The velocity observed at the macroscopic scale is actually a weighted mean of the velocities of many discrete particles of differing masses. The deviations of the velocities of the discrete particles from the mean can be considered essentially as random, but their statistics determine the thermal state of the fluid.

Consider an arbitrary fixed surface *A* in the fluid. The mass
flowing through this surface per unit time will be denoted by *T*.
It is now asserted that there exists a vector function
such that

where is a unit normal to the surface. Strictly
speaking, *T* represents the most probable mass transport through
*A* at the instant *t*. If the surface *A* is closed so as to
contain a finite volume *V*, then *T* is the net mass transport
into or out of the volume *V* at time *t*. If we take
as the outward unit normal to the closed surface *A*
then (1) yields the net outward mass transport (i.e., if the integral
is positive, *T* is outward; if the integral is negative, *T* is
inward).

Equation (1) will be regarded essentially as the definition of the fluid velocity vector . As can be seen, it is dependent upon the definition of the density function given by Eq. (12) of Art. 3.03.

Mon Dec 1 08:50:29 CST 1997