We consider a small portion of fluid, small enough for all
macroscopic gradients to be zero, and large enough to represent a
homogeneous phase. Now consider the quantity

(5.57) |

where the average is an ensemble average, or a time average over a time span long enough to smooth out fluctuations, and short enough to assume that the macroscopic flow is stationary. The sum is over all particles in our control volume, i.e. particles leaving the control volume are from that moment on left out of the sum, and particles entering the control volume will from that moment on contribute to the sum. Since in this case is a bounded quantity, with a well defined average, the average of its time variation must be zero, .

Evaluating the derivative we obtain

(5.58) |

Assuming for simplicity that no body forces are applied, we write

(5.59) |

where the first term results from the presence of all particles, except

Since the control volume is assumed to be of macroscopic size, the second
term may be assumed to be concentrated near the boundary of the volume. In
terms of the stress tensor we have

= | |||

= | (5.60) |

where is a component of the outward normal , and is the average velocity. It has been assumed that gradients in and are negligible. Combining everything we get

This is the result we were after; it expresses the macroscopic stress tensor in terms of averages over microscopic quantities.