# Computing Womersley number

I'd like to know whether the blood flow is pulsatile in small capillaries present in tissues that lie away from the heart. From the literature, I understand that the Womersley number can be computed to find out if the flow profile will be pulsatile. Also, at small values of Womersley number, the flow becomes Poiseuille flow.

Womersley number, $$\alpha = R\sqrt \frac{\rho \omega}{\mu}$$

$$\omega$$ is the frequency

$$\rho$$ is the density and

$$\mu$$ is the viscosity

I am not sure how to determine the value of $$\omega$$ to compute $$\alpha$$.

EDIT: In the discussion below, the following has been suggested

Power spectrum of the cardiogram will give you the frequencies in the pressure with their magnitudes -basically it is a Fourier transform applied to cardiogram data. Then you can transform these frequencies to Womersley number and you have a Womersley number vs Magnitude graph. From there, you can see if there are any frequencies which allow for pulsatile flow

The above-mentioned suggestion will give us the graph for Womersley number and pressure frequencies in the heart. But how do we use this information to infer the pressure frequencies, for example, in the liver or other tissues?

Any suggestion will be highly appreciated.

• $\omega = \frac{\alpha^2 \mu}{R^2 \rho}$ ? – andereBen Aug 13 at 15:18
• @andereBen I would like to compute $\alpha$ – Natasha Aug 13 at 16:18
• Isn't $\omega$ the heart rate in this case? If not, you can measure the distance between pressure spikes at a point to determine a frequency. Note that, I am not really familiar with vascular applications of CFD, so take my suggestions with a tablespoon of salt. – Abdullah Ali Sivas Aug 14 at 21:24
• If you are worried about spikes, shouldn't a power spectrum of a cardiogram converted to a spectrum of Womersley number for the liver parameters give you exactly what you want? With this equipped you can then check if there are frequencies which allow for pulsatile flow. – Bort Aug 15 at 12:16
• @AbdullahAliSivas I by no means know more about the topic. Yes, you describe exactly the idea. I just came up with this because it is a very standard technique in different areas in physics. Practically speaking, if you use a cardiogram of the heart it will be a worst case estimation for frequencies, since pressure waves dissipate. So if the liver is far enough away from the heart, e.g. frequencies will have a reduced magnitude. So if you already have no pulsatile flow with those frequencies, you won't expect a pulsatile flow at the liver as well. – Bort Aug 17 at 8:39