Two-dimensional uniform vortices that rotate without deforming in an isochoric, inviscid fluid are investigated. They have polygonal symmetry and are usually considered of the same kind of the famous Kirchhoff elliptical one. By following this idea, their shapes and rotation speeds are calculated through a new integral approach. It is heuristically proved, by examining the co-rotating streamlines, that the similitude with the elliptical vortex is not complete, due to the appearance of external singularities in the Schwarz functions of the polygonal boundaries. Once the effects of these singularities have been accounted for, exactly steady vortices are obtained. In this way, for any number of “sides”, a continuous family of rotating, steady polygonal vortices is obtained, parametrized in the ratio between minimum and maximum radial coordinates of the boundary. Solutions exist above a critical value of this ratio, at which the boundary ceases to be differentiable.

研究了在等渗，无粘性流体中旋转而不会变形的二维均匀涡旋。它们具有多边形对称性，通常被认为与著名的基尔霍夫椭圆形相同。遵循这个想法，它们的形状和转速是通过一种新的积分方法计算出来的。通过检查同向旋转的流线，通过试探法证明，由于多边形边界的Schwarz函数中出现外部奇异点，因此椭圆形涡旋的相似性并不完整。一旦考虑了这些奇异性的影响，就可以获得完全稳定的涡旋。这样，对于任意数量的“边”，可以获得连续的旋转的稳定多边形涡流族，参数化为边界的最小和最大径向坐标之比。存在高于该比率的临界值的解，在该临界值处边界不再是可微的。