Decomposition

The motion of point P' relative to neighboring point P is given by .

Take the velocity derivatives tensor A apart into symmetric and antisymmetric parts:

The symmetric part is called the strain-rate tensor. By rotating the coordinate system, to x',y',z', it becomes diagonal:

This are three simple straining motions.

The trace of the tensor is the rate of (volume) expansion of the fluid.

The asymmetric part corresponds to solid body motion:

where is the vorticity.

Vorticity is important: when there are no viscous effects, no vorticity will be generated.

The line integral of the velocity along a closed contour is called the circulation of that contour:

Exercise:

What can you say about if ?

Questions: 4.2, 5, 7, 8, 9.

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