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’.

Complex Impedance

This post as­sumes a work­ing knowl­edge of el­e­men­tary cir­cuit the­ory as well as Fourier Analy­sis.

It seems to me that often in an in­tro­duc­tory cir­cuits course com­plex im­ped­ance is a major con­cept but its math­e­mat­i­cal basis is never taught. Thus I'd like to take a mo­ment to dis­cuss some of the math­e­mat­ics sur­round­ing this con­cept.

The re­sis­tor is a purely re­sis­tive el­e­men­tary cir­cuit el­e­ment pos­sess­ing a time in­de­pen­dent cur­rent volt­age char­ac­ter­is­tic de­scribed by Ohm's Law.

$$V(t) = I(t) R$$

The ca­pac­i­tor is a purely re­ac­tive el­e­men­tary cir­cuit el­e­ment that stores en­ergy in its elec­tric field.

$$I(t) = C \frac{dV(t)}{dt}$$

The in­duc­tor is a purely re­ac­tive el­e­men­tary cir­cuit el­e­ment that stores en­ergy in its mag­netic field.

$$V(t) = L \frac{dI(t)}{dt}$$

What we aim to do is to de­rive a lin­ear cur­rent volt­age char­ac­ter­is­tic for the two re­ac­tive cir­cuit el­e­ments by re­duc­ing the dif­fer­en­tial equa­tions to al­ge­braic equa­tions. Using the Fourier Trans­form we can com­pute a com­plex quan­tity in the fre­quency do­main that is anal­o­gous to re­sis­tance in the time do­main.