A. General

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

$$\dot{\bar{\mathcal{G}}}=\langle \frac{1}{V}\frac{d}{dt}\sum_{i}m_{i}\vec{r} _{i}\vec{v}_{i}\rangle$$ (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 $\sum_{i}m_{i}\vec{r }_{i}\vec{v}_{i}$ is a bounded quantity, with a well defined average, the average of its time variation must be zero, $\dot{\bar{\mathcal{G}}}=\bar{0}$.

Evaluating the derivative we obtain

$$\bar{0} = \langle \frac{1}{V} \sum_i m_i \vec{v}_i \vec{v}_i ... ...ac{1}{V} \sum_i m_i \vec{r}_i \frac{d \vec{v}_i}{dt} \rangle .$$ (5.58)

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

$$\frac{d \vec{v}_i}{dt} = \frac{d \vec{v}_i^{\mathrm{ int}}}{dt} + \frac{d \vec{v}_i^{\mathrm{ ext}}}{dt}$$ (5.59)

where the first term results from the presence of all particles, except i, in the control volume, and the second term results from the presence of all particles outside the control volume.

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

$${\langle \frac{1}{V}\sum_{i}m_{i}(\vec{r}_{i}\frac{d\vec{v}_{i}^{ \mathrm{ext}}}{dt})_{\alpha \beta }\rangle }$$

$$\frac{1}{V}\int \sum_{\gamma }r_{\alpha }\{S_{\beta \gamma }-\rho v_{\beta }v_{\gamma }\}t_{\gamma }dA$$ =

$$\frac{1}{V}\int \sum_{\gamma }\frac{\partial }{\partial r_{\gamma... ...\rho v_{\beta }v_{\gamma }\}d^{3}r=S_{\beta \alpha }-\rho v_{\beta }v_{\alpha }$$ = (5.60)

where $t_{\gamma }$ is a component of the outward normal $\vec{t}$, and $\vec{v}$ is the average velocity. It has been assumed that gradients in $\bar{S}$ and $\vec{v}$ are negligible. Combining everything we get 

$$\bar{S}=-\langle \frac{1}{V}\sum_{i}m_{i}(\vec{v}_{i}-\vec{v}... ...{i}\vec{r}_{i}\frac{d\vec{v} _{i}^{\mathrm{int}}}{dt}\rangle .$$ (5.61)

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