The material derivative

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Section 2 of Landau and Lifshitz’s Fluid Mechanics starts by writing down equations of motion for a fluid particle in coordinates attached to that particle, then transforms to coordinates fixed in space. The argument is physically clear, but mathematically opaque. This was bothering me, so I cleaned it up. Introducing a primal, underlying space, without coordinates simplifies everything nicely (of course, this is just introducing a proper manifold structure, but it’s a pretty example of its use).

In coordinates fixed in matter, the velocity of the fluid is governed by

$\mbox{density} \cdot \partial_t \mbox{velocity} = -\nabla \mbox{pressure}$

When we compose each function with a coordinate transformation, density and pressure change in an elementary manner. The only work is in the "$\partial_t \mbox{velocity}$" term. Essentially what we do is,

$\mbox{density} \cdot \partial_t \mbox{velocity} = -\nabla \mbox{pressure}$
$\rightarrow$ { compose with a coordinate transform }
$\mbox{density} \circ \mbox{trans} \cdot \ \partial_t \mbox{velocity} \circ \mbox{trans}\ =\ -\nabla \mbox{pressure} \circ \mbox{trans}$
$\rightarrow$ { density and pressure change coordinates simply }
$\mbox{density} \cdot \partial_t \mbox{velocity} \circ \mbox{trans}\ =\ -\nabla \mbox{pressure}$
$\rightarrow$ { expand $\partial_t \mbox{velocity}$ in terms of the velocity on the new coordinates }
$\mbox{density} \cdot \left[\mbox{expansion of velocity}\right] = -\nabla \mbox{pressure}$

As always, good notation is priceless. First, coordinate systems. We’ll go the manifold route, and immediately invoke the metatheorem that "if I prove $p$ for arbitrary coordinates, and my space is a manifold, I’ve proved $p$ on the manifold."

The very clever might set up both coordinate systems and the transform between them directly. I’m too out of shape, so I’ll start with a primal space (we’ll call it $X$) representing the physical space-time occupied by the fluid, with no coordinates or other structure. Then I construct coordinates as invertible functions from $X$ to open subsets of $\mathbb{R}^n$. The metatheorem lets me brush all the worries about not covering the whole space under the rug. $n = 4$ – three space dimensions (labelled 0, 1, 2) and one time dimension (labelled 3). It’s simpler to keep time with the other coordinates.

Let $\chi : X \rightarrow \mathcal{M}$ be coordinates fixed in matter, where $\mathcal{M} \subset \mathbb{R}^4$, and $\phi : X \rightarrow \mathcal{F}$ be coordinates fixed in space, where $\mathcal{F} \subset \mathbb{R}^4$. Naming the coordinate’s ranges distinctly often makes the question "what function goes here?" trivial: "it must have domain $\mathcal{M}$, and I have only one function like that which makes sense here."

For physical quantities that transform simply (density $\rho$ and pressure $P$), I want to capture what physical quantity I am looking at, and the domain that this particular version of that quantity is defined on in my notation. $P_\mathcal{M}$ for the physical quantity $P$ on the domain $\mathcal{M}$ works nicely. In this notation,

$\rho_X = \rho_\mathcal{M} \circ \chi = \rho_{\mathcal{F}} \circ \phi$

That’s fine for quantities that carry through without major changes, but velocity transforms in a more complicated manner, and we will be using its different forms heavily. Instead of inevitably miscopying a subscript, I assign the versions on different coordinate systems different letters: $\mathcal{V} : X \rightarrow \mathbb{R}^3$, $u : \mathcal{M} \rightarrow \mathbb{R}^3$ and $v : \mathcal{F} \rightarrow \mathbb{R}^3$, all related by

$\mathcal{V} = u \circ \chi = v \circ \phi$

In this notation, the differential equation in coordinates fixed in matter is

$\rho_\mathcal{M} \cdot \partial_3 u^k = -\partial_k P_\mathcal{M}$

The only function we have defined that will make this an equation on $\mathcal{F}$ is $\chi \circ \phi^{-1}$, so we compose the differential equation with it.

$\rho_\mathcal{M} \circ \chi \circ \phi^{-1} \cdot (\partial_3 u^k)\circ \chi \circ \phi^{-1} = -\partial_k P_\mathcal{M} \circ \chi \circ \phi^{-1}$

Pressure and density transform like $\rho_\mathcal{M} \circ \chi \circ \phi^{-1} = \rho_\mathcal{F}$, so

$\rho_\mathcal{F} \cdot (\partial_3 u^k)\circ \chi \circ \phi^{-1} = -\partial_k P_\mathcal{F}$

Now expand the $k$th component of velocity fixed in matter $\partial_3 u^k$ in terms of $v$.

$\partial_3 u^k$
= { $u = v }
$\partial_3 (v \circ \phi \circ \chi^{-1})^k$
= { chain rule }
$\sum_{j=0}^3 (\partial_j v^k) \circ \phi \circ \chi^{-1} \ \cdot \partial_3 (\phi \circ \chi^{-1})^j$
= { $\partial_3 (\phi \circ \chi^{-1})^k = (\delta_{k3} - (1-\delta_{k3})u^k)$ }
$\sum_{j=0}^3 (\delta_{k3} - (1-\delta_{k3})u^k)\ \cdot\ (\partial_j v^k)\circ (\phi \circ \chi^{-1})$
= { split sum over range (j=0 to 2, and j = 3) }
$(\partial_0 v^k) \circ (\phi \circ \chi^{-1})\ -\ \sum_{j=0}^2 u^k \cdot (\partial_j v^k) \circ (\phi \circ \chi^{-1} )$
= { factor out $\phi \circ \chi^{-1}$ }
$\left[ \partial_0 v^k - \left( \sum_{j=0}^2 u^k \cdot \partial_j v^k\right) \right] \circ \phi \circ \chi^{-1}$

Where did the identity $\partial_3 (\phi \circ \chi^{-1})^k = (\delta_{k3} - (1-\delta_{k3})u^k)$ come from? $\partial_3$ is a time derivative. How do the the coordinates fixed in matter change over time with respect to coordinates fixed in space? Coordinate 3 (time) is the same in both, so $\partial_3 (\phi \circ \chi^{-1})^3 = 1$. The change in coordinates 0, 1, 2 is how far the bit of matter we are looking at moves in an infinitesimal time, that is, the velocity. But is it $v$ or $u$, and what is its sign?

$\phi \circ \chi^{-1}$ and its derivatives are functions on $\mathcal{M}$, so we must use $u$. The $\chi$ coordinates move with the fluid, so $\chi \circ \phi^{-1}$ would be positive velocity. We are using the inverse, so it is negative velocity. So for k=0, 1, 2, we have $\partial_3 (\phi \circ \chi^{-1})^k = -u^k$. Combining these with Kronecker $\delta$’s yields the identity.

Finally, put the expansion of $\partial_3 u^k$ into the equation of motion.

$\rho_\mathcal{F} \cdot (\partial_3 u^k)\circ \chi \circ \phi^{-1} = -\partial_k P_\mathcal{F}$
= { substitute for $\partial_3 u^k$ }
$\rho_\mathcal{F} \cdot\left[\partial_0 v^k-\left(\sum_{j=0}^2 u^k \cdot\partial_j v^k \right)\right]\circ \phi \circ \chi^{-1}\circ\chi\circ\phi^{-1}=-\partial_k P_\mathcal{F}$
= { coordinates and inverses cancel }
$\rho_\mathcal{F} \cdot \left[ \partial_0 v^k - \left( \sum_{j=0}^2 u^k \cdot \partial_j v^k\right) \right] = -\partial_k P_\mathcal{F}$

which we all know and love as the simplest equation of motion in fluid mechanics.

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