Abstract
Aims of this exercise: [1] learning to work with some of the fundamental equations of fluid mechanics in terms of vector components and [2] exploiting the mathematical properties (“symmetries”) such as incompressibility and the rotation-free property (i.e. no swirls in the fluid).
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- 1.
Not to be confused with the temperature at the surface of an individual star!
- 2.
So that \(\partial /\partial t = 0\) for any flow quantity.
- 3.
A constant-density flow is by definition incompressible!
- 4.
In this assignment I use r rather than R to denote the cylindrical radius.
- 5.
Defined as usual by \(x = R \cos \phi \), \(y = R \sin \phi \), \(R = \sqrt{x^2 + y^2}\).
- 6.
I use the term ‘universal’ in order to describe a form in which the physical properties of the system, in this case \(\Omega \) and \(C_\mathrm{s}\), no longer appear explicitly. Such a universal form therefore applies to sound waves in all rotating systems, regardless the value of \(C_\mathrm{s}\) or \(\Omega \).
- 7.
Even if that was not the case: we are only interested in the x-component of the relation, and for the front and back surfaces do not even contribute on an individual basis.
- 8.
Here we can neglect the fact that the interface is actually displaced over a distance \(\xi _{z}(z=0)\) since we are doing a linear analysis.
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Achterberg, A. (2016). Selected Problems. In: Gas Dynamics. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-195-6_14
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DOI: https://doi.org/10.2991/978-94-6239-195-6_14
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Online ISBN: 978-94-6239-195-6
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