You need to predict the performance of a large industrial mixer under various operating conditions. You have deduced that power $(P)$ required to operate the mixer depends on the following variables:

Tank diameter D $(L)$

Impeller diameter d $(L)$

Impeller rotational speed $N(M^0{L}^1{T}^{-1})$

Fluid density $(L-2M)$

Fluid viscosity $(\frac{L}{T})$

a$.$ Determine the number of independent variables, fundamental dimensions involved in these variables, and the number of dimensionless groups that can be defined by them.

b$.$ Find an appropriate set of dimensionless groups such that $D$ and $N$ each appear in only one group.