1. Recall the definition of Reynolds number

   For fluid flow in a pipe, the Reynolds number is $R_e = \frac{\text{inertial force}}{\text{viscous force}}$ and its standard formula is $R_e = \frac{\rho v L}{\eta}$ where:

   • $\rho$ = density of fluid
   • $v$ = speed of fluid
   • $L$ = characteristic length
   • $\eta$ = coefficient of viscosity

2. Identify the characteristic length for flow in a pipe

   For flow through a pipe, the characteristic length is the diameter of the pipe, i.e. $L = d$

   Therefore, $R_e = \frac{\rho v d}{\eta}$

3. Match with the given options

   Compare with the options:

   • A: $R_e = \frac{\eta d}{\rho v}$
   • B: $R_e = \frac{\rho v}{\eta d}$
   • C: $R_e = \frac{\rho v d}{\eta}$
   • D: $R_e = \frac{\eta}{\rho v d}$

   The correct match is: $\boxed{C}$

4. Check dimensionlessness

   Reynolds number must be dimensionless.

   For option C: $\frac{\rho v d}{\eta} = \frac{(M L^{-3})(L T^{-1})(L)}{M L^{-1} T^{-1}} = 1$ so it is dimensionless, as required.

Hence, the correct formula is $\boxed{R_e = \frac{\rho v d}{\eta}}$