Momentum Transport?

This post as­sumes that the reader has knowl­edge of basic vec­tor cal­cu­lus and physics.

In the study of fluid dy­nam­ics the term ‘mo­men­tum trans­port’ is thrown around quite often and often in con­junc­tion with the idea of ‘mo­men­tum ac­cu­mu­la­tion’; it is a rather dif­fi­cult con­cept to un­der­stand be­cause it is quite im­pos­si­ble to see mo­men­tum. Hope­fully this post will help to clar­ify the con­cept.

We will start with an easy to vi­su­al­ize con­cept: ‘mass trans­port’. Let us as­sume that there is a lam­i­nar and steady fluid flow of den­sity $\rho$ and ve­loc­ity $\mathbf{v}$. We would like to know the rate $K_T$ at which mass is trans­ported in this flow. The an­swer—

$$K_T = \rho |\mathbf{v}|$$

Makes sense until you con­sider the units of $K_T$ and re­al­ize that they are $\frac{\text{g}}{\text{cm}^2 \cdot \text{s}}$ which def­i­nitely doesn't make much sense. In fact it is dif­fi­cult to de­ter­mine at this point what the cor­rect units of the rate of mass trans­port should even be. To solve this prob­lem we need to con­sider mass trans­port from the other side of the equa­tion: ‘mass ac­cu­mu­la­tion’.