70
Dynamics
of
the
Vascular
System
(3.3.30)
The cylindrical polar coordinates representation, one has
Thus for velocities, u, v, w, we have
u
=
u(z,r,@,t)
v
=
v(z,r,@,t)
w
=
w(z,r,@,t)
(3.3.32)
along the direction
of
flow
z
in a cylindrical blood vessel, along its radius
(r) or radial component, and the rotational component associated with an
angular component
(9).
This representation identifies position within the
blood vessel at any given time t. This coordinates system was originated
by Euler, and is sometimes referred to
as
the Eulerian velocities.
Thus,
it
is clear that when the positions are time-dependent7 i.e. the
fluid element moves from one position to another with changing time,
then we have, for the velocities:
dz
dt
dr
dt
u=-
v=-
(3.3.33)
d6'
w
=
r-
dt
Velocity is clearly here defined as the rate
of
change
of
distance or
position. In the case of irrotational flow, or that the rotational flow
component is negligible, one remains with
u
and v. In the case
of
one-
dimensional flow, i.e. along the longitudinal z-axis
of
the vessel then
only
u
exists.