Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 84

To find the exact solution, we need boundary conditions for the particular flow scenario. Due
to the no-slip boundary condition, we know that the velocity at both walls is zero and that the
shear stress at the centerline is zero. Therefore, our boundary conditions are

u ð 0 Þ 5 0

u ð h Þ 5 0

du ð h = Þ

2

5 0

dy

Using two of these conditions to solve for the integration constants,

0 2
2

@ p

u ð 0 Þ 5

x 1

c 1 0 1 c 2 5 0

μ @

c 2 5 0

du ð h = 2 Þ h = 2

μ

@ p

c 1

5

x 1 dy @

h
2

@ p

c 1 52

μ @ x

Substituting the values for these integration constants into the velocity equation,

y 2
2

@ p h

2

@ p 1

2

@ p

y 2 u ð y Þ 5

x y x ð

hy Þ

x 2

5 2

μ @ μ @ μ @

For this particular flow scenario,

μ 5 3 : 5cP

@ p 40 mmHg 2 50 mmHg

2 mmHg = cm

x 5

52

@ 5cm 0cm

2

h 5 500 μ m

1 3 : 81

Þ 5 2

y 2 y 2

u ð y Þ 5

5cP ð 2

2 mmHg = cm Þð 500 μ my

ms ð

500 μ my Þ

2 2

2 3 : μ

The shear stress profile for this particular flow is equal to

@ u

y 1 @

v

5 μ @

u

τ xy 5 μ

@ @ x @ y

y
μ

@ p h

2

@ p

5 @

p h

2

τ xy 5 μ y

x 2

2

@ μ @ x @ x

Substituting the appropriate known values for this particular scenario,

y
μ

@ p h

2

@ p

5 @

p h

2

τ xy 5 μ

x 2

y 2

@ μ @ x @ x

dyne
cm 2

0 : 267

m ð

y 250 μ m Þ

52 2

μ