Abstract
Dimensional scaling is a process allowing for the specific physical units defining the original form of a problem to be factored out to leave a scaled mathematical problem. The solutions of the scaled problem will depend on a set of nondimensional parameters obtained from combinations of the original given quantities. Two scaling principles are introduced to guide the choices for characteristic scales that are useful for different limits under consideration. The Buckingham Pi theorem, which predicts the number of essential dimensionless parameters based on the form of the problem, is also introduced.
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- 1.
Also called the MKS system from Meters, Kilograms, Seconds.
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This limiting solution could also be obtained using L’Hopital’s rule for the \(\text {St}\rightarrow 0\) limit of (4.15).
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In Chap. 3, we showed that the calculus of variations can be used to prove that, among all smooth simple closed curves, the circle uniquely minimises the ratio \(\varPi _1\).
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The base units, \(\mathsf {U}_k\) in the \(\varPi \) theorem need not be fundamental units, they can be derived units, just as long they form a linearly independent set.
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© 2015 Springer International Publishing Switzerland
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Witelski, T., Bowen, M. (2015). Dimensional Scaling Analysis. In: Methods of Mathematical Modelling. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-23042-9_4
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DOI: https://doi.org/10.1007/978-3-319-23042-9_4
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-23042-9
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