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.