## The material derivative

Last updated: 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

.

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

{ compose with a coordinate transform }

{ density and pressure change coordinates simply }

{ expand in terms of the velocity on the new coordinates }

As always, good notation is priceless. First, coordinate systems. We'll go the manifold route, and immediately invoke the metatheorem that "if I prove for arbitrary coordinates, and my space is a manifold, I've proved 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 ) representing the physical space-time occupied by the fluid, with no coordinates or other structure. Then I construct coordinates as invertible functions from to open subsets of . The metatheorem lets me brush all the worries about not covering the whole space under the rug. -- three space dimensions (labelled 0, 1, 2) and one time dimension (labelled 3). It's simpler to keep time with the other coordinates.

Let be coordinates fixed in matter, where , and be coordinates fixed in space, where . Naming the coordinate's ranges distinctly often makes the question "what function goes here?" trivial: "it must have domain , and I have only one function like that which makes sense here."

For physical quantities that transform simply (density and pressure ), 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. for the physical quantity on the domain works nicely. In this notation,

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: , and , all related by

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

The only function we have defined that will make this an equation on is , so we compose the differential equation with it.

Pressure and density transform like , so

Now expand the th component of velocity fixed in matter in terms of .

= { }

= { chain rule }

= { (see below) }

= { split sum over range (j=0 to 2, and j = 3) }

= { factor out }

Where did the identity come from? 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 . 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 or , and what is its sign?

and its derivatives are functions on , so we must use . The coordinates move with the fluid, so would be positive velocity. We are using the inverse, so it is negative velocity. So for k=0, 1, 2, . Combining these with Kronecker 's yields the identity.

Finally, put the expansion of into the equation of motion.

= { substitute for }

= { coordinates and inverses cancel }

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

Fred Ross
March 2009
Lausanne, Switzerland