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The Quincke rotor : An experimental model for the Lorenz attractor

The Quincke rotor : An experimental model for the Lorenz attractor. E. Lemaire, L. Lobry, F. Peters. LPMC, Groupe « Fluides Complexes ». silica particle (100 m m). transformer oil. E ~ 1 kV/cm. f ~ 100 Hz. E. E. -. -. +. -. +. -. +. -. E. +. -. -. +. -. -. +. +. +.

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The Quincke rotor : An experimental model for the Lorenz attractor

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  1. The Quincke rotor : An experimental model for the Lorenz attractor E. Lemaire, L. Lobry, F. Peters LPMC, Groupe « Fluides Complexes » silica particle (100 mm) transformer oil E ~ 1 kV/cm f ~ 100 Hz E

  2. E - - + - + - + - E + - - + - - + + + Quincke rotation in suspensions : apparent viscosity decrease : Q conductivity enhancement : isolated particle dynamics Chaotic regime

  3. The Quincke rotation GE - - - - - - - - - - - + E E - + + + + + + + + + + + + < 0 e1 s e2  Relaxation equation polarisation coefficient (free charges) Maxwell time equilibrium dipole particle convection

  4. GE wst Gv -Gv E GE GE (E<Ec) 0 E EC w Stationary solution  Electric torque:  Viscous torque : a viscous coefficient  Stationary state :

  5. The Quincke rotor z y cylinder transformer oil E=2 EC E=4 EC Relaxation equation (2D) : Mechanical equation :

  6. flow current 40 40 t masse Maxwell time tM 30 30 r Chaos gravity electric field 20 20 electric dipole gravity center 10 10 0 0 5 5 10 10 15 15 20 20 Pr inertia viscous drag Lorenz equations Variable change : t*=t/tM b=1 Laboratory water wheel (R. Malkus, 1972) : leaky compartment Pr=5 r=31

  7. 50 (a) Z 45 40 -20 0 20 X 12 (c) 11 10 9 8 8 10 12 Chaotic dynamics Poincaré section (plane X=Y) First return map

  8. laser photodiode transformer oil 1 cm 0-15 kV Experimental set-up Rotor : glass capillary length L=5 cm, radius a=1mm permittivity e2=2.4 Transformer oil : conductivity s=2.7.10-10 S.m-1 permittivity e1=2.1 viscosity h=14 mPa.s tM=150 ms Pr=2.5 tmecha=60 ms Ec=0.97 kV/cm

  9. W (rad.s-1) E2 (kV.cm-1)2 Experimental results Bifurcation diagram First bifurcation : Ec exp=2.4 kV/cm (Ec theo=0.97 kV/cm) solid friction Second bifurcation : Echaos exp=6.5 kV/cm (Echaos theo=5.5 kV/cm) W (rad.s-1) t (s) E=6.6 kV/cm

  10. |Wk+1| (rad.s-1) |Wk| (rad.s-1) 80 73.5 |Wk+1| (rad.s-1) 66.5 60 53.5 53.5 66.5 80 |Wk| (rad.s-1) Experimental results First return map experimental Lorenz-type chaos numerical

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