By Barenblatt G.I.
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Extra info for Scaling
4. The dimensionless pressure drop per unit length of pipe, , of fluid passing through a pipe as a function of Reynolds parameter Re = ρUD/μ. With the exception of the transition region between laminar and turbulent flow, the data from different experiments all lie on a single curve. The complicated nature of the curve indicates that the flow regime changes as a function of Re, which is the only parameter that determines the global structure of the flow. with a fluid layer of thickness h between two plates, one of which moves with velocity V and the second of which is at rest.
7) where the quantity α = ∂ L φ(1, 1, 1) is a constant independent of L, M and T . 7), we find that φ(L , M, T ) = L α C1 (M, T ). 6) but with one argument fewer: C1 (M2 , T2 ) = C1 (M2 /M1 , T2 /T1 ). 10) where β = ∂ M C1 (1, 1) is a constant similar to α. Following the same line of reasoning again, we find that C2 (T ) = C3 T γ , so that φ = C3 L α M β T γ . 11) The constant C3 is obviously equal to unity, since L = M = T = 1 means that the fundamental units remain unchanged, so that the value of the quantity a must remain unchanged and φ(1, 1, 1) = 1.
Bm : a = f (a1 , . . , ak , b1 , . . , bm ). The function f is the same for both phenomena because we require them to be similar, though the numerical values of the governing parameters a1 , . . , ak , b1 , . . , bm and the parameter a to be determined may differ. 19) for the prototype takes the form a P = f a1P , . . , akP , b1P , . . , bmP . 40) 1 Dimensional analysis and physical similarity 38 The superscript P will hereinafter be used to refer to the properties of the prototype. 19) for the model is similar in form, but the numerical values of the governing and determined parameters are different: a M = f a1M , .
Scaling by Barenblatt G.I.