Womersley Number formula |
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\( \alpha \;=\; l \cdot \sqrt{ \dfrac{ \omega \cdot \rho }{ \mu } }\) (Womersley Number) \( l \;=\; \sqrt{ \dfrac{ \alpha^2 \cdot \mu }{ \omega \cdot \rho } }\) \( \omega \;=\; \dfrac{ \alpha^2 \cdot \mu }{ l^2 \cdot \rho }\) \( \rho \;=\; \dfrac{ \alpha^2 \cdot \mu }{ l^2 \cdot \omega} \) \( \mu \;=\; \dfrac{ l^2 \cdot \omega \cdot \rho }{ \alpha^2 }\) |
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| Symbol | English | Metric |
| \( \alpha \) (Greek symbol alpha) = Womersley Number | \(dimensionless\) | \(dimensionless\) |
| \( l \) = Approperate Length Scale (for Example the Radius of a Pipe) | \(in\) | \(mm\) |
| \( \omega \) (Greek symbol omega) = Angular Frequency of Oscillation | \(rad\;/\;sec\) | \(rad\;/\;s\) |
| \( \rho \) (Greek symbol rho) = Density | \(lbm\;/\;ft^3\) | \(kg\;/\;m^3\) |
| \( \mu \) (Greek symbol mu) = Fluid Dynamic Viscosity | \(lbf-sec\;/\;ft^2\) | \(Pa-s \) |
Womersley number, abbrerviated as Wo or \(\alpha\), 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
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Low Womersley Number (Wo << 1) - Viscous forces dominate over unsteady inertial forces. The flow behaves more like a steady, Poiseuille-like (parabolic) flow, where the velocity profile adjusts instantaneously to pressure changes. Common in very small vessels (capillaries) or slow oscillations.
- Womersley Number (Wo ≈ 1) - A transition zone where both viscous and inertial effects are significant. The flow profile is neither fully parabolic nor plug-like, and complex interactions occur. Seen in medium-sized arteries or engineered systems with moderate pulsation.
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High Womersley Number (Wo >> 1) - Inertial forces dominate, and the flow exhibits significant unsteady effects. The velocity profile flattens or becomes plug-like, with a phase lag between pressure and flow due to the fluid’s inertia. Typical in larger arteries (the aorta) where pulsatile effects from the heartbeat are pronounced.
Womersley Number Applications
Capillary Flows - In microfluidics or small-scale systems (ink flow in a pen), a low Weber number suggests that surface tension keeps the fluid cohesive and stable.
Aerodynamics - When analyzing the interaction of air with liquid films (raindrops on an airplane wing), the Weber number helps predict whether the liquid will stay intact or fragment.
