Triaxial particles in shear flows : a Floquet analysis using a two-vector formulation. Jean-Régis Angilella, LUSAC - Cherbourg, France

Transcription

1 Triaxial particles in shear flows : a Floquet analysis using a two-vector formulation Jean-Régis Angilella, LUSAC - Cherbourg, France in collaboration with : Jonas Einarsson & Bernhard Mehlig (Physics Department, Gothenburg University, Sweden ) Fabien Candelier (IUSTI Marseille, France) Perspectives: Abdel Zalt (LUSAC-Cherbourg, France) 1 / 24

2 . Introduction Many flows carry tiny solid objects with a very small particle Reynolds number (inertia-free limit). Geophysical flows : aerosols, sediments, etc. Industrial flows : chemical engineering, combustion, etc. In many cases particles are non-spherical: rods, platelets,...complex orientational dynamics. 2 / 24

3 . Axisymmetric ellipsoid or dumbbell in a simple shear flow z e e X y e Z x In the absence of inertia and brownian motion, these particles have periodic orientations (Jeffery 1922). Goal: Investigate the stability of some orbits when triaxiality is present. (See also Hinch & Leal 1979 ; arin et al. 1997). 3 / 24

4 . Motion equations Tiny ellipsoid with semi-axes a ; b; c, in flow ~u. Angular velocity vector with respect to the laboratory frame: (t ~ ) Its components in a cartesian basis attached to the object: X,, Z Jeffery 1922 (inertia-free object): where g = r~u X = b2 g Z c 2 g Z b 2 + c 2 ; (1) = c2 g XZ a 2 g ZX c 2 + a 2 ; (2) Z = a 2 g X b 2 g X a 2 + b 2 (3) 4 / 24

5 . Motion equations The orientational dynamics is extremely complex: periodic motion, quasi-periodic, chaotic (arin et al. 1997). _ sin sin + _ cos = b 2 g Z c 2 g Z b 2 + c 2 ; (4) _ sin cos _ sin = c 2 g XZ a 2 g ZX c 2 + a 2 ; (5) with g ij = g ij (; a 2 g _ cos + _ X b 2 g X = (6) a 2 + b 2 z ; ). Z θ G y flow ψ φ X 5 / 24

6 . Complex dynamics of non-spherical objects: examples a=1, b=2, c=0.05 t n e Z t n 6 / 24

7 . Complex dynamics of non-spherical objects: examples e Z a=1, b=2, c=0.05 t n t n 7 / 24

8 . Complex dynamics of non-spherical objects: examples e Z a=1, b=2, c=0.05 t n t n 8 / 24

9 . Goal: re-visit the dynamics in vector form for triaxial objects. Axisymmetric object (c = b): _~e X = B~e X (~e X :B~e X )~e X where B = (a 2 r~u c 2 r~u T )=(a 2 + c 2 ) (Hinch & Leal 1971). z e X y x 9 / 24

10 . Goal: re-visit the dynamics in vector form for triaxial objects. Axisymmetric object (c = b): _~e X = B~e X (~e X :B~e X )~e X where B = (a 2 r~u c 2 r~u T )=(a 2 + c 2 ) (Hinch & Leal 1971).! to be generalized to triaxial objects. z e e X y e Z x 10 / 24

11 . The dynamics in vector form for triaxial objects. Starting from the rotation vector of non-inertial triaxial object, and since _ ~e X = ~ ~e X : Leads to: = c2 g XZ a 2 g ZX c 2 + a 2 = _ ~e X :~e Z ; (7) Z = a 2 g X b 2 g X a 2 + b 2 = _ ~e X :~e (8) _~e a 2 g X b 2 g X c 2 g XZ a 2 g ZX X = a 2 + b 2 ~e c 2 + a 2 ~e Z Eliminate g XZ ~e Z = r~u T ~e X g XX ~e X g X ~e : 11 / 24

12 . The dynamics in vector form for triaxial objects. Leads to: S = (r~u + r~u T )=2 _~e X = f 1 (~e :S~e X )~e +B~e X {z } "triaxial term" (~e X :B~e X )~e X where B = (a 2 r~u c 2 r~u T )=(a 2 + c 2 ) Same treatment for ~e _ : _~e = f 2 (~e :S~e X )~e X +C~e {z } "triaxial term" and f 1 = 2a 2 (c 2 b 2 ) (a 2 + b 2 )(a 2 + c 2 ). (~e :C~e )~e where C = (b 2 r~u c 2 r~u T )=(b 2 + c 2 ) 2b 2 (c 2 a 2 ) and f 2 = (a 2 + b 2 )(b 2 + c. 2 ) 12 / 24

13 . The dynamics in vector form for triaxial objects. If ~e X = (t ) ~e {z} x +(t )~e y + (t )~e z 2Lab: Frame ~e = (t ) ~e {z} x +(t )~e y + (t )~e z 2Lab: Frame then the dynamics in the plane of shear (; ; ; ) is independent of the the z (vorticity) coordinates (; ). 4 degrees of freedom (not minimal) with cubic non-linearities. 13 / 24

14 . The dynamics in vector form for triaxial objects. Set: ~e X = (t )~e x + (t )~e y + (t )~e z ; (9) ~e = (t )~e x + (t )~e y + (t )~e z (10) then, the exact triaxial inertia-free dynamics is: _ = _ = _ = _ = a 2 a 2 c 2 a 2 + c 2 a 2 + c ( + ); 2 (11) c 2 a 2 c 2 a 2 + c 2 a 2 + c ( + ); 2 (12) b 2 b 2 c 2 b 2 + c 2 b 2 + c ( + ); 2 (13) c 2 b 2 c 2 b 2 + c 2 b 2 + c 2 2 {z } periodic dynamics (J. orbits) + f 2 2 ( + ); (14) 14 / 24

15 . APPLICATION: stability of orbit = z, for all a ; b ; c Set: ~e X (t ) := ~ N 0 (t ) + ~ N 1 (t ) and ~e (t ) := ~e z + ~ P 1 (t ) With: _~N 0 = B ~ N 0 ( ~ N 0 :B ~ N 0 ) ~ N 0 (Jeffery orbit), then: _~N 1 = ( ~ N 0 :B ~ N 0 ) ~ N 1 + f 1 ( ~ P 1 :S ~ N 0 )~e z + (quad : terms);(15) _~P 1 = C ~ P 1 + f 2 ( ~ P 1 :S ~ N 0 ) ~ N 0 + (quad : terms) (16) are the departure from the J. orbit. 15 / 24

16 . APPLICATION: stability of orbit = z, for all a ; b ; c Finally: ~e X (t ) := ~ N 0 (t ) + ~ N 1 (t ) and ~e (t ) := ~e z + ~ P 1 (t ) _~N 1 = ( ~ N 0 :B ~ N 0 ) ~ N 1 + f 1 ( ~ P 1 :S ~ N 0 )~e z ; (17) _~P 1 = C ~ P 1 + f 2 ( ~ P 1 :S ~ N 0 ) ~ N 0 (18) Of the form _ ~q = A(t )~q with A(t ) = A(t + T ) Floquet theory: ~q(t P ) = i C i e i t ~pi (t ), where p i s are T -periodic and exp( i T ) are the eigenvalues of the monodromy matrix M(T ) [i.e. _M = A(t ):M(t ); M(0) = I d :]! Re( i ), if non-zero, indicates stability/instability of the orbit. 16 / 24

17 . Result: Floquet diagram of orbit = z c/a Z x z axisymmetric cube/sphere axisymmetric y X unstable axisymmetric stable stable (To be compared to arin et al. 1997) axisymmetric b/a stable unstable 17 / 24

18 . Result: Floquet diagram of orbit = z a = 1; b = 1; c = 0:05 18 / 24

19 . Result: Floquet diagram of orbit = z z c/a Z x axisymmetric cube/sphere axisymmetric y X unstable axisymmetric stable stable axisymmetric b/a stable unstable 19 / 24

20 . Result: Floquet diagram of orbit = z a = 1; b = 1:2; c = 0:05 20 / 24

21 . Result: Floquet diagram of orbit = z z c/a Z x axisymmetric cube/sphere axisymmetric y X unstable axisymmetric stable stable axisymmetric b/a stable unstable 21 / 24

22 . Result: Floquet diagram of orbit = z a = 1; b = 2; c = 0:05 22 / 24

23 . Result: Floquet diagram of orbit = z a = 1; c = 0:05 b = 6 b = 7 23 / 24

24 . Conclusion: bi-vector formulation for triaxial objects. _~e X = f 1 (~e :S~e X )~e +B~e X {z } "triaxial term" (~e X :B~e X )~e X _~e = f 2 (~e :S~e X )~e X +C~e {z } "triaxial term" B = (a 2 r~u c 2 r~u T )=(a 2 + c 2 ); f 1 = C = (b 2 r~u c 2 r~u T )=(b 2 + c 2 ) ; f 2 = (~e :C~e )~e 2a 2 (c 2 b 2 ) (a 2 + b 2 )(a 2 + c. 2 ) 2b 2 (c 2 a 2 ) (a 2 + b 2 )(b 2 + c. 2 ) Simple formulation, cubic non-linearities only. Allows a straightforward application of Floquet theory. Low computational cost! Perspectives: use the bi-vect formulation to analyze chaotic dynamics (with Abdel Zalt, LUSAC-Cherbourg). Brownian objects, Fokker-Planck Eq. for PDF (~e X ; ~e ) 24 / 24