Deduce the formula of Reynolds number in the form of inertial force and viscous force.

(N/A) The Reynolds number $R_{e}$ is defined as the ratio of inertial force to viscous force.
Inertial force is given by the product of mass and acceleration,which can be expressed as $F_{i} = \rho A v^{2}$,where $\rho$ is the density,$A$ is the cross-sectional area,and $v$ is the velocity.
Viscous force is given by Newton's law of viscosity,$F_{v} = \eta A \frac{dv}{dx}$,which simplifies to $F_{v} = \frac{\eta A v}{d}$ for a characteristic length $d$.
Taking the ratio of these two forces:
$R_{e} = \frac{F_{i}}{F_{v}} = \frac{\rho A v^{2}}{\left(\frac{\eta A v}{d}\right)}$
Simplifying the expression:
$R_{e} = \frac{\rho v^{2}}{\left(\frac{\eta v}{d}\right)} = \frac{\rho v d}{\eta}$
Thus,the Reynolds number represents the ratio of inertial force to viscous force.