Example

Problem: We want to do a computation of a fluid flow in two dimensions. We will use polar coordinates. We chop the flow region up into small volumes and now we must derive equations for each of those small volumes. Write an approximate continuity equation for an arbitrary small volume .

Solution Let us choose the flow quantities at the centers of all the small volumes as the unknowns. We want to write equations for these unknowns. We start with the exact integral mass conservation equation for any arbitrary fixed region:

Now let us restrict ourselves to a finite number of points by making approximations:

Finally we can get rid of the side points A, B, C, and D by approximating further, eg, .There may be further considerations, see Intro to Computational Mechanics.

Exercise:

Derive the continuity equation in polar coordinates by taking the limit .

In the homework, you will do the same in spherical coordinates.

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