Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 83

The following two examples illustrate how to use either the Cartesian form or the cylin-
drical form of the Navier-Stokes equations. Remember that the usefulness of these equa-
tions is that the fully developed velocity profile of any flowing fluid can be determined.
If the flow is one-dimensional, then this can be easily solved by hand.

Example
Find an expression for the velocity profile and the shear stress ( τ xy ) distribution for blood flow-

ing in an arteriole with a diameter of 500 m. Use the Navier-Stokes equations for Cartesian coor-

dinates to solve this problem. The pressure driving this flow is given in Figure 3.20 .

μ

Solution
To solve this problem, assume that v y 5 0 and v x is a function of y only. Assume that the

viscosity is constant and that the flow is incompressible and steady.

v z 5

2 u 2 u 2 u

@ u

@

u @

u

v @

u

w @

u

g x 2 @

p @

x 2 1 @ y 2 1 @

ρ

t 1 x 1 y 1

5 ρ

x 1 μ @ @ @ z @ @ @ @ z 2

2 u

d 2 u
dy 2

52 @

p

x 1 μ @ y 2 52 @

p

0

x 1 μ @ @ @

d 2 u
dy 2 5

1
μ

@ p 1

μ

@ p

d 2 u

x dy 2

x -

5

@ @

ð d 2 u ð 1

μ

@ p

x dy 2

5

@

dy

y
μ

@ p

du c 1

5

x 1 @

ð du ð dy

y
μ

@ p

c 1

5

x 1 @

y 2
2

@ p

u ð y Þ 5 c 1 y c 2

x 1

1

μ @

FIGURE 3.20 Pressure driven flow in an arteriole for example problem.

h

g y

y

x

5 cm

50 mmHg 40 mmHg