On bubble rings and ink chandeliers

We introduce variable thickness, viscous vortex filaments. These can model such varied phenomena as underwater bubble rings or the intricate "chandeliers" formed by ink dropping into fluid. Treating the evolution of such filaments as an instance of Newtonian dynamics on a Riemannian configuration manifold we are able to extend classical work in the dynamics of vortex filaments through inclusion of viscous drag forces. The latter must be accounted for in low Reynolds number flows where they lead to significant variations in filament thickness and form an essential part of the observed dynamics. We develop and document both the underlying theory and associated practical numerical algorithms.

[1]  Bo Ren,et al.  Visual Simulation of Multiple Fluids in Computer Graphics: A State-of-the-Art Report , 2018, Journal of Computer Science and Technology.

[2]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[3]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[4]  Fabrice Neyret,et al.  Simulation of smoke based on vortex filament primitives , 2005, SCA '05.

[5]  Timothy J. Pedley,et al.  The toroidal bubble , 1968, Journal of Fluid Mechanics.

[6]  Alexandre J. Chorin,et al.  Hairpin Removal in Vortex Interactions II , 1990 .

[7]  E. Vouga,et al.  Discrete viscous threads , 2010, SIGGRAPH 2010.

[8]  Ivor Grattan-Guinness,et al.  Landmark writings in western mathematics, 1640-1940 , 2005 .

[9]  Robert Bridson,et al.  Linear-time smoke animation with vortex sheet meshes , 2012, SCA '12.

[10]  Ulrich Pinkall,et al.  Filament-based smoke with vortex shedding and variational reconnection , 2010, SIGGRAPH 2010.

[11]  Peter S. Bernard,et al.  Vortex filament simulation of the turbulent coflowing jet , 2009 .

[12]  Dominique Hulin,et al.  Analysis on Riemannian manifolds and Ricci curvature , 2004 .

[13]  Jeffrey S. Marshall,et al.  A general theory of curved vortices with circular cross-section and variable core area , 1991, Journal of Fluid Mechanics.

[14]  E. Vouga,et al.  Discrete viscous threads , 2010, ACM Trans. Graph..

[15]  Mark J. Stock,et al.  Impact of a vortex ring on a density interface using a regularized inviscid vortex sheet method , 2008, J. Comput. Phys..

[16]  G. Pedrizzetti,et al.  Vortex Dynamics , 2011 .

[17]  Boris Khesin,et al.  Vortex sheets and diffeomorphism groupoids , 2018, Advances in Mathematics.

[18]  Thomas S. Lundgren,et al.  Oscillations of drops in zero gravity with weak viscous effects , 1988, Journal of Fluid Mechanics.

[19]  Zhiyong Yuan,et al.  Animating Wall-Bounded Turbulent Smoke via Filament-Mesh Particle-Particle Method , 2018, IEEE Transactions on Visualization and Computer Graphics.

[20]  H. Bateman,et al.  SOME RECENT RESEARCHES ON THE MOTION OF FLUIDS , 1915 .

[21]  J. J. Sir Thomson,et al.  A Treatise on the Motion of Vortex Rings: An Essay to Which the Adams Prize Was Adjudged in 1882, in the University of Cambridge , 1883 .

[22]  Ching Chang,et al.  The motion of a buoyant vortex filament , 2018, Journal of Fluid Mechanics.

[23]  Uriel Frisch,et al.  From Newton’s mechanics to Euler’s equations , 2008 .

[24]  H. Helmholtz.  Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. , 1858 .

[25]  J. Marsden,et al.  Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids , 1983 .

[26]  V. Arnold,et al.  Topological methods in hydrodynamics , 1998 .

[27]  Ulrich Pinkall,et al.  Filament-based smoke with vortex shedding and variational reconnection , 2010, ACM Trans. Graph..

[28]  Eitan Grinspun,et al.  Surface-only liquids , 2016, ACM Trans. Graph..

[29]  P S Bernard,et al.  Turbulent flow properties of large-scale vortex systems , 2006, Proceedings of the National Academy of Sciences.

[30]  Toshiya Takami,et al.  Breakup and deformation of a droplet falling in a miscible solution. , 2016, Physical review. E.

[31]  Theodor Kaluza.  Zum Unitätsproblem der Physik , 1921 .

[32]  Alexandre J. Chorin.  Regular ArticleHairpin Removal in Vortex Interactions II , 1993 .

[33]  Eitan Grinspun,et al.  Double bubbles sans toil and trouble , 2015, ACM Trans. Graph..

[34]  Joseph John Thomson,et al.  V. On the formation of vortex rings by drops falling into liquids, and some allied phenomena , 1886, Proceedings of the Royal Society of London.

[35]  O. Klein,et al.  Quantentheorie und fünfdimensionale Relativitätstheorie , 1926 .

[36]  J. Turner,et al.  Buoyant vortex rings , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[37]  William E. Schiesser,et al.  Linear and nonlinear waves , 2009, Scholarpedia.

[38]  Uriel Frisch,et al.  Cauchy’s almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow , 2014, 1402.4957.

[39]  Thomas S. Lundgren,et al.  Area-varying waves on curved vortex tubes with application to vortex breakdown , 1989, Journal of Fluid Mechanics.

[40]  Tee Tai Lim,et al.  Motion of a bubble ring in a viscous fluid , 2012 .

[41]  Donald B. Bliss,et al.  Slender-body analysis of the motion and stability of a vortex filament containing an axial flow , 1971, Journal of Fluid Mechanics.

[42]  Thomas S. Lundgren,et al.  Vortex ring bubbles , 1991, Journal of Fluid Mechanics.

[43]  Tatyana S. Krasnopolskaya,et al.  Vortex rings: history and state of the art , 2012 .

[44]  Daniel Bernoulli,et al.  Hydrodynamica, sive, de viribus et motibus fluidorum commentarii. Opus academicum ab auctore, dum Petropoli ageret, congestum , 1973 .

[45]  Louis Rosenhead,et al.  The Spread of Vorticity in the Wake Behind a Cylinder , 1930 .

[46]  G. K. Mikhailov.  Daniel Bernoulli, Hydrodynamica (1738) , 2005 .

[47]  Markus H. Gross,et al.  Lagrangian vortex sheets for animating fluids , 2012, ACM Trans. Graph..

[48]  G. I. Taylor.  Formation of a Vortex Ring by Giving an Impulse to a Circular Disk and then Dissolving it Away , 1953 .

[49]  D. W. Moore,et al.  The motion of a vortex filament with axial flow , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[50]  T. Powers,et al.  The hydrodynamics of swimming microorganisms , 2008, 0812.2887.

[51]  J. Burgers.  A mathematical model illustrating the theory of turbulence , 1948 .

[52]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[53]  R. Wood,et al.  Vortex Rings , 1901, Nature.