Vortex Lines

You think fluid dynamicists are a smart bunch? That after hundreds of years, they have figured out how vortex lines look? Think again.

Shankar Subramaniam, a graduate student working with me, told me that a classical book on fluid dynamics, by Sidney Goldstein of the UK, had a mistake. The book said that vortex lines must be closed curves, as long as they do not end up at the boundary. Truesdell has pointed out that it is wrong: since lines are infinitely thin, they can just keep going without ever returning to the same spot. Even if you give the line some thickness, it may hit itself, but it does not really close up.

So I looked in the text book by Curry that I am using for our graduate Fluid Mechanics class. There was the same error, which I had been repeating uncritically to my students.

Any more text books wrong? You bet. Just look at the list below:

Lamb, Sir H. (1945) Hydrodynamics. 203.
Serrin, J. (1959) Encyclopedia of Physics (Flügge, S. Ed.). 163.
Thwaites, B. (1960) Incompressible Aerodynamics. 32.
Lighthill, M.J. (1963) Laminar Boundary Layers (Rosenhead, L. Ed.) 51.
Ashley, H. & Landahl, M. (1965) Aerodynamics of Wings and Bodies. 8.
Goldstein, S. (1965) Modern Developments in Fluid Mechanics. 27.
Meyer, R.E. (1971) Introduction to mathematical fluid dynamics. 27.
Feynman, R.P. (1975) The Feynman Lectures on Physics, Vol II. 40. (Thanks to Ranjan Muttiah)
Panton, R.L. (1984) Incompressible Flow. 326.
Bertin, J.J. (1984) Engineering Fluid Mechanics. 169.
Bertin, J.J. & Smith, M.L. (1989) Aerodynamics for Engineers. 83.
Saffman, P.G. (1992) Vortex Dynamics. 9.
Batchelor, G.K. (1987) An introduction to fluid mechanics. 75.
Currie, I.G. (1993) Fundamental Mechanics of Fluids. 43.

An author who avoids the issue:

Paterson, A.R. (1989) A First Course in Fluid Dynamics. 88.

In view of the many available texts that make the error, avoiding the issue is not much better than misstating it.

Authors which have a correct (not very clear, though) discussion:

Chorin,A.J. \& Marsden J.E. (1990) A mathematical introduction to fluid dynamics. 28.

I do not have any reference here with a discussion I really like. I lost the Truesdell reference I had, but there should be a discussion in Truesdell (1954) The Kinematics of Vorticity. If I recall correctly, the result is due to Poincare. It is easy to construct an example, by taking vorticity lines to spiral along on toroidal surfaces. Of course, the same observations apply to incompressible streamlines or to electromagnetic fields.

Amazing that so many people who have worked many years in fluid dynamics still are confused how vortex lines should look. It is probably a matter of one textbook author copying it trustingly over from another author. And teachers like me believing it all. Clearly, if someone walks into a fluid mechanics conference and says `Give me your wallet to show that you trust me', he/she will leave a rich person.

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