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By A. J. Chorin, J. E. Marsden (auth.)

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7) allows rather complicated behavior. We shall now discuss the three-dimensional geometry a bit further. A voptex tube consists of a two-dimensional surface tangent to nowhere with vortex lines drawn through each point of the ~, bounding curve C of integral curves of direction. 2-5. 2-5 In fluid mechanics it is customary to be sloppy about this definition and make tacit assumptions to the effect that the tube really "looks like" a tube. e. related to a disc by a one-to-one invertible differentiable transformation) and that the resulting tube is diffeomorphic to the product of the disc and the real line.

As with a line with irrational slope on a torus). Thus our assertion about vortex tubes "ending" is correct if we interpret "ending" properly. But the reader is cautioned that this may not be all that can happen, and that this time honored statement is not at all a proved theorem. The difference between the two-dimensional and three-dimensional conservation laws for vorticity we have discussed, reflects some main differences between two-dimensional and three-dimensional flow. 7) in two-dimension is a helpful tool in establishing a rigorous theory of existence and uniqueness for the Euler (and later Navier-Stokes) equations.

11) as a special case. 7) allows rather complicated behavior. We shall now discuss the three-dimensional geometry a bit further. A voptex tube consists of a two-dimensional surface tangent to nowhere with vortex lines drawn through each point of the ~, bounding curve C of integral curves of direction. 2-5. 2-5 In fluid mechanics it is customary to be sloppy about this definition and make tacit assumptions to the effect that the tube really "looks like" a tube. e. related to a disc by a one-to-one invertible differentiable transformation) and that the resulting tube is diffeomorphic to the product of the disc and the real line.

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