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.