Abstract
The motions of a fluid can be described by using the time rates of change of physical variables defined on the fluid. To reach this end, within the continuum hypothesis, fluid as a continuum should a priori be assumed and the fundamentals of continuum mechanics need to be introduced, including the concepts of material body, reference and present configurations, and motion of a fluid element. Based on these, the material derivative of physical variable and deformation of a material may be defined to obtain the expressions of velocity and acceleration of a fluid element.
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Notes
- 1.
Specifically, \(\aleph \) is a physical variable per unit mass of the fluid element, called the specific variable .
- 2.
The deformation gradient \({\varvec{F}}\) can further be decomposed into a product of two tensors by using the polar decomposition , from which various strain measures of deformable materials can be defined.
- 3.
It is possible to obtain the velocity in the Eulerian description by using the material derivative, viz.,
$$\begin{aligned} {\varvec{u}}=\dot{{\varvec{x}}}=\frac{\partial {\varvec{x}}({\varvec{x}},t)}{\partial t}+\frac{\partial {\varvec{x}}({\varvec{x}},t)}{\partial {\varvec{x}}}\dot{x}_i={\varvec{0}}+{\varvec{I}}{\varvec{u}}={\varvec{u}}, \end{aligned}$$which holds identically.
- 4.
Although \({\varvec{u}}^P\) and \({\varvec{u}}^R\) are in general different in their mathematical forms, the velocity is differentiated with respect to \({\varvec{x}}\) for almost all circumstances, so that the velocity gradient always means the spatial gradient.
- 5.
The stretching tensor does not correspond to the strain rate tensor, for the integration of the latter does not correspond exactly to the former in general.
- 6.
The one-dimensional analogue of Reynolds’ transport theorem is the Leibniz integration rule . The relation between Reynolds’ transport theorem and the material derivative will be discussed in Sect. 5.3.6. Gottfried Wilhelm von Leibniz, 1646–1716, a German polymath, who developed differential and integral calculus independent of Newton.
- 7.
Caution must be made for the formulations of \({\varvec{b}}\) if other body forces present, or the material under consideration is not homogeneous, in which \({\varvec{b}}\) may be different for different material elements, even though \({\varvec{b}}\) is the constant gravitational acceleration.
- 8.
It is noted that
$$\begin{aligned} \sum {\varvec{{\mathcal {F}}}}_{CV}=\displaystyle \int _{V}\rho \,{\varvec{b}}\,\mathrm {d}v,\qquad \sum {\varvec{{\mathcal {F}}}}_{CS}\not =\int _{A}{\varvec{t}}{\varvec{n}}\,\mathrm {d}a, \end{aligned}$$for \(\sum {\varvec{{\mathcal {F}}}}_{CS}\) becomes now the sum of all surface forces external to the fluid body, and the stress traction on CS consists only a part of \(\sum {\varvec{{\mathcal {F}}}}_{CS}\).
- 9.
Equation (5.3.13) and its general form are called the Cauchy equations of motion , which have been derived first by Cauchy, and are applied to study the motions of elastic solid bodies.
- 10.
The balance of angular momentum is one of the basic axioms of the Galilean physics and has been formulated first by Euler for rigid bodies, which is termed the Euler equation of dynamics.
- 11.
The balance of linear momentum can also be formulated in the Lagrangian description, in which the stress is the first Piola-Kirchhoff stress tensor \({\varvec{T}}\) . Formulating the balance of angular momentum in the Lagrangian description shows that \({\varvec{T}}\) is not symmetric, but \({\varvec{TF}}^\mathrm {T}\) is symmetric. Gabrio Piola, 1794–1850, an Italian mathematician and physicist. Gustav Robert Kirchhoff, 1824–1887, a German physicist, who also contributed to the fundamental understanding of electrical circuits and the emission of blackbody radiation by heated objects.
- 12.
The first law of thermodynamics will be explored in a detailed manner in Sect. 11.4.
- 13.
The specific form of Eq. (5.3.27) depends on the definitions of the positivenesses of \({\dot{Q}}\), \({\dot{W}}\), and \(\dot{E}_s\) in thermodynamics. Here they are defined to be positive if they are provided to the system by the surrounding.
- 14.
The entropy of a material is microscopically interpreted as a measure of the disorder of atomic and molecular structures of that material, first proposed by Boltzmann. This topic will be explored in a detailed manner in Sect. 11.5.5. Without loss of generality, a reversible process is that in which the system and surrounding restore to their initial states if the process is reversed without any net change to the surrounding. If it is not the case, the process is referred to as an irreversible process . Ludwig Eduard Boltzmann, 1844–1906, an Austrian physicist, whose contribution was in the development of statistical mechanics and statistical thermodynamics.
- 15.
A physically admissible process is one in which all balances of mass, linear, and angular momentums, energy and entropy are satisfied simultaneously.
- 16.
Another Duhem-Truesdell relation is the relation between entropy supply \(s_\eta \) and energy supply \(\zeta \) given by
$$\begin{aligned} s_\eta =\frac{\zeta }{\theta }. \end{aligned}$$More general formulations on the entropy flux and entropy supply can be accomplished by using the Müller-Liu entropy principle, which will be discussed in Sect. 11.6.1. Pierre Maurice Marie Duhem, 1861–1916, a French physicist and mathematician, who is best known for his works on chemical thermodynamics, hydrodynamics, and the theory of elasticity. Clifford Ambrose Truesdell, 1919–2000, an American mathematician, natural philosopher, and historian of science, who, together with Noll, contributed to foundational rational mechanics.
- 17.
Rudolf Julius Emanuel Clausius, 1822–1888, a German physicist and mathematician, who is considered one of the central founders of the science of thermodynamics.
- 18.
For example, consider a two-dimensional Cartesian coordinate system which is spanned by the fixed orthonormal bases \({\varvec{e}}_1\) and \({\varvec{e}}_2\), and a new coordinate system \(\{{\varvec{e}}'_1,{\varvec{e}}'_2\}\) is obtained by rotating the \(\{{\varvec{e}}_1,{\varvec{e}}_2\}\) counterclockwise by an angle \(30^\circ \). In this case, \({\varvec{Q}}\) is given by
$$\begin{aligned}{}[{\varvec{Q}}]=\left[ \begin{array}{cc} \mathrm {cos}\,30^\circ &{} -\mathrm {sin}\,30^\circ \\ \mathrm {sin}\,30^\circ &{} \mathrm {cos}\,30^\circ \end{array}\right] = \left[ \begin{array}{ccc} \sqrt{3}/2 &{} -1/2 \\ 1/2 &{} \sqrt{3}/2 \end{array}\right] . \end{aligned}$$A point is described by \({\varvec{y}}=[2,2]^\mathrm {T}\) in the \(\{{\varvec{e}}_1,{\varvec{e}}_2\}\) system, the vector \({\varvec{y}}'\) of the same point is then obtained as
$$\begin{aligned}{}[{\varvec{y}}']=[{\varvec{Q}}^\mathrm {T}][{\varvec{y}}]=\left[ \begin{array}{cc} \sqrt{3}/2 &{} 1/2 \\ -1/2 &{} \sqrt{3}/2 \end{array}\right] \left[ \begin{array}{c} 2 \\ 2 \end{array}\right] =\left[ \begin{array}{c} \sqrt{3}+1 \\ \sqrt{3}-1 \end{array}\right] . \end{aligned}$$ - 19.
Euclid of Alexandria, c. Mid-fourth century to Mid-third century BC., a Greek mathematician, whom is often referred to as “Father of Geometry.”
- 20.
Gaspard-Gustave de Coriolis, 1792–1843, a French mathematician, who is best known for his work on the supplementary forces that are detected in a rotating reference frame, leading to the Coriolis effect .
- 21.
A detailed discussion on the topic will be provided in Sect. 5.6.
- 22.
But they are objective under the Galilean transformation.
- 23.
The quantities in the entropy balance and the equation itself are used to accomplish the admissibility of a physical process and are not taken into account at the present stage. A detailed discussion will be provided in Sect. 11.6.2.
- 24.
Every column of the deformation gradient \({\varvec{F}}\) transforms as an objective vector. Hence, \({\varvec{F}}\) transforms as three objective vectors.
- 25.
This requirement is not so universal as the previous two, that it is not addressed as a principle.
- 26.
The definition is given by Noll based on the rules of symmetry transformation. Walter Noll, 1925–2017, an American mathematician, who contributed to the mathematical tools of classical mechanics and thermodynamics.
- 27.
- 28.
Ronald Samuel Rivlin, 1915–2005, a British-American physicist, mathematician and rheologist, who is known for his works on rubber. Eugene Cook Bingham, 1878–1945, an American chemist, whose contributions are mainly in rheology.
- 29.
Jean-Baptiste Joseph Fourier, 1768–1830, a French mathematician and physicist, who contributed to the Fourier series , theory of heat transfer and discovered the greenhouse effect .
- 30.
Claude-Louis Navier, 1785–1836, a French engineer and physicist who specialized in mechanics. The Navier-Stokes equation was first derived by Navier in 1823 and later perfected by Stokes in 1845.
Further Reading
R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics (Dover, New York, 1962)
G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1992)
P. Chadwick, Continuum Mechanics (Dover, New York, 1976)
A.J. Chorin, J.E. Marsden, A Mathematical Introduction to Fluid Mechanics, 2nd edn. (Springer, Berlin, 1990)
I.G. Currie, Fundamental Mechanics of Fluids, 2nd edn. (McGraw-Hill, Singapore, 1993)
K. Hutter, K. Jönk, Continuum Methods of Physical Modeling (Springer, Berlin, 2004)
I.S. Liu, Continuum Mechanics (Springer, Berlin, 2002)
J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, 2nd edn. (Springer, Berlin, 1999)
I. Müller, W.H. Müller, Fundamentals of Thermodynamics and Applications (Springer, Berlin, 2009)
C. Truesdell, A First Course in Rational Continuum Mechanics, Volume 1 (Academic Press, New York, 1977)
C. Truesdell, R.G. Muncaster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas (Academic Press, New York, 1980)
C. Truesdell, W. Noll, The Non-Linear Field Theories of Mechanics (Springer, Berlin, 1992)
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Fang, C. (2019). Balance Equations. 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_5
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