TABLE OF CONTENTS Referring to a frame of reference that is fixed in space, let us observe the flow of a homogeneous fluid that is composed of a single chemical species. Let us consider, in particular, the motion of a fluid parcel that, at the particular observation time t, has a spherical shape with radius ? centered at the point x. We define the velocity of translation of the parcel as the average value of the instantaneous velocity of all molecules that reside within the parcel. It is clear that the value of this average velocity will depend upon the parcel radius ?. Taking the limit as ? tends to zero, we find that the average velocity tends to an asymptotic value, until ? becomes comparable to the typical distance between the molecules, whereupon we observe strong oscillations. These are manifestations of random molecular motions.
We define the velocity of the fluid u at the position x at time t as the apparent or outer limit of the velocity of the parcel as the radius of the parcel ? tends to zero, just before the discrete nature of the fluid becomes apparent. Under normal conditions, the velocity u ( x, t) is an infinitely differentiable function of x and t, but spatial discontinuities may arise under extreme conditions in high-speed flows, or else emerge as the result of mathematical idealizations...
