Abstract
Helium filled carnival balloons take off easily and can fly long distances. How high can they rise? Can they be of concern to airplanes? Now you might say that this latter question sounds a bit artificial and naive. However, helium balloons are regularly sent to the upper atmosphere for observing weather data. When designing and controlling such measurements, one has to know how high the balloons will rise and how much time they need for the journey.
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- 1.
In practice, the functions are only piecewise continuous due to the presence of interfaces between different materials.
- 2.
By thermodynamics, we mainly refer to classical mechanics and heat transfer. In particular, we rule out the relativistic phenomena where the concepts of mass and energy are not separated.
- 3.
When we consider the forces acting at a point x on the surface, we first notice that the force must be a vector. Let us denote it by F. On the other hand, the surface can be defined in the neighbourhood of x through its unit normal n. Thus, F = F(x, n). It turns out that there exists a matrix τ(x), such that \(F(x,n)_{i} =\tau _{ij}(x)n_{j}\) for all possible normal directions n. This matrix is called stress tensor in the sequel. At this stage, we cannot go into more details of the tensor concept.
References
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van Groesen, E., Molenaar, J.: Continuum Modelling in the Physical Sciences. SIAM (Mathematical Modelling and Computation), Philadelphia (2007)
Hämäläinen, J., Kuzmin, D.: Finite Element Methods for Computational Fluid Dynamics: A Practical Guide. SIAM (Computational Science & Engineering), Philadelphia (2014)
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Tiihonen, T. (2016). Continuum Models. In: Pohjolainen, S. (eds) Mathematical Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-27836-0_9
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