Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 34

In solid mechanics, stress and strain are related by the modulus of elasticity.
Fundamentally, viscosity is defined in the same way, as a value that completes the follow-
ing proportionality:

y 1 @

v y
@ y 1 @

v y
@

2d xy 5 @

v x
@

@ v x

@

τ xy ~

x - τ xy 5μ

ð 2 : 33 Þ

x

) is the dynamic viscosity of the fluid. This is the
internal resistance to flow that was discussed previously. The common unit for viscosity is
Poise (where 1P

In this formulation, the viscosity ( μ

g
cm s ). In fluid mechanics, the kinematic viscosity (

) is a standardized
version of the viscosity. The kinematic viscosity is defined as the dynamic viscosity
divided by the fluid density. The unit for kinematic viscosity is Stoke (where 1St

1 ν

1 cm 2
s

).

Example
Calculate the shear stress on the upper and lower boundary of the following fluid, given that
the viscosity of the fluid is 3.5cP. The distance between the two plates is 10 mm and the velocity
is 0 mm/s at the lower plate and 30 mm/s at the upper plate.

FIGURE 2.19 Y Figure for the Example Problem.

U upper = 30 mm/s
U lower = 0 mm/s

d = 10 mm

Solution

y 1 @

v y
@

@ v x

@

5μ @

v x
@

τ upper 5μ

x y

There is no velocity in the y-direction so the second term within the parenthesis drops out of
the relationship. Since the velocity profile is linear in the y-direction, the shear stress equation
simplifies to

τ upper 5μ Δ

v x
Δ

U upper 2 U lower U upper

d upper y 5μ d lower 5μ d upper 2

1g = cm s 30mm s

10mm 5

=

105 dyne
cm 2

10 2 2 P

τ upper 5 3 : 5 3 0 : 0105Pa 5 0 :

1P

Similarly, the shear stress on the lower plate is
τ lower 5μ Δ

U lower 2 U upper U upper

d upper

v x
Δ

y 5μ d upper 5μ d lower 2