Mark Stremler

The fluid motion produced by a spatially periodic array of identical, axisymmetric, thin-cored vortex rings is investigated. It is well known that such an array moves uniformly without change of shape or form in the direction of the central axis of symmetry, and is therefore an equilibrium solution of Euler's equations. In a frame of reference moving with the system of vortex rings, the motion of passive fluid particles is investigated as a function of the two nondimensional parameters that define this system: $\varepsilon = a/R$, the ratio of minor radius to major radius of the torus-shaped vortex rings, and $\lambda=L/R$, the separation of the vortex rings normalized by their radii. Two bifurcations in the streamline topology are found that depend significantly on $\varepsilon$ and $\lambda$; these bifurcations delineate three distinct shapes of the ``atmosphere'' of fluid particles that move together with the vortex ring for all time. Analogous to the case of an isolated vortex ring, the atmospheres can be ``thin-bodied'' or ``thick-bodied''. Additionally, we find the occurrence of a ``connected'' system, in which the atmospheres of neighboring rings touch at an invariant circle of fluid particles that is stationary in a frame of reference moving with the vortex rings.

The classic problem of three point vortex motion on the plane is revisited by using the interior angles of the vortex triangle, $\theta_{j}$, $j=1,2,3$, as the key system variables instead of the lengths of the triangle sides, $s_j$, as has been used classically. Similar to the classic approach, the relative vortex motion can be represented in a phase space, with the topology of the level curves characterizing the motion.  In contrast to the classic approach, the alternate formulation gives a compact, consistent phase space representation and facilitates comparisons of vortex motion in a co-moving frame. This alternate formulation is used to explore the vortex behavior in the two canonical cases of equal vortex strength magnitudes, $\Gamma_{1} = \Gamma_{2} = \Gamma_{3}$ and $\Gamma_{1} = \Gamma_{2} = -\Gamma_{3}$.

We investigate the finite-time collapse of three point vortices in the plane utilizing the geometric formulation of three-vortexmotion from Krishnamurthy, Aref and Stremler (2018) Phys. Rev. Fluids 3, 024702. In this approach, the vortex system is described in terms of the interior angles of the triangle joining the vortices, the circle that circumscribes that triangle, and the orientation of the triangle. Symmetries in the governing geometric equations of motion for the general three-vortex problem allow us to consider a reduced parameter space in the relative vortex strengths. The well-known conditions for three-vortex collapse are reproduced in this formulation, and we show that these conditions are necessary and sufficient for the vortex motion to consist of collapsing or expanding self-similar motion. The geometric formulation enables a new perspective on the details of this motion. Relationships are determined between the interior angles of the triangle, the vortex strength ratios, the (finite) system energy, the time of collapse, and the distance traveled by the configuration prior to collapse. Several illustrative examples of both collapsing and expanding motion are given.