$\alpha = R\sqrt{\omega\rho/\mu}$, $\omega = 2\pi f$. Aortic defaults: $R = 10$ mm, $f = 1$ Hz, $\rho = 1060$ kg/m³, $\mu = 3.5$ mPa·s give $\alpha \approx 13.8$ (inertia dominant).
Blue lines = vessel wall at $\pm R$ / four curves = axial velocity $u(r,t)$ at four phases ($t = 0$, $T/4$, $T/2$, $3T/4$) / orange band = viscous penetration depth $\delta$. Small $\alpha$ gives Poiseuille parabolas, large $\alpha$ produces a central piston with a thin wall boundary layer and a phase lag.
X axis = $\alpha$ (log scale 0.1 to 100) / green band = quasi-steady ($\alpha < 1$) / yellow band = transitional ($1 \le \alpha < 10$) / red band = inertia dominant ($\alpha \ge 10$) / yellow dot = current alpha. The curve is the normalised inertial index (phase lag).
Definition of the Womersley number:
$$\alpha = R\,\sqrt{\dfrac{\omega\,\rho}{\mu}},\quad \omega = 2\pi f$$Viscous penetration depth (Stokes layer) and its relation to $\alpha$:
$$\delta = \sqrt{\dfrac{\mu}{\rho\,\omega}},\quad \alpha = \dfrac{R}{\delta}\,\sqrt{2}\;\;(\text{approx.})$$Peak Reynolds number (assuming peak velocity $U_\mathrm{peak}=1$ m/s):
$$\mathrm{Re}_\mathrm{peak} = \dfrac{\rho\,U_\mathrm{peak}\,(2R)}{\mu}$$$R$ is vessel radius [m], $f$ is heart-rate [Hz], $\omega = 2\pi f$ is angular frequency [rad/s], $\rho$ is blood density [kg/m³] and $\mu$ is dynamic viscosity [Pa·s]. The flow is quasi-steady (Poiseuille-like) for $\alpha < 1$, transitional for $1 \le \alpha < 10$ and inertia dominant for $\alpha \ge 10$ (aorta at 1 Hz heart rate).