Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 71

Solution
The inertial reference frame is chosen at the aortic valve and the non-inertial reference frame
is chosen to coincide with the ball. With the volume of interest chosen to coincide with the flow
direction around the ball, Figure 3.16 represents the flow situation. We will only analyze half of
the equation, because the flow is symmetrical around the uniform ball.

ð ð ð

5 @
@

- bx 1 - sx 2 d -

- rx ρ - xyz U dV u xyz ρ dV u xyz ρ

1

t

V V area

If we make the assumption that the blood flow is steady and uniform, the equation reduces to

ð ð

d - - rx ρ - xyz U

dV u xyz ρ

2 5

V area

because there are no external forces acting on the system. Substituting known values into this
equation, we get

ð

2 ρ

v 1 A 1
2

- rx ρ dV u 1 u 2 ðρ v 2 A 2 Þ

2 5 1

V

We will make the assumption that as blood flows around the ball, there is no loss of velocity
(and the area does not change) due to friction between the ball and blood. We will also assume
that there is no change in velocity between the aortic valve location and the ball. This makes the
magnitude of the following quantities
v 1
2 5

u 1 5 u 2 5 v 1 5 v 2 5 v inlet 2 v ball

which is the relative velocity, the same. Furthermore, the area can be defined as

A 1
2 5

A 2 5 A 3 5 A

Simplifying each term of the momentum equation and only considering the top half of the
ball:

ð

dv ball
dt

- rx ρ - rx ρ - rx m dV V m

5 5 5

V

FIGURE 3.16 Location 2

A 2 , v 2

Free body diagram for preceding example problem.

y

x

Loc ation 1
A 1 , v 1

m

Location 3
A 3 , v 3