On the Force acting on a Vortex in a Non-uniform Stream of a Viscous Fluid

[摘要] In the present short note, we investigate the force acting on a vortex in a non-uniform stream of a viscous fluid. According to Bernoulli's theorem the pressure at a point in a certain stream-line becomes larger where the stream-lines converge. Hence, a vortex standing perpendicularly to the stream is forced to the side where the stream-lines converge. By the same reason two vortices revolving in the same sense repulse each other and vice versa. But actually, we observe facts that contradict with the above law, e.g. two circular cylinders revolving in the same sense attract each other if the viscosity is large and two vortices in water attract each other and amalgamate. The latter fact is called Okada's law in the meteorological circle of Japan. In order to remove the contradiction, we give here a note. Assuming a two-dimensional stationary state and integrating the equation of motion of an incompressible viscous fluid along a stream-line, we get: where the notations are those commonly used and C is a constant. The integral-term becomes effective, when μ becomes sufficiently large. Investigating this term, we obtain the result that does not contradict with the fact.

[学科分类] 大气科学