Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 41

d - -

@

- -

@

-
@

-
@

-
@

-
@

-
@

dt 5 @

dt

dt 1 @

dx

dt 1 @

dy

dt 1 @

dz

dt 5 @

v x @ v y @ v z @

- 5

t 1 x 1 y 1

t @ x y z z

Temporal and spatial driving forces can arise within the vascular system. The angular
velocity of a fluid can be defined as

- - -

1
2

@ v z

@ y 2 @

v y
@

@ v x

@ z 2 @

v z
@

@ v y

@ x 2 @

v x
@

1
2 curl -

- 5 1 1 5

z x y

The deformation tensor takes the form of

2
4

3
5

@ v y

@

2 @

v x
@ x 1 @

v x
@

@ v z

@ x 1 @

v x
@ x y z

@ v y

@ x 1 @

v x
@

2 @

v y
@

@ v z

@ y 1 @

v y
@

1
2

d 5

y y z

@ v z

@ x 1 @

v x
@

@ v z

@ y 1 @

v y
@

2 @

v z
@ z z z

2.7 Viscosity is a quantity that relates the shear rate of a fluid to the shear stress. This relation-
ship takes the general form of

x .
2.8 The definition of a fluid as Newtonian depends on whether or not the viscosity is constant
at various shear rates. Newtonian fluids are classified with constant viscosities for a range
of shear rates, whereas non-Newtonian fluids have a non-constant viscosity. For most bio-
fluids applications, we can assume that the fluid behaves as a Newtonian fluid. An inviscid
fluid has no viscosity, and this is useful in some formula derivations.
2.9 A two-phase flow consists of a fluid in both a gas and liquid phase or two fluids (with dif-
ferent viscosities) within the same flow conditions. Blood is a two-phase flow because the
cellular matter may have a different viscosity than the plasma component.
2.10 Many fundamental fluid relationships change on the microscale. One of the most applicable
for biofluids problems is the viscosity, which is described by an apparent viscosity within
the microcirculation.
2.11 Fluid structure interaction modeling is important if the fluid can affect and cause a defor-
mation on the flow boundary. This is important in the cardiovascular system, in which the
blood vessel wall is deformable and cellular matter can interact with the wall.

1 @ v y
@

@ v x

@

τ xy 5μ

y

HOMEWORK PROBLEMS
2.1 For the velocity distribution v x 5

0, determine the acceleration vector.
Also, determine whether this velocity profile has a local and/or convective acceleration.
2.2 Consider a velocity vector v

5x, v y 52 5y, v z 5

- . (a) Determine whether this flow is
steady (hint: no changes with time). (b) Determine whether this is an incompressible flow
(hint: check if

- 1 ð 5 ð xt 2 2 y Þ xt 2 y 2 Þ

r ￿ v 5 0).

- 1 ð - , determine whether it is irrotational. 2.3 Given the velocity v 5 ð 2x 2 y Þ x 2 2y Þ