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Projection matrix in cylindrical geometry

In polar coordinates equation (B.6) becomes

 (B.15)

Substituting in equation (2.48) for and performing the integration with respect to gives:
 (B.16)

This is in the form of the standard integral from equation (A.1) of appendix A. Substituting in the variables: , , and gives

When the evaluation is carried out the contribution when is zero giving:
 (B.17)

Now noticing from equation (A.2) that and using the fact that is a zero of the second term vanishes:
 (B.18)

Expressing this in terms of the ratio of the radii, we get
 (B.19)

hence we have proved equation (2.85).

The integration used to obtain the analytical expression for is identical to that for except that the labels are interchanged for surface 1 and surface 2. Interchanging and means that will be replaced with giving .

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Next: Inductance method Up: Projection at a discontinuity Previous: Volume velocity   Contents
Jonathan Kemp 2003-03-24