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

apparent viscosity decrease : isolated particle dynamicsChaotic regime conductivity enhancement : E Quincke rotation in suspensions : E Q

The Quincke rotation + E E EE 11    Relaxation equation equilibrium dipole particle convection Maxwell time polarisation coefficient (free charges) < 0

 Electric torque:  Viscous torque :  Stationary state : Stationary solution E EE  E (E<E c ) vv EE   v  viscous coefficient 0 ECEC E  st

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

Laboratory water wheel (R. Malkus, 1972) : Pr=5 r=31 flow  masse current Maxwell time  M gravity electric field gravity center electric dipole inertia viscous drag Lorenz equations Variable change : b=1 t * =t/   leaky compartment

Chaotic dynamics Poincaré section (plane X=Y)First return map

Experimental set-up 0-15 kV 1 cm transformer oil laser photodiode Rotor :glass capillary length L=5 cm, radius a=1mm  2 =2.4 Transformer oil : conductivity  = S.m -1 viscosity  =14 mPa.s permittivity  1 =2.1 permittivity  M =150 ms E c =0.97 kV/cm  mecha =60 ms Pr=2.5

Experimental results  (rad.s -1 ) E 2 (kV.cm -1 ) 2 Bifurcation diagram E c exp =2.4 kV/cm First bifurcation : (E c theo =0.97 kV/cm) solid friction Second bifurcation : E chaos exp =6.5 kV/cm (E chaos theo =5.5 kV/cm)  (rad.s -1 ) t (s) E=6.6 kV/cm

Experimental results  k+1  (rad.s -1 )  k  (rad.s -1 ) First return map Lorenz-type chaos numerical experimental