Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 79

per unit mass). Substituting these values into Equation 3.22 , we have a statement for the
conservation of energy related to the second law of thermodynamics:

ð ð

dS

dt 5 @

d - - s ρ dV s ρ ð : Þ

1 U 3 51

@ t

V area

Substituting Equation 3.50 into 3.51 ,

ð ð

Q
T

@
@

d - - s ρ dV s ρ ð 3 : 52 Þ

1 U $

t

area

V

Equation 3.52 is a statement of the second law of thermodynamics, which can be directly
applied to fluid mechanics problems. In some instances, it may be useful to know that

!

ð

Q
T

Q
AT dA

ð 3 : 53 Þ

V 5

area

Q
A is the heat flux along one particular area. This is normally constant for one particular
surface area of interest.

3.7 THE NAVIER-STOKES EQUATIONS

In the previous sections, we have applied various physical laws to a fluid volume of
interest. However, to obtain an equation that describes the fluid motion at any time or
location within the flow field, it is easier to apply Newton's second law of motion to a par-
ticle. For a system such as this, Newton's law becomes

dm d -
dt

d -

ð : Þ

5 3 54

The derivation for the acceleration of a fluid particle has already been shown in
Chapter 2. Using that relationship for particle acceleration, Newton's law becomes

!

-
@

-
@

-
@

-
@

@

u @ v @ w @ d - dm ð 3 : 55 Þ

5

t 1 x 1 y 1 z

As discussed before, the forces on a fluid particle can be body forces or surface forces.
To define these forces, let us look at the forces that act on a differential element with mass
dm and volume dV dxdydz . As done previously to describe the pressure acting on a dif-

ferential element, assume that the stresses acting at the cubes center (denoted as p ) are
ω xx ,

5

τ yx , and τ zx ( Figure 3.19 ). Note that all of these stresses act in the x-direction on this

figure.