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Yasuhide Fukumoto Last modified date:2018.06.11

Graduate School
Undergraduate School
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Administration Post

Academic Degree
Field of Specialization
fluid mechanics
Outline Activities
I am engaged in the mathematical study of problems arising in fluid
mechanics. Fluid motion governed by the Navier-Stokes equations is a
nonlinear dynamical system of infinite degrees of freedom, with a
hierarchy of modes of various scales coupling with each other, and
exhibits a complicated irregular behavior. From a viewpoint vortices and
waves as fundamental elements, I make an effort to understand fluid
phenomena and then to build mathematical and numerical models for
analyzing them. My interest is also in solar magnetic fields and
magnetohydrodynamics in space.
Recently, I have been working on nonlinear analyses of the
three-dimensional vortex motion. By invoking ideas and machinery in
integrable systems and Hamiltonian dynamical systems, I derived an
accurate formula of translation speed of a vortex ring and discovered
its new instability mode.
I have been appointed as an Editor of Fluid Dyn. Research. Also I have
been a member of organizing committee of a couple of domestic Conferences.

My graduate students are encouraged to tackle with original problems
for their Master Theses. I have a foreign student. I invited a few
foreign scholars from Italy, Croatia and Russia for short terms.

Based on my experience, I wrote the Chapter "Vortex Dynamics" in the
"Dictionary of Flows" (Maruzen, 2004). Since 2005, I have been writing a
series of expository articles ``Fundamentals of Vortex Motion" in
``Nagare", a bimonthly Journal edited by the Japan Society of Fluid
Mechanics. I gave an introductory lecture on "topological aspect of
vortex and its relation to a variational principle" at Mathematical
Physics Summer School 2003, held at Tokyo University, September 2003,
and published its Lecture Note (2005).
Research Interests
  • Three-domentional stability of vortices in a fluid
    keyword : Elliptic vortex, Vortex ring, Three-dimensional stablility, Krein's theory of Hamiltonian spectra, WKB method
    1999.04I investigate three-dimensional instability of a vortex tube to infinitesimal perturbations of long and short wavelengths. An elliptic vortex column serves as a universal model for strained vortices. Three-dimensional instability of a vortex tube subjected to a weak strain is investigated from the viewpoint of Krein's theory for Hamiltonian spectra. Energetic aspect of instability is clarified. Kirchhoff's elliptic vortex is subjected to finite strain. Three-dimensional linear stability of Kirchhoff's elliptic vortex, based on the Euler equation, is numerically investigated. To supplement this result, we made a local stability analysis of potential flows, using the WKB method. We verified rigorously that the parametric resonance is absent in this case. A new instability mode originating from vortex-line curvature is found for a vortex ring. This finding is supported by a local stability analysis using the WKB method..
  • Motion of a vortex ring in a viscous fluid
    keyword : Vortex ring, Navier-Stokes equations, Matched asymptotic expansions, Dyson's method
    1996.08A large-Reynolds-number asymptotic solution of the Navier-Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. The method of matched asymptotic expansions is extended to a third order in a small parameter, the ratio of core to ring radii. It is demonstrated that viscosity acts to expand, linearly in time, the ring radius. By exploiting the formulas of kinetic energy and impulse, Fraenkel-Saffman's first-order formula (1970) for translation speed is successfully extended to third order..
  • Three-dimensional dynamics of a vortex filament in a fluid
    keyword : Vortex filament, Biot-Savart law, Localized induction approximation, Integrable system, Laser-matter interaction
    1998.01Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated based on the Euler equations. The method of matched asymptotic expansions in a small parameter, the ratio of core to curvature radii, is extended to a higher order and thereby torsion and nonuniformity in curvature along the center curve is taken into consideration. We devise a systematic method for an asymptotic development of the Biot-Savart law. The velocity of a vortex filament is derived to third order. In the localized induction approximation, the resulting equation of filament motion is reducible to a completely integrable evolution equation among the localized induction hierarchy. We have studied the string-like structure generated by irradiating nano-second XeCl excimer laser pulses on Co-coated substrate. An attempt is made at gaining, from a frozen picture, information on the vortex-filament dynamics by estimating the hydrodynamic parameters of the shear layer of molten metal surface..
Current and Past Project
  • Topological vorticity dynamics in the physical and biological sciences
  • We study the stability of vortices with swirl. The effects of swirl on a helical vortex tube which is a model of rotating wingtip vortices is investigated by linear stability analysis and DNS.
Academic Activities
1. Handbook of Fluid Mechanics, Ver.2, Chap.5 "Vortex" (ed. by the Japan Society of Fluid Mechanics, Maruzen)
by Yasuhide Fukumoto, Takeshi Miyazaki.
1. Yasuhide Fukumoto, M. Hirota, Elliptical instability of a vortex tube and drift current induced by it, Physica Scripta, Vol.T132, 014041 (9 pages), 2008.10.
2. Yasuhide Fukumoto, H. K. Moffatt, Kinematic variational principle for motion of vortex rings, Physica D, Vol.237, No.14-17, pp.2210-2217, 2008.08.
3. Yasuhide Fukumoto, V. L. Okulov, The velocity field induced by a helical vortex tube, Physics of Fluids, 10.1063/1.2061427, 17, 10, Vol. 17, No.10, pp. 107101-1-19, 2005.10.
4. Yasuhide Fukumoto, The three-dimensional instability of a strained vortex tube revisited, Journal of Fluid Mechanics, 10.1017/S0022112003006025, 493, 287-318, Vol.493, pp.287-318, 2003.10.
5. Yuji Hattori, Yasuhide Fukumoto, Short-wavelength stability analysis of thin vortex rings, Physics of Fluids, 10.1063/1.1606446, 15, 10, 3151-3163, Vol.15, No.10, pp.3151-3163, 2003.10.
6. Yasuhide Fukumoto, Three-dimensional motion of a vortex filament and its relation to the localized induction hierarchy, European Physical Journal B, 10.1140/epjb/e2002-00279-5, 29, 2, 167-171, Vol.29, No.2, pp.167-171, 2002.09.
7. Yasuhide Fukumoto, H. K. Moffatt, Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity, Journal of Fluid Mechanics, 10.1017/S0022112000008995, 417, 1-45, Vol.417, pp.1-45, 2001.08.
8. Yasuhide Fukumoto, Stationary configurations of a vortex filament in background flows, Proceedings of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences, Vol.453, No.1961, pp.1205-1232, 1997.06.
9. Yasuhide Fukumoto, Mitsuharu Miyajima, The localised induction hierarchy and the Lund-Regge equation, Journal of Physics A - Mathmatical and General, 10.1088/0305-4470/29/24/025, 29, 24, 8025-8034, Vol.29, No.24, pp.8025-8034, 1996.12.
1. Yasuhide Fukumoto, Rong Zou, Azimuthal and helical magnetorotational instabilities to non-axisymmetric perturbations, Turbulent Mixing and Beyond Workshop: Mixing in Rapidly Changing Environment - Probing Matter at the Extremes, 2014.08.
2. Yasuhide Fukumoto, Are all the topological invariants representable as cross helicities?, JSPS/UK Meeting "Topological Vorticity Dynamics in the Physical Sciences", 2013.09.
3. Yasuhide Fukumoto, Hirofumi Sakuma, A unified view of topological invariants of barotropic and baroclinic fluids and their application to formal stability analysis of three-dimensional ideal gas flows, IUTAM symposium "Topological Fluid Dynamics", 2012.07, Integrals of an arbitrary function of the vorticity, two-dimensional topological invariants of an ideal barotropic fluid, take different guise from the helicity. Noether's theorem associated with the particle relabeling symmetry group leads us to a unified view that all the topological invariants of a barotropic fluid are variants of the cross helicity. Baroclinic fluid flows admit, as the Casimir invariants, a class of integrals including an arbitrary function of the entropy and the potential vorticity. A consideration is given to them from the view point of Noether's theorem. We then develop a new energy-Casimir convexity method for a baroclinic fluid, and establish a novel linear stability criterion, to three-dimensional disturbances, for equilibria of general rotating flows of an ideal gas without appealing to the Boussinesq approximation. By exploiting a larger class of the Casimir invariants, we have succeeded in ruling out a term including the gradient of a dependent variable from the energy-Casimir function. For zonally symmetric flows, the resulting criterion is regarded as an extended Richardson number criterion for stratified rotating shear flows with compressibility taken into account..
Educational Activities
I direct students in the Master and Doctor Courses. Seminar classes are
held for Graduate students, the third and fourth year students of the
Undergraduate Course. In seminars, students read textbooks in the field
of fluid mechanics and dynamical systems chosen in conformity with their
knowledge and interest. The students are trained to make original
research for their Master Theses.
I give lectures for graduate and undergraduate students of mathematics,
and have courses on "industrial mathematics" in Faculty of Engineering
and on "calculus" in general education. For the fourth year students of
mathematics, I give an introductory lecture on "fluid mechanics" and
"dynamical systems". For graduate students, topics are chosen from the
latest topics in "vortex dynamics" and "magnetohydrodynamics".
I gave an intensive course in Hiroshima University (2003).