Microvascular Beds - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 185

stream of red cells. Red cells that are following this first packet of cells also push the plasma
back between the wall and cells, which again causes red cells to congregate into a single-file
packet followed by a plasma packet. Another way to think of this is with the no-slip bound-
ary condition. Because the cells stream toward the centerline, effectively pushing the
plasma toward the walls, the plasma will have a velocity that is close to zero. From the cel-
lular point of view, this looks like the plasma is being pushed backward, but in reality the
plasma is just holding relatively steady along the wall. Eventually enough of this plasma
builds up to cause a break in the stream of red cells, which forms the plasma packet.
The spatial difference in velocity in the radial direction is relatively negligible in
capillaries because on average, the entire cell will travel with the same velocity. To model
the flow of blood through these small vessels, one can use the assumption that the fluid is
invisicid and is composed of elastic particles. However, this will not accurately account for
the portion of blood that solely consists of plasma, and the red blood cells do exhibit some
viscous properties (the red blood cell membrane is a viscoelastic material). To develop a
more accurate model, we will discuss fluid flow at low Reynolds numbers (the Reynolds
number will be discussed in Chapter 13, with numerical methods). Without going into the
details of the Reynolds number here, using typical blood flow properties found in the
microcirculation, the Reynolds number is within the range of 0.001 and 0.01. At a
Reynolds number this low, the viscous effects of the blood flow dominate over the inertial
effects (i.e., velocity). In this situation, the Navier-Stokes equations simplify to

0 1

@ @

2 u
@ x 2 1 @

2 u

y 2 1 @

2 u

p

@ A

x 5 ρ

g x 1 μ

@ @ @ z 2

0
@

1
A

2 v 2 v 2 v

@ p @

x 2 1 @ y 2 1 @ y 5 ρ

1 μ

ð 6 : 1 Þ

g y

@ @ @ @ z 2

0
@

1
A

2 w
@

2 w
@

2 w
@

@ p @

x 2 1 @ y 2 1 @ z 5 ρ

g z 1 μ

@ z 2

If gravitational effects can be ignored (which is probably a good assumption within the
microvasculature because the length scales are relatively small), then Equation 6.1 can be
simplified even further to

0 1

2 u
@

2 u 2 u

@ p @

x 2 1 @ y 2 1 @

@ A 5 μr

2 u

x 5 μ

@ @ @ z 2

0 1

@ p @

2 v
@ x 2 1 @

2 v

y 2 1 @

2 v

@ A 5 μr

2 v

y 5 μ

ð 6 : 2 Þ

@ @ @ z 2

0
@

1
A 5 μr

@ @

2 w
@ x 2 1 @

2 w
@ y 2 1 @

2 w
@

p
@

2 w

z 5 μ z 2