About existence of stationary points for the Arnold-Beltrami-Childress (ABC) flow

Transcription

1 bout eistence of stationary points for the rnold-eltrami-hildress () flow Sergey V. Ershkov Institute for Time Nature Eplorations M.V. Lomonosov's Moscow State University Leninskie gory - Moscow 999 Russia sergej-ershkov@yande.ru The eistence of stationary points for the dynamical system of -flow is considered. The -flow a three-parameter velocity field that provides a simple stationary solution of Euler's equations in three dimensions for incompressible inviscid fluid flows is the prototype for the study of turbulence (it provides a simple eample of dynamical chaos). ut nevertheless between the chaotic trajectories of the appropriate solutions of such a system we can reveal the stationary points the deterministic basis among the chaotic behaviour of -flow dynamical system. It has been proved the eistence of point for two partial cases of parameters { }: ) = = ; ) = (² + ² = ). Moreover dynamical system of -flow allows 3 points of such a type depending on the meanings of parameters { }. Keywords: rnold-eltrami-hildress () flow helical flow stationary points.

2 . Introduction the -flow. The rnold-eltrami-hildres flow [-] a well known helical steady solution of Euler equations for ideal incompressible flow of Newtonian fluids should be presented in the artesian coordinates as below ( = const): u y u u 3 y (.) - where u = {u₁ u₂ u₃} is the vector of flow velocity field a particular simple (steady) case of helical flow: w u u w 0 - here is the constant coefficient given by the initial conditions ( 0). Such a type of solution (.) is also known as eltrami flow [3-4] i.e. a fluid motion in which the vorticity vector is parallel to the velocity vector at every point of the fluid. s for the domain in which the flow occurs let us consider the periodic boundary conditions and the limited domain. If we remember the originating of denotation for the components of velocity field: u d d y u u 3 d t d t d d t - it should yield a system of ordinary differential equations as below d d t d y d t d d t y y.

3 3 - which is proved to have not an analytical solutions but moreover it reveals a dynamical chaos among the trajectories of appropriate solutions of such a system [5-6]. The -flow a three-parameter velocity field that provides a simple stationary solution of Euler's equations in three dimensions for incompressible inviscid fluid flows can be considered to be a prototype for the study of turbulence: the -flow provides a simple eample of dynamical chaos in spite of the simple analytical epression for each of the components of a solution [7].. nalysis of the eistence of stationary points in -flow. Let us eplore the fied (stationary) points of -flow system (.) which could be associated with the local stability of the dynamical trajectories for such a system. It means that there should be valid appropriate conditions d/dt = dy/dt = d/dt = 0 [8]; so we obtain from the equations of -flow dynamical system (.) as below: System of Eqs. (.) yields three types of solutions depending on the meanings of parameters { } as presented below (.4) ) ( (.3) ) ( (.) ) ( y y y (.). y y

4 4 For the solutions of a type (.) we could obtain from system (.) as below If we assume = = the last epressions for could be simplified properly Solutions (.) (.6) should be substituted (under assumptions = = ) to the system (.) for the resulting checking of such a solution. 3. nalysis for the eistence of stationary points in -flow. So if there eists a fied (stationary) point of -flow dynamical system (.) for the case = = the condition below should be valid for the range of meanings of parameter : - which is obviously valid for all the range of meanings of parameter from inequality shown in (.6 ); besides we should choose the sign - in a left part of epression (3.) above. Indeed let us present the appropriate plots of the left and right parts of Eq. (3.) at Fig.- below: (.6) arc С С С (.5) arc (3.) arc С С

5 Fig.. Plot of the left part of Eq. (3.) here we denote =. Fig.. Plot of the right part of Eq. (3.) here we denote =. In general case (.5) if there eist a fied (stationary) points of -flow dynamical system (.) the condition below should be valid for the appropriate ranges of meanings of parameters { }: 5

6 6 - where we could choose = (moreover we could choose any meanings of which should be a scale-parameter for the meanings of parameters { } in (.)). Let us present the appropriate plots of the left and right parts of Eq. (3.) at Fig.3-4 below (we should choose the sign - in a left part of epression (3.) above): Fig.3. Plot of the left part of Eq. (3.) here we denote = = y. (3.) arc

7 7 Fig.4. Plot of the right part of Eq. (3.) here we denote = = y. We can see at the Figs. 3-4 the only surface for the coinciding of the left and right parts of Eq. (3.) which should apparently be the intersecting circle (see Fig.3) at ero plane: Moreover we could obviously assume that an equality below is valid for such an intersecting circle - for which the inequalities from (.) should be obviously valid as shown below: ) (3.3 arc 0. 0 ) ( ) ( (3.4)

8 So ug Eq. (3.4) we obtain from (3.3) (3.5) - where (3.5) is obviously valid for chosen meanings of { } according to (3.4). 4. Discussion. The main result which should be outlined is the eistence of stationary points for the dynamical system (.) of -flow. The -flow a three-parameter velocity field that provides a simple stationary solution of Euler's equations in three dimensions for incompressible inviscid fluid flows can be considered to be a prototype for the study of turbulence - the -flow provides a simple eample of dynamical chaos [7]. ut nevertheless between the chaotic trajectories of the appropriate solutions of such a system we can reveal the stationary points the stationary kernel or deterministic basis among the chaotic behaviour of -flow dynamical system. If we assume = = the epressions for coordinates y (just for one of 3 types of solution (.)-(.4)) should be as below С y 4 arc С С (4.) - so at least point is possible in case = = for each from the range of meanings in (4.) just for one of 3 types of solution (.)-(.4). 8

9 If we assume = the epressions for coordinates y should be as below y arc (4.) - where according to (3.4) we should choose - so at least point is also possible in case = for each meaning of { } above just for one of 3 types of solution (.)-(.4). 5. onclusion. The eistence of stationary points for the dynamical system of -flow is considered. The -flow a three-parameter velocity field that provides a simple stationary solution of Euler's equations in three dimensions for incompressible inviscid fluid flows is the prototype for the study of turbulence (it provides a simple eample of dynamical chaos). ut nevertheless between the chaotic trajectories of the appropriate solutions of such a system we can reveal the stationary points the deterministic basis among the chaotic behaviour of -flow dynamical system. It has been proved the eistence of point for two partial cases of parameters { }: ) = = ; ) = (² + ² = ). Moreover dynamical system of -flow allows 3 points of such a type depending on the meanings of parameters { }. 9

10 References: []. Landau L.D.; Lifshit E.M. (987) Fluid mechanics ourse of Theoretical Physics 6 (nd revised ed.) Pergamon Press ISN []. Lighthill M. J. (986) n Informal Introduction to Theoretical Fluid Mechanics Oford University Press ISN [3]. Saffman P. G. (995) Vorte Dynamics ambridge University Press ISN X. [4]. Milne-Thomson L.M. (950) Theoretical hydrodynamics Macmillan. [5]. ogoyavlenskij O. Fuchssteiner. (005) Eact NSE solutions with crystallographic symmetries and no transfer of energy through the spectrum Journal of Geometry and Physics Volume 54 Issue 3 July 005 Pages [6]. rnold V.I. (965) Sur la topologie des 'ecoulements stationnaires des fluids parfaits. R cad Sci Paris [7]. T.Dombre et al. (986) haotic streamlines in the flows. J Fluid Mech 67 p [8]. Kamke E. (97) Hand-book for Ordinary Differential Eq. Moscow: Science. 0