Microvascular Beds - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 186

Equation 6.2 shows the famous Stokes equations, which hold for relatively slow, viscous
flows that are only pressure driven. It is important to emphasize that the validity of these
equations are only for flows where the viscous terms dominate the inertial terms by at
least one order of magnitude, or that the length scales are so small that the fluid inertial
effects do not have time to overcome the viscous effects and that gravitational effects can
be neglected. However, we will see in Chapter 9 that many gas flows (e.g., air flow in the
lungs) and blood flow within the microcirculation can be characterized as a Stokes flow,
because they obey the simplified Navier-Stokes equations shown in Equation 6.2 . The
Stokes equations in cylindrical coordinates are

2
4

0
@

1
A 1

3
5

2 v r
@θ

2 v r
@

@ p @

@

1

r @ r 2 @

1

r 2 @

2

@θ 1 @

v

θ

r 5 μ r ð

rv r Þ

2 2

z 2 @ r @

2 0 1 3

2 v z
@θ

2 v z
@

@ p 1

r @

r @

v z
@ r 2 @

1

2 1 @

4 @ A 1 5

z 5 μ

ð 6 : 3 Þ

@ @ r r z 2

2 0 1 3

2 v 2 v

@ p

@θ 5 μ

@
@

1

r @ r 2 @

1

r 2 @

2

@θ 1 @

v r

4 @ A 1

θ θ

5

r ð

rv

θ Þ

2 2

r @ @ z 2

@θ

Example
Consider the flow of a blood cell within the microcirculation (see Figure 6.4). Determine the
velocity profile within a 16

μ μ

m diameter red cell (assume
that it is cylindrical, Figure 6.4) centered within the blood vessel, flowing with a uniform velocity
v z 2 max . Assume that the viscosity of the blood is

m diameter blood vessel with an 8

μ and that the pressure gradient only acts

within the z direction and is @ p
@ z . Solve for the velocity profile at the blood cells centerline.

Solution

1

@ p 1

r @

r @

v z
@ r 2 @

1

2 v z
@θ

1 @

2 v z
@ z 5 μ

2

z 2

@ @ r r

@ p

z 5 r

@
@

r @

v z
@ @ r r

FIGURE 6.4 Schematic of a red blood cell flowing through
a centerline. This is associated with the in-text example.

r

8 µ m

4 µ m

z