Supplemental reading:

Holton (1979), chapters 2 and 3 deal with equations, section 2.3 deals with spherical coordinates, section 2.4 deals with scaling, and section 3.1 deals with pressure coordinates. Hxoughton (1977), chapter 7 deals with equations, and section 7.1 deals with spherical coordinates.

Serrin (1959)

As has been mentioned in the Introduction, it is expected that almost everyone reading these lecture notes (and despite TEXification, these are only notes) will have already seen a derivation of the equations. I have, therefore, decided to cover the equations using Serrin's somewhat less familiar approach.

Coordinate systems and conservation

Let x = (x1, x2, x3) be a fixed spatial position; this will be referred to as an Eulerian coordinate system. Now, at some moment t = 0 let's look at a fluid and label each particle of the fluid X = X(t, x) = (x1, x2, x3) where Xi∥t=0 = Xi; that is, we label each particle by its position at t = 0; this will be referred to as a Lagrangian coordinate system. In general, each coordinate system may, in principle, be transformed into the other:

Let the velocity of a fluid ‘particle’ be ū = (u1, u2, u3).

Similarly, let ā be the acceleration of a fluid ‘particle’:

where the summation convention is used; that is, we sum over repeated indices.

The laws of physics are fundamentally conservation statements concerning D/Dt of something following the fluid. Let us, for the moment, deal with some unspecified field f(xi, t) (per unit mass):

Now consider some region of space R(x).