It's common for finite volume method to calculate the derivatives in viscous flux according to divergence theorem, but I do not known the detail formula in axisymmetrical cylinderical coordinates. for example, how to caluclate the derivatives of temperature according to divergence theorem? Thank a lot in advance!

Get a copy of "Transport Phenomena", by Bird, Stewart and Lightfoot. This shows all the vector operations in the three major coordinate systems - along with lots of other details.

A bit less expensive, Schwamm's outline series has a soft-cover summary on Vector Analysis.

Thank Park, I know how to calculate the derivatives of a vector, such as velocity, using divergence theorem according to any Vector Analysis book, but I don not known how to calculate the derivatives of a scalar, such as temperature, using divergence theorem in axisymmetrical coordinates. Can you shed light on it?

Peter,

Sorry, I did answer the wrong question!

You might take a look at

C. W. Hirt, A. A. Amsden, and J. L. Cook, "An arbitrary Langrangian-Eulerian Computing Method for All Flow Speeds," J. Comp. Phys, v. 14, pp. 227-253 (1974).

Also the reports for the KIVA series of codes developed at Los Alamos - if you can get them.

Finally, try the U. of Wisconsin web sites for research simulating combustion in a diesel engine that was spun off from the KIVA-series codes.

I don't have the details readily available. You might try contacting Hans Ruppel at Los Alamos, who at one time had a nice set of notes working out the details of the surface integral around a control volume.

Good luck!