Step 5. Construct Dimensionless Groups
Use exponent algebra: select
r parameters as repeating parameters
│
│
└─ must span \(r\)-dimensional space \(M, L, T\) (det. of matrix formed from these non-repeating parameters must be \(\neq 0\))
★ For many fluid problems: velocity, length, and a fluid property
★ Each dimensionless group is formed by combining the repeating parameters, raised to unknown powers, with one of the nonrepeating variables
$$
[\Pi_1] = M^a L^b T^c = [\Delta p^a U^b d^c \rho^c] = (M L^{-1} T^{-2})^a (L T^{-1})^b (L)^c (M L^{-3})^c
= M^{a+c} L^{b+c-3c-a} T^{-a-2}
$$
$$
\Rightarrow a = -2,\; b = 0,\; c = -1
\Rightarrow \Pi_1 = \Delta p \, \rho^{-1} U^{-2} \;\equiv\; \frac{\Delta p}{\rho U^2}
$$
Similarly
$$
\Pi_2 = \frac{\Delta x}{d}, \quad \Pi_3 = \frac{\varepsilon}{d}, \quad \Pi_4 = \frac{\mu}{\rho U d}
$$
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