Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 33

Solution

d xx 5 @

v x
@ x 5 @ð

Cxy Þ

5

Cy

@ x

d yy 5 @

v y
@ y 5 @ð

sin ð y ÞÞ

5 cos ð y Þ

@ y

d zz 5 @

v z
@ z 5 @ð

Cz Þ

5 C

@ z

@

y 1 @

v y
@

@ð Þ

1 @ð

ð ÞÞ

1
2

v x
@

1
2

Cxy sin y Cx

2

d xy 5 d yx 5 5 5

x @ y @ x

1
2

@ v x

@ z 1 @

v z
@

1
2

@ð Cxy Þ

1 @ð

Cz Þ

d xz 5 d zx 5

5 5

0

x @ z @ x

1
2

@ v y

@ z 1 @

v z
@

1
2

@ð sin ð y ÞÞ

1 @ð

Cz Þ

d yz 5 d zy 5 5 5 0

y @ z @ x

2
4

3
5

Cx
2

Cy 0

d 5 Cx

2

cos ð y Þ 0

0 0 C

2.7 VISCOSITY

Viscosity is a property of fluids that relates the shear stress acting on a fluid to the shear
rate of the fluid. Fundamentally, viscosity arises because of inter-molecular interactions
within the fluid. For instance, the inter-molecular interactions in water are mostly hydro-
gen bonds (i.e., adhesion and/or cohesion; see Figure 2.12 ). These bonds are relatively
weak; therefore, water does not have a high viscosity. Fluids that possess stronger inter-
molecular bonds will have a larger viscosity (e.g., syrup). A fluid with a very large viscos-
ity is glass. If a 50-year-old house still has the original window panes, you might notice
that these window panes have changed their shape slightly. There will be bulges towards
the bottom of the window. The top will be narrower than when the windows were
installed. This is because the force of gravity acting on the window gradually caused the
flow of glass molecules towards the bottom of the pane. Another way to consider viscosity
is as the fluid's internal resistance to motion. (Recall that a body's resistance to acceleration
is its mass, and the body's resistance to rotation is its mass moment of inertia.) Fluids with
a larger viscosity will resist motion more easily than those with a lower viscosity.