# Womersley Number

on . Posted in Dimensionless Numbers

Womersley number, abbrerviated as Wo,  a dimensionless number, is used in fluid dynamics to characterize the pulsatile flow of a fluid within a conduit or tube.  The Womersley number is particularly relevant in the study of blood flow in arteries, where the periodic pumping action of the heart generates pulsatile flow.  The Womersley number provides information about the relative importance of inertia and viscous effects in a pulsatile flow.  In simple terms, it indicates whether the flow is dominated by inertial forces (high Womersley number) or viscous forces (low Womersley number).

In the context of blood flow in arteries, a high Womersley number indicates that the pulsatile nature of blood flow is significant, which has implications for the distribution of blood velocity and pressure throughout the cardiac cycle.  The Womersley number helps researchers and engineers understand the behavior of pulsatile flows in various applications, including cardiovascular studies, where it's important to comprehend how blood flow dynamics might impact the health and functioning of arteries and other blood vessels.

### Womersley number Interpretation

• Low Wo < 1  -  The viscous effects dominate over inertial effects.  Flow is characterized by a parabolic velocity profile, similar to steady laminar flow.
• Intermediate 1 < Wo < 20  -  Both inertial and viscous effects are significant.  The flow may exhibit a combination of inertial and viscous effects, and the velocity profile may deviate from the simple parabolic shape.
• High Wo > 1  -  Inertial effects dominate over viscous effects.  The flow is influenced more by the pulsatile nature, and the velocity profile may become more plug-like.

## Womersley Number formula

$$\large{ \alpha = l \; \sqrt{ \frac{ \omega \; \rho }{ \mu } } }$$   (Womersley Number)

$$\large{ l = \sqrt{ \frac{ \alpha^2 \; \mu }{ \omega \; \rho } } }$$

$$\large{ \omega = \frac{ \alpha^2 \; \mu }{ l^2 \; \rho } }$$

$$\large{ \rho = \frac{ \alpha^2 \; \mu }{ l^2 \; \omega } }$$

$$\large{ \mu = \frac{ l^2 \; \omega \; \rho }{ \alpha^2 } }$$

Symbol English Metric
$$\large{ \alpha }$$ (Greek symbol alpha) = Womersley number $$\large{dimensionless}$$
$$\large{ l }$$ = approperate length scale (for example the radius of a pipe) $$\large{in}$$ $$\large{mm}$$
$$\large{ \omega }$$  (Greek symbol omega) = angular frequency of oscillation $$\large{\frac{rad}{sec}}$$ $$\large{\frac{rad}{s}}$$
$$\large{ \rho }$$   (Greek symbol rho) = density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ \mu }$$  (Greek symbol mu) = dynamic viscosity of the fluid $$\large{\frac{lbf-sec}{ft^2}}$$ $$\large{ Pa-s }$$

Tags: Flow Fluid