Vortex Filament Equation for some Regular Polygonal Curves

Field: Mathematics (Physical Sciences)

One of the most interesting phenomena in fluid literature is the occurrence and evolution of vortex filaments. Some of their examples in the real world are smoke rings, whirlpools, and tornadoes. For an ideal fluid, there have been several models and governing equations to describe this evolution; however, due to its simplicity and geometric properties, the vortex filament equation (VFE) has recently gained attention. As an approximation of the dynamics of a vortex filament, the equation first appeared in the work of Da Rios at the beginning of the twentieth century. This model is usually known as the local induction approximation (LIA). In this work, we examine the evolution of VFE for regular polygonal curves both from a numerical and theoretical point of view in the Euclidean as well as hyperbolic geometry. In the first part of the thesis, we observe the evolution of the Vortex Filament equation taking $M$-sided regular polygons with nonzero torsion as initial data in the Euclidean space. Using algebraic techniques, backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; moreover, the multifractal trajectory of the point X(0, t) is not planar and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as the one already established for circles. This is accomplished by showing the existence of variants of the so-called Riemann’s non-differentiable function that are as close to smooth curves as desired when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the self-similar solutions of VFE have finite renormalized energy. In the rest of the work, we delve into the hyperbolic setting and examine the evolution of VFE for a planar $l$-polygon, i.e., a regular planar polygon in the Minkowski 3-space $\mathbb{R}^{1,2}$. Unlike in the Euclidean case, a planar $l$-polygon is open which makes the problem more challenging from a numerical point of view. After trying several numerical methods, we conclude that a finite-difference discretization in space combined with an explicit Runge--Kutta method in time, gives the best numerical results both in terms of efficiency and accuracy. On the other hand, using theoretical arguments, we recover the evolution algebraically, and thus, we show the agreement between the two approaches. During the numerical evolution, it has been observed that the trajectory of a corner is multifractal and as the parameter $l$ goes to zero, it converges to the Riemann’s non-differentiable function. Furthermore, as in the Euclidean case, we provide strong numerical evidence to show that at infinitesimal times, the evolution of VFE for a planar $l$-polygon as an initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only we can compute the speed of the center of mass of the planar $l$-polygon theoretically, the relationship also reveals important properties of the trajectory of its corners which we compare with its equivalent in the Euclidean case. Finally, a nonzero torsion in the hyperbolic case, yields two different kinds of helical polygonal curves, however, with the numerical and theoretical techniques developed so far, we are able to address them as well. This remains part of the future work of the thesis.