E.1 Buckingham Π-theorem [nondimensionalization top]

The Buckingham Π-theoremis a theorem in dimensional analysis that quantifies how many nondimensional parameters are required to specify a problem. It also provides a process by which these nondimensional parameters can be determined. The Buckingham Π-theorem states that a system with n independent physical variables that are a function of m fundamental physical quantities can be written as a function of n-m nondimensional quantities. As an example, the steady Navier-Stokes equations have four parameters: a characteristic length ℓ, a characteristic velocity U, the viscosity η, and the fluid density ρ. These are a function of three fundamental physical quantities: mass, length, and time. Thus the system can be described in terms of 4-3 = 1 nondimensional quantity, and it can be shown that the nondimensional quantity must be proportional to ρUℓ∕η to some power.

The Buckingham Π-theorem does not define unique nondimensional quantities, nor does it help identify which nondimensional quantities are most physically meaningful or useful; however, it does frame general problems from a dimensional standpoint.

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