Visualizing Morse Flows and Elliptic Umbilic Catastrophes in N-Roll Mill Fluids Simulated by Smoothed Particle Hydrodynamics
Leandro Tavares da Silva1; Gilson Antônio Giraldi1
1 Laboratório Nacional de Computação Científica - LNCC
doi:10.20906/CPS/CILAMCE2015-0185
Resumo
Smoothed Particle Hydrodynamics (SPH) is a meshfree, Lagrangian method based on particle systems and interpolation theory. Since its invention , SPH has been extensively studied and extended to address scientific and engineering problems in material science, free surface flows, explosion phenomena , heat transfer, mass flow and many other applications [1]. On the other hand, in the 1970's the René Thom's work in structural stability (insensitivity to small perturbations) of dynamical systems allow the born of the subject now known as catastrophe theory. The key idea of this field is to apply topological theory of dynamical systems to model abrupt changes in natural phenomena when changing the parameters that appear in the equations. The remarkable consequence of such development is that, in same contexts, the system can be locally described by one of seven standard forms named the elementary catastrophes [2]. That is the case of the N-roll mill apparatus where N symmetrically rollers are surrounding by a fluid and they rotate at equal speeds and alternating directions. Qualitative analysis of the flow patterns have been performed in the context of catastrophe theory using frames extracted from experimental realizations [2]. In this paper we simulate the N-roll mill flows, for N=2,4,6 using the SPH technique. We revise, by computational experiments, the stability of Morse functions respect to the roll velocity and fluid viscosity for two dimensional flows. Our motivation for this work is to demonstrate the power of the combination of catastrophe theory with modern computational tools in the understanding of scientific and technological problems. The analysis are supported by quantitative results and visual inspection of the flow patterns. References [1] Liu, G.R. and Liu, B. "Smoothed Particle Hydrodynamics: A Meshfree Particle Method", World Scientific, 2003. [2] Tim Postom and Ian Stewart. "Catastrophe Theory and its Applications", Pitman Publishing Limited, London, 1978.
Palavras-chave: meshfree; catastrophe theory; dynamical system