WSEAS Transactions on Fluid Mechanics
Print ISSN: 1790-5087, E-ISSN: 2224-347X
Volume 11, 2016
Binormal Evolution of Curves with Prescribed Velocity
Authors: ,
Abstract: At the end of the 19th century, Kirchhoff studied dynamical problems involving vortex flows of inviscid incompressible fluids focusing on flows having the shape of a vortex tube (vortex filaments). In 1906, Da Rios, a student of Levi-Civita, analyzed the motion of a vortex filament and obtained the remarkable equation describing its evolution, which, under mild conditions, is equivalent to the so called binormal evolution equation. Motivated by this, in this work we use fundamental facts of the theory of submanifolds to analyze the evolution of curves under binormal flows with curvature dependent velocity in pseudo-riemannian 3-space forms. The compatibility conditions for these systems are given by the Gauss-Codazzi equations, which here are expressed with respect to a geodesic coordinate system in terms of the Frenet curvatures of the evolving curves. Then, an existence result is derived from the Fundamental Theorem of submanifolds. Moreover, we show the connection between travelling wave solutions of the Gauss-Codazzi equations and the Frenet-Serret dynamics of curves. In fact, travelling wave solutions of the Gauss-Codazzi equations are shown to lead to the Euler-Lagrange equations of extremal curves for curvature dependent energies with a penalty on the total torsion and the length (generalized Kirchhoff centerlines). A characterization of generalized Kirchhoff centerlines in terms of Killing vector fields allows us to construct binormal evolution surfaces with prescribed velocity by using them as initial conditions for the evolution. Binormal surfaces obtained in this way evolve without change in shape. Finally, we particularize the previous findings to three significant cases which give rise to Hasimoto surfaces, Hopf tubes, and constant mean curvature surfaces.
Search Articles
Keywords: binormal flow, curve evolution, Frenet-Serret dynamics, extremal curves, submanifolds, real space forms
Pages: 112-120
WSEAS Transactions on Fluid Mechanics, ISSN / E-ISSN: 1790-5087 / 2224-347X, Volume 11, 2016, Art. #14