Physical Concepts
and
Basic
Fluid
Mechanics
73
deformation, or the velocity gradient is small. And for a cylindrical
vessel with velocity v, and diameter d, this becomes:
(3.3.37)
Blood
has plasma, blood cells and other formed elements. In most
common analysis of blood flow
in
vessels, the assumption of blood as a
Newtonian fluid seems to work well. Except in the case of very small
vessels, such as the small arterioles capillaries where red blood cell size
actually approaches that of the vessel lumen diameter, one needs to be
concerned, not only of fluid shear, but also
of
shear stress on the flowing
red blood cell and of the differential velocity gradients generated by the
formed elements.
3.3.6.2
No-Slip Boundary Conditions
The “no-slip” condition refers to the assumption that concerns the fluid-
solid interface
or blood-vessel wall
interface.
No-slip
boundary
condition refers to the condition when the flow velocity at the tube wall
is
the same as the wall velocity, such that there is no “jump’ or a step
change in velocity to cause discontinuity. The general assumption is that
the fluid
in
contact with the wall does not move at all. This assumption
is
generally true for the large vessels. In small vessels, plasma dominates
as fluid and this no-slip condition generally applies to the plasma in
contact with the wall, rather than red blood cells or the formed elements
in blood.
3.3.6.3
Laminar and Turbulent Flow
Reynolds apparently was the first to use dye of visible color to
investigate the manner in which fluid flows in a tube and provided a
quantitative relation between the viscosity of the fluid and the mass of
the fluid. Reynolds number as it is called, and as we derived earlier
is
given by
