and astronomer, who contributed to the fields of analysis, number theory, and both classical and celestial mechanics. The Lagrangian derivative is the convection part of material derivative, which will be discussed in Sect. 5.1.3.
- 8.
Johann Carl Friedrich Gauss, 1777–1855, a German mathematician, who contributed to many fields such as statistics, analysis, differential geometry, geophysics, mechanics, electrostatics, astronomy, matrix theory, and optics.
- 9.
George Green, 1793–1841, a British mathematical physicist, whose main contributions were summarized in “An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism” in 1828.
- 10.
Sir George Gabriel Stokes, 1819–1903, an Irish physicist and mathematician. He made not only contributions to fluid dynamics and physical optics, but also to the theory of asymptotic expansions .
- 11.
Augustin-Louis Cauchy, 1789–1857, a French mathematician and physicist. Georg Friedrich Bernhard Riemann, 1826–1866, a German mathematician. Cauchy almost singlehandedly founded complex analysis. Riemann contributed to complex analysis by the introduction of the Riemann surfaces.
- 12.
Édouard Jean-Baptiste Goursat, 1858–1936, a French mathematician. He sets a standard for the high-level teaching of mathematical analysis, especially of complex analysis.
- 13.
Brook Taylor, 1685–1731, a British mathematician with his most known works as the Taylor theorem and Taylor series. Colin Maclaurin, 1698–1746, a Scottish mathematician, with contributions to geometry and algebra.
- 14.
Pierre Alphonse Laurent, 1813–1854, a French mathematician and military officer. He is best known as the discoverer of the Laurent series.
- 15.
From the mathematical view point, the residue theorem is a generalization of the Cauchy integral theorem and Cauchy integral formula.
Further Reading
R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics (Dover, New York, 1962)
R.V. Churchill, J.W. Brown, Complex Variables and Applications, 5th edn. (McGraw-Hill, Singapore, 1990)
I.G. Currie, Fundamental Mechanics of Fluids, 2nd edn. (McGraw-Hill, Singapore, 1993)
F.B. Hildebrand, Methods of Applied Mathematics, 2nd edn. (Prentice-Hill, New Jersey, 1965)
M. Itskov, Tensor Algebra and Tensor Calculus for Engineers: With Applications to Continuum Mechanics (Springer, Berlin, 2015)
J.P. Keener, Principles of Applied Mathematics (Addison-Wesley, New York, 1988)
W.M. Lai, D. Rubin, E. Krempl, Introduction to Continuum Mechanics, 3rd edn. (Pergamon Press, New York, 1993)
J.E. Marsden, Basic Complex Analysis (W.H. Freeman and Company, San Francisco, 1973)
D.E. Neuenschwander, Tensor Calculus for Physics (Johns Hopkins University Press, New York, 2014)
K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University Press, Cambridge, 1998)
I.S. Sokolnikoff, Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua, 2nd edn. (Wiley, New York, 1969)
D.V. Widder, Advanced Calculus, 2nd edn. (Prentice-Hill, New Jersey, 1961)
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Fang, C. (2019). Mathematical Prerequisites. In: An Introduction to Fluid Mechanics. Springer Textbooks in Earth Sciences, Geography and Environment. Springer, Cham. https://doi.org/10.1007/978-3-319-91821-1_1
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