Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 72

2

u 1 ð 2 ρ v 1 A Þ 5 ð v inlet 2 v ball Þð 2 ρð v inlet 2 v ball Þ A Þ 52 ρ A ð v inlet 2 v ball Þ

45 Þðρð

2 cos

45 Þ u 2 ðρ v 2 A Þ 5 ð v inlet 2 v ball Þ cos ð v inlet 2 v ball Þ A Þ 5 ρ A ð v inlet 2 v ball Þ ð

Substituting these values into the simplified conservation of momentum equation,

dv ball
dt

2 2 cos

45 Þ 5 ð 45 Þ 2

2

m

52 ρ

A ð v inlet 2 v ball Þ

1 ρ

A ð v inlet 2 v ball Þ ð cos ð 1 Þρ A ð v inlet 2 v ball Þ

2

To solve this differential equation, we must separate the variables as follows:

45 ÞÞρ dv ball

5 ð

1 2 cos ð A

dt

2

m

ð v inlet 2 v ball Þ

Integrate this equation as shown:

v ball 2 max

ð ð

t

45 ÞÞρ

dv ball ð 1 cos ð A

2

dt

5

2

m

ð v inlet 2 v ball Þ

0 0

v ball 2 max t

0 5 ð

45 ÞÞρ 45 ÞÞρ 1

5 ð

1 cos ð A 1 cos ð At

2 2

ð v inlet 2 v ball Þ m m

0

45 ÞÞρ 1 1

v inlet 5

v ball

v ball Þ 5 ð

1 cos ð At

2

v ball Þ 2 ð v inlet 2 v inlet ð v inlet 2 m

Solving this equation for v ball ,

45 ÞÞρ v ball

v inlet 2 v inlet v ball 5 ð

1 cos ð At

2

m

45 ÞÞρ ð 1 cos ð At

2

v inlet 2

v ball 5 ð v inlet v ball Þ

m

45 ÞÞρ 45 ÞÞρ ð 1 cos ð At ð 1 cos ð At

2 2

v inlet

v inlet v ball

5 2

m m

45 ÞÞρ 45 ÞÞρ

ð ð At ð ð At

1 2 cos 1 2 cos

v ball 1 v inlet v ball 5 v ball 1 1 v inlet

m m

45 ÞÞρ

ð 1 cos ð At

2

v inlet

5

m

0
@

1
A 5

45 ÞÞρ 45 ÞÞρ

ð 1 cos ð At ð 1 cos ð At

2 2

v inlet v inlet v ball 5

45 ÞÞρ

m v inlet ðð 1 cos ð At Þ

45 ÞÞρ

1 2

ð 1 2 cos ð At

m 1 v inlet

1

m

0
@

1
A

2 t

45 ÞÞð

ð ð =

m 3

Þ 2

27 mm
2

1 2 cos 1050 kg

2

5 ð

150 cm = s Þ

2 t

45 ÞÞð Þ 2 25g

1 ð

150 cm = s Þð 1 cos ð 1050 kg = m 3

27 mm
2

2

s 2 198 gm = t

5

25 g 132 g = s t

1