Examine the $1$D NS Eq$.$ in a pipe flow: $\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}=-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{x}}+\frac{de{l}^{2}u}{del{y}^2}$

If we consider $\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}$ the "transient inertia force" and, as in the discussion of

Reynolds number, $\frac{de{l}^{2}u}{del{y}^2},$ the convective shear force; please use $u(y,t)=u(y)e^{it}$

and approximation in the order of magnitude analysis, $\frac{de{l}^{2}u}{del{y}^2}$~~$\frac{u}{D^2},$ where D is the

characteristic pipe dimension, to demonstrate that Womersley number is the ratio of

transient inertial force over the shear force.