In fluid dynamics, when I have to compute Reynolds number, I was taught to differentiate between two cases: $$ 1)\space \frac{\rho\cdot v\cdot d}{\mu}$$ or $$2)\space \frac{v\cdot d}{\mu}$$
but, why do I have to do this? Why shouldn't I simply use the first formula? what is the criteria to use in order to include or omit the density (i.e $\rho$)?
I've never seen the second formula. It's incorrect, unless it's a specific application like water where density is numerically 1 g/cc and they just don't write it.
The first formula
$$\mathrm{Re} =\frac{\rho vd}{\mu}$$
is the definition of Reynolds number and is always true. But you may sometimes see it written in terms of kinematic viscosity $\nu$ which is equal to $\mu/\rho$, and so you will see:
$$\mathrm{Re} =\frac{vd}{\nu}$$
In other more specialized geometries like a porous bed or perforated plate, you may see the Reynolds number defined in terms of other parameters like the mass flow rate and some dimension. But for flow in a pipe, the first formula is always true.
