Well, since no one else has tried –

(a) With three units (length, mass, and time), assuming powers of the four variables, \eta, rho, p, and d, leaves some degeneracy in the exponents,

R ~ [\sqrt{\rho p} d / \eta]^\alpha \sqrt{p/\rho} d^2

where the factor in brackets is dimensionless and \alpha is arbitrary. \sqrt{p/\rho} is dimensionally a velocity, and the bracketed quantity is dimensionally density times velocity times diameter divided by viscosity, a Reynolds-number sort of thing. In full generality, any function of the bracketed factor works.

(b) For laminar flow, the choice \alpha = 1 or R ~ p d^3/\eta seems right, leaves no dependence on density. One-third the viscosity, three times the flow rate.

9√2 change in R

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