Abstract
Fluids at rest or in motion exhibit distinct characteristics from those of solids. Fundamental concepts which are essential to the understanding of fluid motions are explored in this chapter. First, distinctions between common fluids and solids with their underlying physical features are discussed. The Deborah number is introduced in order to take into account the rheological characteristics of matter under different external excitations. Equations in applied mechanics and fluid mechanics are classified into two categories to demonstrate their intrinsic features, followed by the method of analysis used in describing physical process. The assumption of fluid as a continuum plays a crucial role in defining fluid properties, with which theory of fluid motions may be established. Among the properties of a fluid are the viscosity and pressure relatively important. While the former is explored by using Newton’s law of viscosity, the latter is discussed by using Pascal’s law. Characteristics of fluid flows such as ideal flows versus viscous flows, incompressible flows versus compressible flows, and laminar flows versus turbulent flows are introduced, with their detailed discussions provided in the forthcoming chapters. A structural classification is given at the end to show the main topics of the book, which will be discussed separately in different chapters.
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Notes
- 1.
Gedankenexperiment is a German word, which means a thought experiment. This idea was first introduced by Galileo Galilei, 1564–1642, an Italian polymath, in his “Discourses and Mathematical Demonstrations” in 1638. Pioneered thought experiments are Schrödinger’s cat in quantum mechanics and Maxwell’s demon in second law of thermodynamics. Erwin Rudolf Josef Alexander Schrödinger, 1887–1961, a Nobel Prize-winning Austrian physicist. James Clerk Maxwell, 1831–1879, a Scottish scientist in mathematical physics, who formulated the classical theory of electromagnetic radiation.
- 2.
The applied shear force is the external excitation, while the angular deformation is the response of system in the context of mechanics. Material is classified according to the relations between external excitations and system responses. This experiment is called a simple plane shear, which is a standard method of mechanics to test material performance.
- 3.
The latest research outcome of particle physics indicates that all matters are composed of the Higgs bosons. However, to simplify the discussions, it is assumed that all matters are composed of atoms and molecules.
- 4.
A direct scientific evidence of solid structure is, for example, the scanning electron microscope image of copper. The indirect evidence of gas molecular structure is provided by using the technique entitled “Development of methods to cool and trap atoms with laser light”, proposed by Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips, who were the winners of The Nobel Prize in Physics 1997.
- 5.
Robert Hooke, 1653–1703, a British polymath. He came near to an experimental proof that gravity follows an inverse square law and hypothesized that such a relation governs the motions of planets.
- 6.
This number was originally proposed by Markus Reiner, 1886–1976, an Israeli scientist and a major figure in rheology. The name was inspired by a verse in the Bible, which reads: “The mountains flowed before the Lord” in a song by the prophet Deborah.
- 7.
In the limiting case of \(T\rightarrow \infty \), all materials behave like fluids. This idea was first introduced by Heraclitus of Ephesus, c. 535–475 BC. a pre-Socratic Greek philosopher. A related proverb reads: “Everything flows if you wait long enough”, so that “It is impossible to step twice into the same river”, which is stated in another motto.
- 8.
- 9.
Leonhard Euler, 1707–1783, a Swiss mathematician, physicist, and engineer, who made influential discoveries in many branches of mathematics and is also known for the work in mechanics, fluid dynamics, optics, and music theory.
- 10.
However, if the number of spheres approaches infinite, statistical methods can be applied to conduct the calculations, giving rise to the theory of statistical mechanics or statistical thermodynamics.
- 11.
Region I in Fig. 2.3b is very close to the vertical axis in real scale. It is enlarged here to simplify the discussions.
- 12.
Martin Hans Christian Knudsen, 1871–1949, a Danish physicist, who is known for his study of molecular gas flow and the development of the Knudsen cell, which is a primary component of molecular beam epitaxy systems.
- 13.
An exception emerges for nano-structures, in which the physical lengths are of an order of \(10^{-9}\) m, yielding \(k_n\sim \mathcal {O}(1)\).
- 14.
Based on the molecular structures of fluids, the simplification of two-dimensional flow is exact and physically justified. On the other hand, two-dimensional formulations of solids, e.g. plane stress and plane strain theories, are only approximations, for non-vanishing out-of-plane strain and stress exist due to the conservation of mass.
- 15.
On the contrary, a quantity is a constant if its time rate of change vanishes in the Lagrangian description.
- 16.
An inconsistency is the term uniform flow field , which is used to denote a flow whose velocity is constant throughout the entire space.
- 17.
The Cauchy stress principle and the Cauchy lemma will be discussed in a detailed manner in Sect. 5.2.2.
- 18.
A Newtonian fluid is that satisfies Newton’s law of viscosity, to be discussed in Sect. 2.6.1.
- 19.
Blaise Pascal, 1623–1662, a French mathematician, physicist, and Catholic theologian, who contributed to the study of fluids and clarified the concepts of pressure and vacuum by generalizing the work of Evangelista Torricelli.
- 20.
There exists another pressure, called the mechanical pressure. The difference between thermodynamic and mechanical pressures of the Newtonian fluids will be discussed in Sect. 5.6.3.
- 21.
Sir Isaac Newton, 1642–1726, a British mathematician, astronomer, and physicist. His book entitled “Mathematical Principles of Natural Philosophy”, first published in 1687, laid the foundations of classical mechanics. In the same book, the property of viscosity was also defined, and the original statement reads: “The resistance which arises from the lack of slipperiness of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another”. Instruments for viscosity measurements are called viscometers.
- 22.
The classification is based on the relation between \(\mu \) and \(\dot{\gamma }\). The non-Newtonian fluids can also be classified as thixotropic and rheopectic (antithixotropic) fluids . In thixotropic fluids, the dynamic viscosity decreases with time under a constant applied shear stress, while rheopectic fluids exhibit a reverse tendency.
- 23.
Simple compressible substances are a subset of simple materials , whose states are determined by prescribing the values of two independent intensive properties. A detailed discussion will be provided in Sect. 11.1.4.
- 24.
Rigorous mathematical conditions of incompressibility of fluids will be provided in Sect. 5.3.1.
- 25.
Thomas Young, 1773–1829, a British polymath and physician, who made contributions to the fields of vision, light, solid mechanics, energy, etc., and has been described as “The Last Man Who Knew Everything”.
- 26.
Sir William Thomson, or Lord Kelvin, 1824–1907, a Scots-Irish mathematical physicist and engineer, who contributed not only to the mathematical analysis of electricity and formulation of first and second laws of thermodynamics, but also did much to unify the emerging discipline of physics in its modern form.
- 27.
The exact definition of the Mach number is the square root of the ratio of inertia force divided by compressibility force of a fluid, as will be discussed in Sect. 6.5.2. Ernst Waldfried Josef Wenzel Mach, 1838–1916, an Austrian physicist and philosopher, who contributed to the study of shock waves. Through his criticism of Newton’s theories of space and time, he foreshadowed Einstein’s theory of relativity.
- 28.
Influence of surface tension on flow behavior is described by the dimensionless Weber number , which will be discussed in Sect. 6.5.2. Moritz Gustav Weber, 1871–1951, a German engineer and university professor, who is known for his work on the systematic study of model similarity.
- 29.
Data quoted from Blevins, R.D., Applied Fluid Dynamics Handbook, Van Norstrand Reinhold Co. Inc., New York, 1984; and Handbook of Chemistry and Physics, 69th ed., CRC Press, New York, 1988.
- 30.
Jacques Alexandre César Charles, 1746–1823, a French scientist, who formulated the original law in his unpublished work from the 1780s. Joseph Louis Gay-Lussac, 1778–1850, a French chemist and physicist. This law can refer to several discoveries made by Gay-Lussac and other scientists in the late eighteenth and early nineteenth centuries. Robert William Boyle, 1627–1691, an Anglo-Irish natural philosopher. This law was first noted by Richard Towneley and Henry Power in the seventeenth century and was confirmed by Boyle through experiments.
- 31.
Johannes Diderik van der Waals, 1837–1923, a Dutch theoretical physicist and thermodynamicist, who is known for his work on an equation of state for gases and liquids.
- 32.
Jean-Baptiste le Rond d’Alembert, 1717–1783, a French mathematician, mechanician, and physicist, who also contributed to d’Alembert’s equation for obtaining solutions to the wave equations.
- 33.
Ludwig Prandtl, 1875–1953, a German aerodynamicist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlying the science of aerodynamics and is recognized as “Father of Modern Fluid Mechanics”.
- 34.
Henri Pitot, 1695–1771, a French hydraulic engineer, who invented the original pitot tube in the early eighteenth century.
- 35.
Osborne Reynolds, 1842–1912, a British prominent innovator in the understanding of fluid dynamics. He most famously studied the conditions in which the fluid state in pipes transitioned from laminar to turbulent flows.
Further Reading
H.A. Barnes, J.F. Hutton, K. Walters, An Introduction to Rheology (Elsevier, Amsterdam, 1989)
G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1992)
C. Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations (Cambridge University Press, Cambridge, 2000)
D.F. Elger, B.C. Williams, C.T. Crowe, J.A. Roberson, Engineering Fluid Mechanics, 10th edn. (Wiley, New York, 2014)
R.W. Fox, P.J. Pritchard, A.T. McDonald, Introduction to Fluid Mechanics, 7th edn. (Wiley, New York, 2009)
P.M. Gerhart, R.J. Gross, Fundamentals of Fluid Mechanics (Addison-Wesley, New York, 1985)
L.D. Landau, E.M. Lifshitz, Fluid Mechanics, 2nd edn. (Elsevier, Amsterdam, 2005)
E.A. Moelwyn-Hughnes, States of Matter (Oliver and Boyd, New York, 1961)
B.R. Munson, D.F. Young, T.H. Okiishi, Fundamentals of Fluid Mechanics, 3rd edn. (Wiley, New York, 1990)
P. Oswald, Rheophysics: The Deformation and Flow of Matter (Cambridge University Press, Cambridge, 2009)
R.H.F. Pao, Fluid Mechanics (Wiley, New York, 1961)
W.R. Schowalter, Mechanics of Non-Newtonian Fluids (Pergamon Press, Oxford, 1978)
A.J. Smith, A Physical Introduction to Fluid Mechanics (Wiley, New York, 2000)
D. Tabor, Gases, Liquids and Solids, and Other States of Matter, 3rd edn. (Cambridge University Press, Cambridge, 1993)
R.I. Tanner, Engineering Rheology, revised edn. (Oxford University Press, Oxford, 1992)
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Fang, C. (2019). Fundamental Concepts. 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_2
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