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
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.