Question

can someone give me an example of relative motion with vertices travelling in the same direction?

Answer 1

@Loser66 help?

Answer 2

@sleepyjess @Astrophysics @alibaby can anyone help?

Answer 3

@Love_Ranaa @sweetburger can you guys assist?

Answer 4

ABSTRACT

We solve the equations governing the relative motion of three point vortices of arbitrary strength moving on the surface of a sphere of radius R. The system is more general than the corresponding one in the plane [1,3,5,19,28,33,34,37,38] and reduces to it in the limit R → ∞, as long as the three vortices remain sufficiently close to each other during the course of their motion. Instead of using spherical coordinates, we use cartesian coordinates in which the vector χi points from the center of the sphere to the vortex with strength Γi. An important conserved quantity is the center of vorticity vector, which, with no loss of generality, we align with the z-axis. Based on the size of this vector relative to the radius of the sphere, we classify the motion into one of five types: super-radial, radial, sub-radial, degenerate, or a limiting super-radial case. This categorization allows us to draw several conclusions about the qualitative motion of the vortices. We then fully characterize all fixed and relative equilibria on the sphere. For fixed equilibria, the vortices must lie on great circles (geodesics). If the strengths are equal, they form an equilateral triangle. Otherwise, the triangle shape is specified once the strength of the vortices is given. The relative, equilibria are classified as either degenerate (c = 0) or non-degenerate (c ≠ 0). For each type, the shape of the vortex triangle is described and the frequency of rotation is computed. As in the planar problem, it is possible to introduce trilinear coordinates and study the motion in a phase plane, which allows us to locate all the equilibria, as well as to characterize more complex relative dynamics. We then describe self-similar vortex collapse on the sphere, stating necessary and sufficient conditions for collapse to occur, computing the collapse times and vortex trajectories on the route towards collapse. Comparisons are made with the collapse formulas in the plane derived in [1,30].

Answer 5

grr i got it from a website lets hope it helps

Answer 6

@alibaby thanks

Answer 7

welcome