Biomedical Transport Processes - Introduction to Biomedical Engineering - page 965

conservation of fluid momentum. These equations arise from applying Newton's second
law to fluid motion, together with the assumption that the fluid stress ( in the preceding

section describing viscosity) is the sum of a diffusing viscous term (proportional to the gra-
dient of velocity), plus a pressure term. As was the case for the continuity equation, the
Navier-Stokes equations may be written in various coordinate systems, but the cylindrical
coordinate system is most appropriate for blood flow, since blood vessels are cylinders.

t

0
@

1
A ¼ @ p

2
4

0
@

1
A þ

3
5 þr g r

2
y
r

2 2

r @ u r

@ t þ u r @ u r @ r þ u y

@ u r

@y þ u z @ u r @ z u

1
r

@
@ r

r @ u r
@ r

1
r

2 @

y 2 @

u r u r

@ z

u r
r

2
r

2 @ u y

@ r þm r

2 2

@y

@

0
@

1
A ¼

2
4

0
@

1
A þ

3
5 þr g y

@ u y

@ t þ u r @ u y @ r þ u y

@ u y

@y þ u z @ u y @ z u r u y

1
r

@ p
@y þm

1
r

@
@ r

r @ u y
@ r

1
r

2 @

@y 2 þ @

2

u y u y

@ z

2

2
r

2 @ u r

@y u y

r þ

r r

2

r

2

0
@

1
A ¼ @ p

2
4

0
@

1
A þ

3
5 þ

2 2

r @ u z

@ t þ u r @ u z @ r þ u y

@ u z
@ y þ u z @ u z

1
r

@
@ r

r @ u z
@ r

1
r

2 @

y 2 @

u z u z

@ z @ z þm

r g z

r @ z

2

@

The Navier-Stokes equations also assume a constant density as well as a constant viscosity,
indicating not only an incompressible fluid but a Newtonian one as well. Note that there
are three fluid flow equations, since fluid momentum is a vector quantity, the product of
mass and velocity, with velocity being a vector quantity with three components. Also note
that each of the three fluid flow equations also contain multiple elements owing to the three
derivatives of the fluid velocity.

14.2.4 Poiseuille Flow
As we just saw, even with the assumption of a Newtonian fluid, these equations are time
varying with partial derivatives and are difficult to solve. As such, the Poiseuille assump-
tions, named after Jean Louis Marie Poiseuille, who formulated Poiseuille's law of viscous
flow (as mentioned previously), are as follows:

1. The flow is steady ( 0).

2. The radial and swirl components of the fluid velocity are zero (

@ (

... Þ=@ t ¼

u r ¼ u y ¼ 0).

3. The flow is axisymmetric ( @ (

... Þ=@

y ¼ 0) and fully developed ( @ u z =@ x ¼ 0).

As a result, the first two of the three Navier-Stokes momentum equations and the continuity
equation are identically satisfied. The third (z) momentum equation reduces to

¼

1
r

@
@ r

r @ u z
@ r

1

m @ p @ z

The two boundary conditions for the flow of a viscous fluid (such as blood) are (dV z /dr) ¼ 0,

when r R (edge of tube), which are the result of

axisymmetric flow and viscous “no slip at the wall,” respectively.
The solution is

¼ 0 (center of blood vessel), and V z ¼ Oatr ¼

4m @ p

1

2

u z ¼

þ c

1 ln

r þ c

@ z r

2