Physical Concepts
and
Basic Fluid Mechanics
63
3.3
Fluid Mechanics and Rheology
3.3.1
Steady Flow and Poiseuille Equation
The flow of viscous blood in a relatively cylindrical elastic arterial vessel
has borrowed much
of
the quantitative treatment from fluid mechanics.
Navier-Stokes equations are the fundamental equations describing fluid
motion. Poiseuille equation is a special case of the general solution of
the Navier-Stokes equations.
Steady pressure-flow relations are commonly described by the
Poiseuilleâ€™s equation:
(3.3.1)
where
Q
is the mean or steady flow in
ds,
r is the inner radius
of
the
vessel,
I
is the length through which blood flows and
q
is the viscosity
of
the fluid, in this case, blood
(0.03
poise
or
3 centi-poise) and Ap is the
pressure drop across the vessel. Thus, the amount of flow is critically
dependent on the size
of
the lumen radius and
is
proportional to its fourth
power. This equation has also been used to determine fluid viscosity, by
measuring flow and pressure drop over a known geometry
of
the tube.
The force opposing the flow of a viscous fluid with surface area
A,
is
proportional to the viscosity and the velocity gradient (v/d) across the
fluid layers with separation d. This defines the fluid viscosity as,
FIA
q=-
vld
(3.3.2)
It
is clear that the numerator is the applied pressure and the denominator,
velocity gradient. Thus, the more viscous the fluid, the greater amount
of
pressure is needed
to
apply to the fluid to generate the same amount
of
velocity gradient.
For a constant vessel geometry and fluid viscosity, it can be seen that
the pressure gradient governs the flow. From Fig. 3.3.1, it is clear that the
pressure gradient
is