Abstract
The ultimate goal of this chapter is to deduce the mass and linear momentum balance laws in the topography-fitted coordinates introduced in Chap. 2, and to be prepared with some mathematical prerequisites when we formulate constitutive equations for the flowing material down arbitrary topography.
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Notes
- 1.
This is the rule when dealing with physical quantities, e.g. density \(\rho \), velocity \(\varvec{v}\), otherwise we would have used the “tilde” notation for f, i.e., \(\tilde{f}(\xi ^{1},\xi ^{2},\xi ^{3},t)\).
- 2.
Clearly, in these relations there is an abuse of notations consisting in the way \(\Gamma \) and \({\varvec{\Gamma }}\) have been used.
References
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H. Viviand, Formes conservatives des équations de la dynamique des gaz. Rech. Aosp. 1, 65–66 (1974)
M. Vinokur, Conservation equations of gasdynamics in curvilinear coordinate systems. J. Comp. Phys. 14, 105–125 (1974)
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Luca, I., Tai, YC., Kuo, CY. (2016). Differential Operators and Balance Laws in the Topography-Fitted Coordinates. In: Shallow Geophysical Mass Flows down Arbitrary Topography. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-02627-5_3
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DOI: https://doi.org/10.1007/978-3-319-02627-5_3
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