Some notes on what I have learned about fluid dynamics and what I have found interesting as well as a selection of links.

- Stress and The Material Continuum
- The Material derivative and Cauchy's Momentum Equation
- Conservation of mass and the Absance of no Deviatoric stress
- The Action of Incompressible Flow
- The Gross-Pitaevskii Equation, $\phi^4$-theory and Super Fluids

The Material derivative asks the question, how fast is some given field changing relative to the current.

More precisely, if we have a test particle $P$ whose 3 Cartesian spacial coordinates are functions of time $P_i:\mathbb{R}\to \mathbb{R}$ for $i \in \{1,2,3\}$, that moves with the current, so that $$ \partial_t P_i = u_i $$ then what is the rate of change of the field in time observed by this particle.

The scaler field $\phi:\mathbb{R}^4 \to \mathbb{R}$ measured by the test particle throughout time would be given by $$f(t) \stackrel{_{def}}{=} \phi(t,P_1(t),P_2(t),P_3(t)) \equiv \phi(t,P_i(t)).$$

There for the material derivative would be given by $d_t f(t)$ evaluated at for a particle at time $t$ and position $\bf x$ , \begin{align} d_t \phi(t,P_i(t)) &= \partial_t \phi(t,P_i(t)) + \sum_i \partial_i \phi(t,x_i) \partial_t P_i(t) \\&= \partial_t \phi(t,P_i(t)) + \sum_i u_i \partial_i \phi(t,x_i). \end{align}

Now evaluating this for a test particle that is at position $\bf x$ at time $t$, \begin{align} d_t \phi(t,P_i(t))\|_{P_i(t) = x_i} &= \partial_t \phi(t,x_i)) + \sum_i u_i \partial_i \phi(t,x_i) \\&= (\partial_t + {\bf u}\cdot \nabla) \phi(t,x_i). \end{align} To summarise, if we which to find out the rate of change of a scaler field $\phi$ in time observed by a particle moving with the current for any time $t$ and position $\bf x$ we simply apply the operator $$ D \stackrel{_{def}}{=} \partial_t + {\bf u}\cdot \nabla $$ to the field $\phi$. Thus $D$ is our material derivative.

*Nb: this derivation has taken place in cartisan coordinates, In general $\nabla$
is the covariant derivative, not simply the spacial partial derivatives $\partial_i$
as I have taken it to be. I will save the generalisation for when I approch this
derivation with the differential forms.*

If there is no flow ${\bf u} = {\bf 0}$ or the field is not distinguished in space $\phi = f(t)$, then the meterial derivative is simply the partial time derivative $D = \partial_t$.

If the material derivative is zero $D\phi = 0$, then the field must be a constant pattern moving with the flow, that is $\partial_t \phi = - u\cdot\nabla \phi$.