7. Conditions where the surface streamlines meet
So far, we have used the idealization of a point source or sink within the context of a 2-dimensional flow field. The consequence of this is that the velocity increases to very high values as the source or sink centre is approached.
In order to connect with a proper 3-dimensional analysis let us write:
and introduce the surface divergence:
still using the spherical approximation r = R for the surface itself.
We can now invert Eq. (54) and regard the quantity S as being the source-function for the 2-dimensional flow pattern on the surface. In particular we can, within a certain small distance measured along the surface from the source or sink centre, take S to be non-zero and finite (S0 for source and S0 for sink). Outside this `near zone' we can assume S = 0 as previously and well outside it we can expect the solutions to be substantially the same as before.
Inside the `near zones' however the situation has changed relative to that in the 2-dimensional case. Instead of containing a singularity the central portion of the source (or sink) region will now contain a stagnation point at which the `open' surface streamlines can meet. At this meeting point the Jacobi energy will be the same for the various streamlines. Hence if it were possible to connect every point on the object surface by means of a streamline to the source or sink then it would be correct, as noted in K 97, to infer from Bernoulli's equation that the Jacobi energy must be constant over the whole surface.
However when a change of topology occurs closed streamlines cover a large part of the surface and the above argumentation can not be used. The Coriolis forces, being the cause of the topological change, are also the cause of the non-constancy (in general) of the Jacobi energy over the surface.
© European Southern Observatory (ESO) 1999
Online publication: December 4, 1998