If $$p$$ is the density and $$\eta$$ is coefficient of viscosity of fluid which flows with a speed $$v$$ in the pipe of diameter $$d$$, the correct formula for Reynolds number $$R_e$$ is

The Reynolds number is a dimensionless quantity used to predict the flow pattern in a fluid.

The Reynolds number is defined as the ratio of inertial forces to viscous forces:

$$R_e = \frac{\text{Inertial forces}}{\text{Viscous forces}} = \frac{\rho v d}{\eta}$$

where:

• $$\rho$$ = density of fluid
• $$v$$ = velocity of flow
• $$d$$ = diameter of pipe
• $$\eta$$ = coefficient of viscosity

$$[\rho] = ML^{-3}, \quad [v] = LT^{-1}, \quad [d] = L, \quad [\eta] = ML^{-1}T^{-1}$$

$$[R_e] = \frac{ML^{-3} \times LT^{-1} \times L}{ML^{-1}T^{-1}} = \frac{ML^{-1}T^{-1}}{ML^{-1}T^{-1}} = M^0L^0T^0$$

This confirms $$R_e$$ is dimensionless.

Hence, the correct answer is Option C.