Blood Flow in Arteries and Veins - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 160

Solution
Section 1 and Section 3 should be relatively easy to solve because there is no taper in these
sections. Also, the sections can be solved independently because we know that the outflow of
one section will be the inflow of the next section.
For section 1,

@ u

u @

u

v @

u

w @

u

g x 2 @

p @

2 u

x 2 1 @

2 u

y 2 1 @

2 u

ρ

5 ρ

x 1 μ

t 1 x 1 y 1 @ @ @ @ z @ @ @ @ z 2

2 u

52 @

p @

0

x 1 μ @ @ y 2

2 u

@ 1

μ

@ p

y 2 5 @ @ x

@ u y

μ

@ p

y 5 x 1

c 1

@ @

y 2
μ

@ p

u ð y Þ 5 c 1 y c 2

x 1

1

@

Using the boundary conditions of

u ð 0 Þ 5 0 and u ð 2 r 1 Þ 5 0

the solution of the velocity profile becomes

y 2
μ

@ p r 1

μ

@ p

u ð y Þ 5

x y x 2 @ @

Solving the Navier-Stokes equations for section 3,

2 v 2 v 2 v

@ v

@

u @

v

v @

v

w @

v

g y 2 @

p @

x 2 1 @ y 2 1 @

ρ

5 ρ

y 1 μ t 1 x 1 y 1 @ @ @ z @ @ @ @ z 2

g y 2 @

p @

2 v

5 ρ 0

y 1 μ

@ @ x 2

2 v

@ 1

μ

@ p

y 2 ρ

x 2 5

g y

@ @ μ

FIGURE 5.21 Cross section of a tapered curving blood
vessel.

Section 1

r 1

Section 2

g y

Section 3

r 3