[A] (a) $$ \begin{array}{c|cccccccc} & \quad \Delta \quad & \quad m \quad & \quad d \quad & \quad \rho \quad & \quad \mu \quad & \quad \omega \quad & \quad U \quad & \quad L \quad \\ \hline \text{M} & \quad 0 \quad & \quad 1 \quad & \quad 0 \quad & \quad 1 \quad & \quad 1 \quad & \quad 0 \quad & \quad 0 \quad & \quad 0 \quad \\ \text{L} & \quad 1 \quad & \quad 0 \quad & \quad 1 \quad & \quad -3 \quad & \quad -1 \quad & \quad 0 \quad & \quad 1 \quad & \quad 1 \quad \\ \text{T} & \quad 0 \quad & \quad 0 \quad & \quad 0 \quad & \quad 0 \quad & \quad -1 \quad & \quad -1 \qquad & \quad -1 \quad & \quad 0 \quad \\ \end{array} $$ The rank of this matrix is 3, 8 parameters - 3 dimensions = 5 groups
Construct the dimensionless groups $$ \Pi_1 = \frac{\Delta}{d}, \quad \Pi_2 = \frac{m}{\rho d^3}, \quad \Pi_3 = \frac{\rho U d}{\mu}, \quad \Pi_4 = \frac{\omega d}{U}, \quad \Pi_5 = \frac{L}{d} $$ \(\therefore\) the dimensionless law for $\Delta$ is $ \frac{\Delta}{d} = \Phi \!\left( \frac{m}{\rho d^3}, \, \frac{\rho U d}{\mu}, \, \frac{\omega d}{U}, \, \frac{L}{d} \right) $ where $\Phi$ is an undetermined function
(b) When the viscosity is no longer a parameter, the Reynolds number must drop out $$ \frac{\Delta}{d} = \Psi\!\left(\frac{m}{\rho d^3}, \; \frac{\omega d}{U}, \; \frac{L}{d}\right) $$ where $\Psi$ is an undetermined function
(c) When the rotation rate is no longer a parameter, the Strouhal number must drop out $$ \frac{\Delta}{d} = \Theta\!\left(\frac{m}{\rho d^3}, \; \frac{L}{d}\right) $$ where $\Theta$ is an undetermined function
(d) $\Delta$ does not depend on $U$ at all

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