Pi Theorem
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Although named for Edgar Buckingham, the π theorem was first proved by French mathematician J. Bertrand [1] in 1878. 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 clear indication of its utility for modelling physical phenomena. The technique of using the theorem (“the method of dimensions”) became widely known due to the works of Rayleigh (the first application of the πtheorem in the general case [2] to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892, [3] a heuristic proof with the use of series expansion, to 1894 [4] ). Formal generalization of the π theorem for the case of arbitrarily many quantities was given first by A. Vaschy in 1892, [5] then in 1911—apparently independently—by both A. Federman, [6] and D. Riabouchinsky, [7] and again in 1914 by Buckingham. [8] It was Buckingham's article that introduced the use of the symbol "πi" for the dimensionless variables (or parameters), and this is the source for the theorem's name. Fluid Mechanics[edit] Common dimensionless groups in fluid mechanics include: Reynolds number (Re) generally important in all types of fluid problems. Re = ρVd/μ Froude number (Fr) modeling flow with a free surface. Fr = V/√(gL) Euler number (Eu) used in problems in which pressure is of interest. Eu = V/(p/ρ)
description
liquids
Transcript of Pi Theorem
Although named for Edgar Buckingham, the π theorem was first proved by French mathematician J. Bertrand[1] in 1878. 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 clear indication of its utility for
modelling physical phenomena. The technique of using the theorem (“the method of dimensions”) became widely known due to the works of Rayleigh (the first application of
the πtheorem in the general case[2] to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892,[3] a heuristic proof with the use of
series expansion, to 1894[4]).
Formal generalization of the π theorem for the case of arbitrarily many quantities was given first by A. Vaschy in 1892,[5] then in 1911—apparently independently—by both A.
Federman,[6] and D. Riabouchinsky,[7] and again in 1914 by Buckingham.[8] It was Buckingham's article that introduced the use of the symbol "πi" for the dimensionless variables
(or parameters), and this is the source for the theorem's name.
Fluid Mechanics[edit]
Common dimensionless groups in fluid mechanics include:
Reynolds number (Re) generally important in all types of fluid problems.
Re = ρVd/μ Froude number (Fr) modeling flow with a free surface.
Fr = V/√(gL)
Euler number (Eu) used in problems in which pressure is of interest.
Eu = V/(p/ρ)
