In essence, we can regard the application of a net force
to a given mass M as the rate of transfer of momentum
per unit time, and since momentum cannot be destroyed within the
mass M its momentum must accordingly be altered at the
rate . This is essentially Newton's law of motion.
From it we can draw two corollaries which are sometimes referred
to rather inappropriately as separate laws:
The first of these corollaries is an obvious special case of the general law because if the net force is zero then the momentum of the body must be constant, and since the mass is stipulated as constant it follows that the absolute vector velocity must be invariant under these conditions. The static situation is of course a further speical case.
The second is in effect a statement that within a given medium or at its boundary, the force (or what amounts to the same thing, the rate of exchange of momentum) must be continuous. This is also a consequence of the general law when it is considered that if the rate of exchange of momentum were not continuous then at those points where a discontinuity exists in a given medium, there must be an infinite acceleration since a finite rate of accumulation of momentum would occur in a region of zero mass. By excluding the infinite acceleration on physical grounds, the second corollary must follow.
In regard to a fluid we must ascribe a further property to the
already defined functions 
and 
in order to
formulate the equation of motion. This is that the functions
and are conditioned by the requirement that
the total momentum 
of the fluid in a volume of constant
mass M moving with the fluid at time t is
There is nothing obviously unreasonable with regard to this
stipulation from the physical point of view. The conditioning
of the functions and is one of a statistical
nature related of the ensemble of fluid particles.
The Newtonian law of motion asserts that
the units being so chosen for M and that this is in
fact an equality. Eq. (I-3) applies to each of the components
of 
and hence applies to the vector sum as well.
Consequently, Eq. (27) can be expressed in the form
In general we may divide the force into two major
categories: the volume force component and the surface force
component. The simplest volume force is that of gravitational
attraction. If 
, 
, 
are the absolute gravitational attractions on a unit mass due
to the earth, moon and sun, respectively, then the total
gravitational force on mass M is
Other volume forces, such as electromagnetic forces in the sea, are relegated to a role of very minor importance in regard to their influence on the dynamics of the fluid.
Next consider the surface forces on the boundaries of the
volume V. These are distinct from the so called volume forces
in that the latter are, more broadly speaking, the long range forces
while the surface forces are the short range forces between the
fluid particles immediately adjacent to one side of the surface
and those particles immediately adjacent to the other side.
Consider a small area increment of the surface 
.
The force exerted by the fluid particles exterior to the volume
upon this element will be denoted by 
.
The vector 
is the stress or force per unit area at the
point in question. This stress may be inclined with respect to a normal
to the surface; in particular it can have both a tangential and normal
component. An example of a normal component is the fluid pressure.
In fact, this is the only surface force under hydrostatic conditions
by definition of the fluid. However, if the fluid is in motion
then shearing stresses can exist. The sign convention with respect
to the normal component of the stress will be taken
such that tension is positive and pressure is negative.
The hydrostatic pressure, as was proven in article 3.02, is independent
of the orientation of the surface . However, it is
clear that will depend in general upon both the position
and orientation of as well as on time.
Let 
represent the outward unit normal to the surface
element. Then 
where
the space-time dependency is understood.
The total surface force over the entire surface area A of the volume V is given by
Thus Eq. (28) can be expressed as
where 
is used as an abbreviation of thesum of the three
separate gravitational forces , and
. Eq. (29) may be simplified only to the point
In order to go further one must determine the dependence of
on . This can be done by considering a small
volume of fluid such that Eq. (30) may be approximated by
Now consider the volume in the form of a tetrahedron, three of
whose faces are normal to the three coordinate axes and the fourth
determined by the unit normal and of area
as shown in Fig. 4.07-2.
The various stress components acting on the three coordinate
faces are shown in the figure. There exist three components on each
face, given a total of nine coordinate stresses which, acting upon
the coordinate surfaces, must balance the force
exerted upon the inclined face when the volume V is reduced to an
arbitrariliy small value. The component stresses acting on the
positive sides (outside of the volume) of the three coordinate
surfaces define a stress tensor 
, where
i, j take on the values x, y, z independently.
The components of the stresses are taken as positive in the direction
of the outward normal to the coordinate surface. Thus the total
force on the face normal to the x axis is
.
However, from geometrical considerations the area
is equal to 
where
is the x component of the unit normal to the
inclined surface .
Similar expressions hold for the other two component surfaces
and the forces on these three surfaces can be written therefore as
and the sum is simply 
following
the usual addition convention of the indicial notation.
The negative sign is needed since the components of the
vector 
(i.e. ) are opposite in sense to the
directions of the respective coordinate axes.
The total surface force is consequently
and therefore Eq. (32) becomes
If the volume is reduced keeping the ratios 
the same then in the limit as the volume is shrunk to the point
x, y, z, 
also goes to zero and therefore
at the point x, y, z.
In the special case of hydrostatic conditions of the
fluid, P is always normal to the surface
and is independent of its orientation and consequently the
hydrostatic stress 
can be expressed as
where the negative sign is required since the hydrostatic
pressure p is takien as a positive quantity, and the
convention for P is that tension is positive. It must
be required then that hydrostatic stress tensor
is composed of the terms
in the usual systematic array. It is convenient to introduce
the special symbol 
known as the
Kronecker delta, which has the components
Then we can write
Finally, we will define 
as the anomaly of the
stress tensor from the hydrostatic condition such that
or
Returning now to the general equation of motion (Eq. 30) and introducing Eq. (35) yields
But since the volume V is arbitrary, it follows that
In view of Eq. (41) this can be written finally as
This is the equation of absolute motion where both
the operator D/Dt and are regarded in the absolute
sense, being determined with reference to fixed coordinates
in space.
In we add Eq. (I-1) multiplied by to this equation and
represent in indicial notation, we obtain
The term 
represents the momentum density,
is the advective flux of momentum, and
is the non-advective flux of momentum. The latter is
composed of the non-advective flux due to the hydrostatic pressure
p and the non-advective flux due to the anomalous stresses
. Essentially, the latter represent the viscous
stresses of the fluid, and are finite only in the presence
of velocity gradients.