Fluid Dynamics (Draft)

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

Content

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

Stress and The Material Continuum

The Material derivative

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 $\partial_t \phi = - u\cdot\nabla \phi$.

Cauchy's Momentum Equation

$$ D \vec u = \frac{1}{\rho} \nabla \cdot \bar {\bar \sigma} + \vec g $$

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