64
Dynamics
of
the
Vascular System
(3.3.3)
PI
P2
I
i
I
1
Artery
Fig.
3.3.1:
A cylindrical tube containing steady flow
of
a fluid with viscosity
q.
For pulsatile flow, this pressure gradient changes with time. In fact,
before the advent of electromagnetic and ultrasonic flow transducers,
derivation of flow from pressure gradient was common. This latter was
applied to obtain pulsatile flow information, rather the steady or mean
flow.
The velocity profile for a steady, laminar flow is based on the general
derivation of Poiseuille’s equation.
Consider a cylindrical vessel with
internal radius ri and length
1,
the viscous force exerted on the cylindrical
unit of fluid is
(3.3.4)
The viscous force retarding the motion of the cylindrical volume of flow,
is, from
(3.3.2),
the product of cross-sectional area and viscosity,
multiplied
by
the velocity gradient,
(3.3.5)
