Hemodynamics
of
Large Arteries
125
Navier-Stokes equations for a Newtonian and incompressible fluid in
cylindrical coordinates (r,
8,
z)
and assuming irrotational flow, i.e. the
angular
8
components are negligible, can be written as:
fir
hr
av
1
ap
'I
a2vr
1
av
d2Vr
v,
-
-)
-+v
-fV
L=---++(-+-L+-
at
dr
dz
p
az
p
ar2
r
dr
az2
r2
(4.5.18)
where v,
=
longitudinal velocity component
v,
=
radial velocity component
q
=
viscosity of blood
and
q/p
=
kinematic viscosity of blood
Notice that when the fluid is assumed to be ideal, i.e.
q/p
=
0,
the
equations reduce to:
ld
--(rv,)
=
--
r
dr
dZ
(4.5.19)
For
linearized Navier-Stokes equations, the second and third terms of
the left-hand side are negligible for small velocities.
For long
wavelength (h>>r), one obtains solutions as follows, for a periodic
sinusoidally varying function
@(t-z/c):
(4.5.20)
'D(1-z
I
c)
p
=
-Aupe'
