What is it about?

Finding analytical and numerical solutions to the Navier-Stokes equations has been a longstanding and will be an everlasting theme in fluid dynamics because of its intrinsic complexity as well as its physical, mathematical and applicational importance. In the article, the author's interest is on a vortex in free space, which bears close relevance to such phenomena in nature as tornado, typhoon, galaxy, turbulence and so on. Fifty years ago, Bellamy-Knights derived an ordinary differential equation reduced from the Navier-Stokes equations and successfully employed his equation to find various types of unsteady axisymmetric vortex solutions. However, two riddles have been left. One is that his equation and the solutions do not connect with the known steady vortex solutions, which contradicts our intuition. The other is that the exact solutions to the Bellamy-Knights equation were isolated from the numerical solutions in the space of boundary conditions. Are the exact solutions really peculiar? The work in the article was started aiming at resolving these mysteries.

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Why is it important?

By appropriately redefining the variables, the Bellamy-Knights equation is transformed to an equation equivalent to the equation of one-dimensional motion of a particle subject to conserved and non-conserved forces. The new form of equation involves a new parameter - equation parameter - whose continuous variation deforms the Bellamy-Knights equation to the one for steady vortices, thereby resolving the first riddle. The new form of the equation also enables us to find pseudo-sinusoidal oscillations inherent in newly discovered vortex solutions, each of which are labelled by the cycles of oscillations. For a given equation parameter, the boundary conditions for these solutions form an interesting self-similar set of curves, which is called a branches - that connect separate boundary conditions for different solutions. The branches are densely packed and two arbitrarily close boundary conditions on different branches give two finitely different solutions. After all, the boundary conditions cannot be arbitrary. This is an intriguing example of the Cauchy problem.


Unveiling the whole structure of the manifold formed by the branches, although very important, is quite a troublesome work that is beyond the capability of a single desktop computer. I hope many researchers to have interests in this topic and explore the field in their own ways. I also expect that the the results of the present work may provide a hint in tackling the well-known Millennium Problem on the Navier-Stokes equations.

Dr. Koichi Takahashi
Reasearch Institute for Data Science, Tohoku Gakuin University

Read the Original

This page is a summary of: Three-dimensional unsteady axisymmetric vortex solutions to the Bellamy-Knights equation and the distribution of boundary conditions, AIP Advances, August 2022, American Institute of Physics,
DOI: 10.1063/5.0106368.
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