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This is a list of well-known dimensionless quantities illustrating their variety of forms and applications. The tables also include pure numbers, dimensionless ratios, or dimensionless physical constants; these topics are discussed in the article.
Dimensionless quantities, or quantities of dimension one, [1] are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. [ 2 ] [ 3 ] Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units .
The Knudsen number is a dimensionless number defined as =, where = mean free path [L 1], = representative physical length scale [L 1].. The representative length scale considered, , may correspond to various physical traits of a system, but most commonly relates to a gap length over which thermal transport or mass transport occurs through a gas phase.
Setting the three dimensions of to zero, we obtain 3 independent linear constraints, so the solution space has 1 dimension, and it is spanned by the vector (,,,). Thus, any dimensionless quantity constructed out of ρ , u , L , μ {\displaystyle \rho ,u,L,\mu } is a function of ρ u L μ − 1 {\displaystyle \rho uL\mu ^{-1}} , the Reynolds number.
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Although named for Edgar Buckingham, the π theorem was first proved by the French mathematician Joseph Bertrand in 1878. [1] Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena.
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. [1] [2] The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.
Let () represent the matter overdensity, a dimensionless quantity defined as: = ¯ ¯, where ¯ is the average matter density over all space.. The power spectrum is most commonly understood as the Fourier transform of the autocorrelation function, , mathematically defined as: = (′) = (), for = ′.