Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 32

The other two normal deformations are represented as
d yy 5 @

v y
@

and d zz 5 @

v z
@ y ;

ð 2 : 28 Þ

z

The other six strain tensor components are derived in a similar manner using the rela-
tionship that these stress components arise from shear stress (or shear forces) on the fluid
elements and not from velocity changes. These shear stresses cause a change in the angle
of our fluid element (which is normally represented as a cube). Similar to our angular
velocity calculations, the strain will be calculated as the average of the time rate of change
of the two angles

that the element makes with the fixed coordinate axis (these
angles are shown in Figure 2.15 ).

α and β

Δ Δ Δ Δ

v y 1 @ v y =@ x x 2 v y t v x 1 @ v x =@ y y 2 v x t

Δα D ; and Δβ D ð 2 : 29 Þ

Δ x Δ y

Using these formulas, the rate of deformation simplifies to

0
@

1
A 5

1
Δ

1
2 Δα1Δβ

1
2

@ v y

@ x 1 @

v x
@

d xy 5 d yx 5 lim

Δ

lim
Δ

ð Þ ð 2 : 30 Þ

t y t - 0

x 0

-

Δ y 0

-

The remaining components of deformation are defined in a similar manner, making the
deformation tensor

y 1 @

v y
@

1
2

@ v z

@ x 1 @

v x
@

1
2

@ v z

@

d xz 5 d zx 5 ; and d yz 5 d zy 5 ð 2 : 31 Þ

z z

2
4

3
5

2 @

v x
@

@ v y

@ x 1 @

v x
@

@ v z

@ x 1 @

v x
@ x y z

2 3

d xx d xy d xz

@ v y

@ x 1 @

v x
@

2 @

v y
@

@ v z

@ y 1 @

v y
@

1
2

4 5 5

d 5 d yx d yy d yz ð 2 : 32 Þ

y y z

d zx d zy d zz

@ v z

@ x 1 @

v x
@

@ v z

@ y 1 @

v y
@

2 @

v z
@ z z z

In fluid mechanics the deformation tensor is known as the shear rate (similar to strain
in solid mechanics). Again, this is the rate of change of the angles or changes in the ele-
ment's geometry within a fluid and can be calculated from the velocity changes within the
fluid.

Example
Calculate the shear rate of a fluid with the following velocity profile:

Cxy - - Cz -

- 5 1 sin ð y Þ 1