Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 31

- .An
irrotational fluid has a vorticity of zero. This simplifies the governing equations of the
fluid (to be derived later) and makes the computational analysis simpler. In most exam-
ples in this text, the vorticity will be zero. However, in biofluid mechanics, due to the large
pressure and velocity gradients, recirculation zones, turbulence and shed vortices can
be generated which can cause a fluid to have a non-zero vorticity; hence, the fluid would
be rotational.

Vorticity ( ζ ) is a measure of the rotation of fluid elements and is defined as 2

Example
Calculate the vorticity of the following velocity field. Is the flow rotational or not?

- - - - Cx 2 zcos 5 ½ Czsin ð xy Þ 1 Cxyzcos ð xy Þ 1 ð xy Þ 1 Cxsin ð xy Þ

Solution

- - -

y 2 @

v y
@

@ v y

@

@ v z

@

@ v x

@ z 2 @

v z
@ x 2 @

v x
@

ξ 5 1 1

z x y

- -

Cx 2 zcos @ð Cxsin ð xy ÞÞ

2 @ð

ð xy ÞÞ @ð½ Czsin ð xy Þ 1 Cxyzcos ð xy ÞÞ

2 @ð

Cxsin ð xy ÞÞ

5 1

@ y @ z @ z @ x

-

@ð Cx 2 zcos ð xy ÞÞ

2 @½

Czsin ð xy Þ 1 Cxyzcos ð xy Þ

1

@ @

x y

ÞÞ i - ÞÞ j - Cx 2 cos Cx 2 cos 5 ð ð xy Þ 2 ð xy 1 ð Csin ð xy Þ 1 Cxycos ð xy Þ 2 Csin ð xy Þ 2 Cxycos ð xy

ÞÞ -

Cx 2 yzsin Cx 2 yzsin 1 ð 2Cxzcos ð xy Þ 2 ð xy Þ 2 Cxzcos ð xy Þ 2 Cxzcos ð xy Þ 1 ð xy

0 -
The fluid is not rotational.

0 i - 0 j -

5 1 1

In solid mechanics, deformations were described by the change in length of a small ele-
ment (i.e., strain). Here we will discuss the rate of change of the extension/compression of
fluid elements. Strain has nine components that comprise the strain tensor (similar to
stress components that comprise the stress tensor). Six of these components are indepen-
dent of each other using the same assumptions for developing the stress tensor analysis.
The rate of extension is defined by velocity changes through the flow field. If at some loca-
tion in time, the velocity of a fluid is v x and this changes to v x 1Δ v x at time t 1Δ t

then the fluid elongates by t. The first term in the elongation formulation arises

from the original length between two points within the fluid. The second term is derived
from the velocity component that may cause a change in the fluid element length. Similar
to the angular velocity formulations, deformation in one dimension becomes

Δ x 1Δ v x Δ

1
Δ

Δ x 1Δ v x Δ t 2Δ x

5 @

v x
@

5 ð : Þ

d xx lim

Δ

2 27

t Δ x x

t 0

-