The Womersley number () is a -group given by the ratio of [pulsatile transient force]/[viscous force]. Biomedical

The Womersley number (α) is a π-group given by the ratio of [pulsatile transient force]/[viscous force]. Biomedical engineers have applied this to flow in blood vessels. The Womersley number is given by:where r = blood vessel radius, and ω = frequency, typically the heart rate. Just as does Re, α has different practical implications in critical ranges. In the range a α ≤ 1, a parabolic (laminar) velocity distribution has time to develop in a tube during each heartbeat cycle. When α ≥ 10, the velocity profile is relatively flat (plug flow) in the blood vessel. For a human research subject, assume the heart rate is 70 beats/s, radius of the aorta is 17 mm, density of blood is 1060 kg/m³, and radius of a capillary is 7 μm. The viscosity of blood is normally 3 × 10^–3 P_a∙s.

a. Find α for the aorta of this subject.

b. Find α for the capillary of this subject.

c. Does either the aorta or the capillary have an α that would predict plug fl ow? Does either have an α indicating a parabolic velocity distribution?

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ωρ a = r.