Summary
We show that if the equations governing the fluid motion in a trailing vortex is linearized as by Batchelor, more than one solution can be constructed. Within the framework of the linear theory, there is no criterion to determine which solution is to be used. To clarify the situation, we formulate the Navier-Stokes equations in parabolic coordinates and seek asymptotic solutions valid far downstream. By insisting that the interaction of the swirl with the uniform stream be a first order effect, we obtain the first two terms in the asymptotic expansions for the Stokes stream function and the angular momentum. The result thus obtained differs from that given by Batchelor in that the axial velocity defect decays algebraically.
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References
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On leave from Department of Mathematics, McGill University, Montreal 110, Quebec, Canada.
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Tam, K.K. A note on the flow in a trailing vortex. J Eng Math 7, 1–6 (1973). https://doi.org/10.1007/BF01535263
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DOI: https://doi.org/10.1007/BF01535263