In Silico Biofluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 351

but the actual relationship between the groups would need to be determined experimentally

(e.g., are there constants of proportionality?). However, the usefulness of this analysis technique
is that we can obtain dimensionless parameters that can relate fluid properties. In our example,
drag force was related to the fluid's velocity, density, and a characteristic length, and the viscos-
ity was related to the same three parameters.

Π

There are many different dimensionless parameters that are important in biofluid flows,
and we will take the time to describe some of these parameters. The first parameter is the
Reynolds number (Re), which is defined as

5 ρ vL
μ

vL
ν

Re 5 ð 13 : 24 Þ

where L is some characteristic length which is typically the blood vessel diameter in bio-
fluids parameter (see example for Reynolds number formulation). The Reynolds number
relates the inertial forces to the viscous forces and also is a criterion to describe the flow
regime as either laminar or turbulent. As the Reynolds number becomes large, the flow is
more likely to be turbulent. As the Reynolds number approaches 1, the inertial forces and
the viscous forces balance and as the Reynolds number approaches zero, the viscous forces
dominate and the flow is normally slow and laminar.
Another important dimensionless number is the Womersley number ( ), which relates

the pulsatility of the flow to the viscous effects. The Womersley number is defined as
α5L ω
ν

α

1 = 2

ð : Þ

13 25

where ω is the angular frequency. When α is less than 1, the fluid can become fully devel-

oped during each flow cycle. As 10), the velocity profile is typically repre-

sented as a plug-flow scenario. The Womersley number is important in biofluid mechanics
because nearly all flow in larger blood vessels has some pulsatility to it. It is important to
be able to determine whether the flow can fully develop in between each pressure pulse
or whether the flow never develops fully.
The Strouhal number is another dimensionless parameter that is important in pulsatile
flows. The Strouhal number is defined as

α increases ( .

fL
v

5 ð : Þ

St 13 26

where f is the frequency of vortex shedding, L is a characteristic length, and v is the fluid
velocity. For a Strouhal number greater than 1, the fluid will move as a plug following the
oscillating frequency of the driving force. As Strouhal numbers decrease past 10 2 4 , the
velocity of the fluid dominates the oscillation and little to no vortices are shed. For inter-
mediate Strouhal numbers, there is a rapid vortex formation and many vortices can be
shed into the mainstream fluid. Interestingly, the Womersley parameter can be defined
using the Reynolds number and the Strouhal number as

p

2 α5 π ReSt ð 13 : 27 Þ