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Symmetry transformations of an ideal steady fluid flow determined by a potential function. (English) Zbl 1349.76027

Summary: First, we consider a transformation {\(\Xi\)} of 3D trajectories (fluid particle paths) of inviscid steady flows using the dual stream function approach for the (local) representation of velocity fields \(\vec{u}(x, y, z) = \nabla \lambda \times \nabla \mu\). This enables to derive the equation governing the deformation of trajectories by the gradient field \(\vec{\xi} = \nabla \mu\) along the surface \(\lambda(x, y, z) = \lambda_{0}\). In fact, \(\Xi\) is a symmetry transformation and it looks formally like the filament motion which preserves the curvature. Then, we investigate in detail a fine structure of a Lie algebra associated with an extension of the transformation \(\Xi\) which creates a visual appearance of sliding stream surfaces \(\lambda(x, y, z) = \lambda_{0}\) along itself. The minimal set of generating differential invariants is found. This set consists of a single invariant which coincides with a Hamiltonian function.{
©2016 American Institute of Physics}

MSC:

76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
17B81 Applications of Lie (super)algebras to physics, etc.
35Q31 Euler equations
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[1] Frewer, M.; Oberlack, M.; Grebenev, V. N., The dual stream function representation of an ideal steady fluid flow and its local geometric structure, Math. Phys. Anal. Geom., 17, 1-2, 3-25 (2014) · Zbl 1302.76073 · doi:10.1007/s11040-014-9138-5
[2] Bushgens, S. S., Geometry of an ideal incompressible steady fluid flow, Izv. Math., 12, 481-512 (1948) · Zbl 0031.04402
[3] Yih, C. S., Stream functions in three-dimensional flows, La Houlle Blanche, 3, 445-450 (1957)
[4] Dubrovin, B. A.; Fomenko, T. A.; Novikov, S. P., Modern Geometry - Methods and Applications, Part 1 (1984) · Zbl 0529.53002
[5] Barbarosie, S. S., Representation of divergence-free vector fields, Q. Appl. Math., 69, 309-316 (2011) · Zbl 1220.26008 · doi:10.1090/S0033-569X-2011-01215-2
[6] Hasimoto, H., A soliton on a vortex filament, J. Fluid. Mech., 51, 3, 477-485 (1972) · Zbl 0237.76010 · doi:10.1017/S0022112072002307
[7] 7.I. S.Gromeka, “Some cases of incompressible fluid motion,” Proc. Kazan. State Univ.3, 107 (1952);reprint in: I. S.Gromeka, Collected Works (USSR Academy of Sciences Publication, Moscow, 1952) (in Russian).
[8] Arnold, V. I.; Khesin, B. A., Topological Methods in Hydrodynamics (1998) · Zbl 0902.76001
[9] Arnold, V. I., Sur la topologie des e’coulements stationnaires des fluides parfaits, C. R. Acad. Sci. Paris, 261, 17-20 (1965) · Zbl 0145.22203
[10] Zeitunyan, R. K., Theory of three-dimensional vorticity flows of ideal fluids, Numer. Methods Continuum Mech., 5, 71-101 (1977)
[11] Keller, J., A pair of stream fucntions for three-dimensional vortex flows, ZAMP, 47, 821-836 (1996) · Zbl 0887.76013 · doi:10.1007/BF00920036
[12] Zakharov, V. E.; Kuznetsov, E. A., Hamiltonian formalism for nonlinear waves, Phys.-Usp., 40, 11, 1087-1116 (1997) · doi:10.1070/PU1997v040n11ABEH000304
[13] Bolsinov, A. V.; Borisov, A. V.; Mamaev, I. S., Hamiltonisation of non-holonomic systems in the neighborhood of invariant manifolds, J. Nonl. Dyn., 6, 4, 829-854 (2010)
[14] Marsden, J.; Weinstein, A., Coadjoint orbits, and Clebsch variables for incompressible fluids, Physica D, 7, 305-323 (1983) · Zbl 0576.58008 · doi:10.1016/0167-2789(83)90134-3
[15] Lamb, H., Hydrodynamics (1932) · JFM 26.0868.02
[16] Borisov, A. V.; Mamaev, I. S., Isomorphism and Hamilton representation of some nonholonomi systems, Sib. Math. J., 48, 1, 26-36 (2007) · Zbl 1164.37342 · doi:10.1007/s11202-007-0004-6
[17] Kida, S., A vortex filament moving without change of form, J. Fluid Mech., 112, 397-409 (1981) · Zbl 0484.76030 · doi:10.1017/S0022112081000475
[18] Fomenko, A. T., Integrability and Nonintegrability in Geometry and Mechanics (1988) · Zbl 0675.58018
[19] Yaholom, A., Using fluid variational variables to obtain new analytic solutions with non zero helicity, Proc. IUTAM Topol. Fluid Dyn., II, 10 (2012)
[20] Kuznetsov, E. A., Mixed Lagrangian-Eulerian description of vortical flows for ideal and viscous fluids, J. Fluid Mech., 600, 167-180 (2008) · Zbl 1151.76404 · doi:10.1017/S0022112008000281
[21] Sabitov, I. K., Isometric transformation of a surface inducing conformal maps of the surface into itself, Sb. Math., 189, 1, 111-132 (1998) · Zbl 0914.53006
[22] Schottenloher, M., A Mathematical Introduction to Conformal Field Theory, 759 (2008) · Zbl 1161.17014
[23] Ovsyannikov, L. V., Group Analysis of Differential Equations (1978) · Zbl 0484.58001
[24] Olver, P. J., Generating differential invariants, J. Math. Anal. Appl., 333, 450-471 (2007) · Zbl 1124.53006 · doi:10.1016/j.jmaa.2006.12.029
[25] Simon, U., The Pick invariant in equiaffine differential geometry, Abh. Math. Semin. Univ. Hamburg, 53, 225-228 (1983) · Zbl 0508.53006 · doi:10.1007/BF02941321
[26] Olver, P. J.; Pohjanpelto, J., Differential invariant algebra of Lie pseudo-groups, Adv. Math., 222, 5, 1746-1792 (2009) · Zbl 1194.58018 · doi:10.1016/j.aim.2009.06.016
[27] Pinskiy, D., Sliding deformation: Shape preserving per-vertex displacement, Eurographics, 2010, 4
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