Level set equation in axisymmetric 2-D flow
 

I am solving spherical compressible flows under the assumption of symmetry. Before this task, I have solved 2-D cylindrical fluid flows with r and z. To extend it, I attempted axisymmetric fluid equations but I failed to continue the computation after a certain instant. I am guessing that this is due to the difference between the cylidrical level set equation and the axisymmetric fluid equaiton.

To remedy this difference between fluid flow and interface motion, I realized that I need to solve the axisymmetric level set equation rather than the 2-D level set equation. I am looking for the axisymmetric expression of the level set equation. Can anyone see that expression before in the literature? If so, please let me know what I have to refer or change for my purpose.

Thanks in advance.