The material derivative

March 27, 2009

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} = -\
abla \\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} = -\
abla \\mbox{pressure}
\\rightarrow { compose with a coordinate transform }
\\mbox{density} \\circ \\mbox{trans}\\ \\cdot \\ \\partial_t \\mbox{velocity} \\circ \\mbox{trans}\\ =\\ -\
abla \\mbox{pressure} \\circ \\mbox{trans}
\\rightarrow { density and pressure change coordinates simply }
\\mbox{density} \\cdot \\partial_t \\mbox{velocity} \\circ \\mbox{trans}\\ =\\ -\
abla \\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] = -\
abla \\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 kth component of velocity fixed in matter \\partial_3 u^k in terms of v.

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

Fred Ross
March 2009
Lausanne, Switzerland

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