Abstract
This chapter on gas dynamics illustrates a technically important example of a fluid field theory, where the information deduced by the second law of thermodynamics delivers important properties, expressed e.g. by the thermal and caloric equations of state of, say, ideal and real gases. Problems of acoustics, steady isentropic flow processes and their stream filament theory are briefly touched. The description of the propagation of small perturbations in a gas serves in its one-dimensional form ideally as a model for the propagation of sound e.g. in a flute or organ pipe, and it can be used to explain the Doppler shift occurring when the sound source is moving relative to the receiver. Moreover, with the stream filament theory, the sub- and supersonic flow through a nozzle can be explained. In a final section the three dimensional theory of shocks is derived as the set of jump conditions on surfaces for the balance laws of mass, momentum, energy and entropy. Their exploitation is illustrated for steady surfaces for simple fluids under adiabatic flow conditions. This leads to the well-known Rankine–Hugoniot relations. These problems are classics; gas dynamics, indeed forms an important advanced technical field that was developed in the 20th century as a subject of aerodynamics and astronautics and important specialties of mechanical engineering.
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Notes
- 1.
This statement remains valid also for a heat conducting viscous fluid , provided the dissipative working is ignored, as was shown in Sect. 18.2.
- 2.
Hermann Ludwig Ferdinand von Helmholtz (1821–1894) physician and physicist, was professor of physiology in Königsberg (Kaliningrad), Bonn, Heidelberg and since 1871 professor of physics in Berlin. For a brief biography, see Fig. 4.10 in Vol. 1.
- 3.
For a brief biography of Jean Baptiste le Rond d’Alembert (1717–1783), see Fig. 19.1 .
- 4.
This is e.g. the appropriate condition if \(\varPhi \) is the horizontal velocity.
- 5.
The index \((\cdot )_{n}\) in (19.39) will shortly become understood below.
- 6.
For a brief biography of Christian Johann Doppler (1803–1853), see Fig. 19.4 .
- 7.
For a short biography of Ernst Waldfried Josef Wenzel Mach , see Fig. 19.9 .
- 8.
The concept of the Laval nozzle was developed in 1878 by the German engineer and manufacturer Ernst Körting (1842–1921) and independently in 1883 by the Swedish engineer Carl Gustav Patrik de Laval (1845–1913) for use in steam turbines and rocket engine nozzles. It is named after Laval only.
- 9.
We must restrict ourselves to orientable surfaces, because physically a body or body part has always an outside and inside, which are always separated by an orientable surface. Examples of non-orientable surfaces are the Möbius strip or the Klein bottle, see Fig. 19.16 . August Ferdinand Möbius (17. Nov. 1790–26. Sept. 1868) was a German mathematician and theoretical astronomer and Felix Klein (25. April 1849–22. June 1925) was a mathematician, who chiefly established the distinguished mathematical school in Göttingen in the late 19th and early 20th century.
- 10.
It is assumed here that snow particles turning into water particles fall immediately into the pore space of the snow space, otherwise the singular surface is material. Alternatively, one may interpret melting as formation of a water layer between the snow cover and the sole of the ski. In this interpretation, two singular surfaces must be introduced, a material surface separating the ski sole and upper water layer and a non-material surface separating the dry snow from the water in the thin layer above it.
- 11.
Actually, slightly more general dependences are allowed, see e.g. K. Hutter and K. Jöhnk [8].
- 12.
- 13.
One obtains
$$\begin{aligned} \begin{array}{l} \dfrac{\kappa }{\kappa - 1}(\hat{p} \rho - p \hat{\rho } ) = \dfrac{1}{2}(\hat{\rho } + \rho )(\hat{p} - p), \\ \dfrac{\kappa }{\kappa - 1}\left\{ (\hat{p} - p)(\hat{\rho } + \rho ) - (\hat{\rho } - \rho )(\hat{p} + p) \right\} = (\hat{\rho } + \rho )(\hat{p} - p), \\ (\hat{p} - p)(\hat{\rho } + \rho )\left\{ \dfrac{\kappa }{\kappa - 1} - 1\right\} - \dfrac{\kappa }{\kappa - 1}(\hat{\rho } - \rho )(\hat{p} + p) = 0, \\ (\hat{p} - p)(\hat{\rho } + \rho ) = \kappa (\hat{\rho } - \rho )(\hat{p} + p). \end{array} \end{aligned}$$This relation leads to the first of the von Kármán relations.
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Hutter, K., Wang, Y. (2016). Gas Dynamics. In: Fluid and Thermodynamics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-33636-7_19
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