Pulsatile blood flow in the body is a response to periodic variations in velocity. These pulsating characteristics have been shown to be a result of two pumps. As the primary pump, the heart causes the blood flow and velocity to oscillate from zero to very high rates as the valves at the entrances and exits to the ventricles intermittently close and open with each beat of the heart. The second pump is a result of the respiratory and skeletal systems, which exert their greatest action on venous flow. Specifically pulsation that result from the release of blood from the left ventricle show that they exhibit non-liner, transient pulsations in pressure and flow. These create complex pulse patterns which are further propagated through the rest of the network. This results in variations in the applied shear stress to the layer of enodthelial cells which separate blood flow from the vessel wall. Depending upon the amount of force, the ECs will release chemicals that either induce dilation or constriction of the smooth muscle surrounding the vessel.

It is nearly impossible to mathematically model such a flow using the standard Navier-Stokes equations

Navier-Stokes equations

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...

. Rather than give an equation that can model the flow, which has proven to be near impossible; the Womersley number

Womersley number

The Womersley number is a dimensionless number in biofluid mechanics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscous effects. It is named after John R. Womersley . The Womersley number is important in keeping dynamic similarity when scaling an experiment. An...

 is used. This dimensionless number has been developed to give a measure of the frequency and magnitude of pulsations rather than a model of the actual flow.

As you can see, the equation can take on two forms by substituting mu/rho for nu. It can also be shown that Wormesley number is primarily influenced by the size of the vessel which can be shown in the table below. Since the density of blood and blood viscosity remain fairly constant (with slight variations throughout) the value of the square root will be similar for all, thus vessel size is most important.

Section	Radius (cm)	alpha
Ascending Aorta	0.75	14.628
Descending Aorta	0.65	12.677
Abdominal Aorta	0.45	8.777
Femoral Artery	0.2	3.901
Cartoid Artery	0.25	4.876
Arteriole	0.0025	0.049
Capillary	0.0003	0.006
Venule	0.002	0.039
Inferior Vena Cava	0.5	9.752
Main Pulmonary artery	0.85	16.578

These values were calculated using a cardiac frequency of 2 Hz, a blood density of 1060 kg/m^3 at 37 C, and a dynamic viscosity of 0.035 Pa-s