On the orientation of flat triaxial objects in shear flows.

Transcription

1 On the orientation of flat triaxial ojects in shear flows. J.R. Angilella Octoer 13, 011 (LAEGO, group Transfer in Porous Media, Nancy) Goal: predict the orientation of nonspherical particles in shear flow. Analyse the effect of non-axisymmetry Hypotheses: particles are isolated, low-re, ellipsoidal rigid ojects. Method: use Jeffery (19) Hinch & Leal (1979) Yarin et al. (1997) Gauthier, Gondret & Raaud (1998) V 0f x Context: clays: 1 V. Bezuggly, P. Adrian-Marie, C. Baravian (LEMTA), L. Michot, I. Bihanic (LEM)

2 Jeffery 19: Ellipsoid in a non-uniform flow V 0f x Rotation vector of the oject in La. frame: 1 = = g 3 c g 3 = sin sin cos = sin cos sin c c g 13 a g31 3 = a c a g 1 g 1 a = cos Oj / Lao= 1 e X e Y 3 e Z = gij =gij,, : components of V 0f x grad V 0f in e X, e Y, e Z a,, c : half axes

3 Complex ehaviour of elongated non-axisymmetric ojects 3

4 Axisymmetric rod with finite aspect ratio: a=1, =1, c=1/, =0.05 tip of rod y flow x t Initial nutation = Pi/ 4

5 Axisymmetric rod with finite aspect ratio: a=1, =1, c=1/, =0.05 tip of rod z flow y x t Initial nutation = Pi/3 5

6 Triaxial rod with finite aspect ratio: a=1, =, c=1/, =0.05 tip of rod z flow y t x See Yarin et al. JFM

7 Complex dynamics of flat ojects. Under which conditions? 7

8 Asymptotic expansion of Jeffery's equations: 0 Let: and: 0f =k y e x, k =1, a=1, c=, V = X precession nutation intrinsic rotation 1 dx F X.. = F 0 X dt 8

9 Axisymmetric disk with zero thickness: =1, c= =0 Stale equilirium position : vertical flow-tangent platelet =0 = / = intrinsic rotation = 0 Preferential position oserved in sheared clays: Baravian, Michot, Adrian-Marie, Bihanic,... 9

10 Axisymmetric disk with zero thickness: =1, c= =0 Stale equilirium position : vertical flow-tangent platelet =0 = / = intrinsic rotation = 0 Is it stale when > 1 (triaxial oject) and finite thickness? 10

11 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : c=, t = r t 1 Effect of non-axisymmetry = sin t 1 cos cos cos 1 r r t = sin cos sin 1 1 r r t = cos sin sin cos cos

12 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : t = r t c=, 1 (1) self-induced precession = sin t 1 cos cos cos 1 r r t = sin cos sin 1 1 r r t = cos sin sin cos cos 1 4 1

13 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : = sin t c=, t = r t 1 cos 1 cos cos 1 r r t = sin cos sin 1 1 r r t = cos sin sin cos cos 1 4 () precession affects intrinsic rotation 13

14 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : = sin t c=, t = r t 1 cos 1 cos cos 1 r r t = sin cos sin 1 1 r r t = cos sin sin cos cos 1 4 (3) intrinsic rotation causes nutation 14

15 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : = sin t t = r t 1 cos c=, 1 cos cos 1 r r t = sin cos sin 1 1 r r t = cos sin sin cos cos 1 4 (4) nutation modifies intrinsic rotation 15

16 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : t = r t c=, 1 (5) intrinsic rotation affects precession = sin t 1 cos cos cos 1 r r t = sin cos sin 1 1 r r t = cos sin sin cos cos

17 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : t = r t c=, 1 (5) intrinsic rotation affects precession = sin t 1 cos cos cos 1 r r t = sin cos sin 1 1 r r t = cos sin sin cos cos 1 4 Full 3D dynamics (precession,nutation,intrinsic rot ) if finite thickness and non-axisymmetric oject. 17

18 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : c=, t = r t t = sin cos cos O 1 r r t = sin cos sin 1 1 r r t = cos sin sin cos cos 1 4 NB: precession almost decoupled from nutation and intrinsic rot. For all. 18

19 Non-axisymmetric disk with finite thickness: 1, 1 c=, Approximate solution of de-coupled precession equation: Precession equation: t = sin cos t 1 t O 1 4 cos O t tan t 19

20 Non-axisymmetric disk with finite thickness: 1, 1 c=, Approximate solution of de-coupled precession equation: Precession equation: t = sin cos t 1 t O 1 4 cos O t tan t t 0 Consequence #1: when thickness is finite, precession appears. 0

21 Non-axisymmetric disk with finite thickness: 1, c=, 1 Approximate solution of de-coupled precession equation: Consequence #: Approximate, dynamics as a periodically forced D dynamical system 1 r r t = f t g t = t 1 1 f t 1 r g t = A t. X X A t T = A t, T = where: f t =sin t and g t =cos t Nearly vertical platelet : t = r t 1

22 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : Floquet exponents : c=, t = r t 1 1, t t X =C 1 e p 1 t C e p t 1 e i / = eigenvalues of monodromy matrix: B T A t. B t, B 0 =Id B= i 0 vertical position is unstale i 0 vertical position is stale

23 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : r = = e t X pi t i=1 i c=, t = r t Floquet exponents 1 1 =0.05 =0.03 =

24 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : r = = e t X pi t i=1 i c=, t = r t Floquet exponents 1 1 =0.05 =0.03 Vertical rotating position is unstale when: =0.01 finite thickness AND non-axisymmetric platelet with: 0 * * 4 6

25 Non-axisymmetric disk with finite thickness: 1, Nearly vertical platelet : t = r t c=, 1 Gloal view: stale 1=0 a unstale 1 0 c (= ) a 5

26 Numerical verification: =1, c= = =0, 0 =, 0 =10 = intrinsic rotation = 0 e Z flow t y x 6

27 Numerical verification: =1., c= = =0, 0 =, 0 =10 intrinsic rotation e Z z flow y t x 7

28 Numerical verification: =, c= = =0, 0 = 6, 0 =10 intrinsic rotation z e Z flow t y x 8

29 Numerical verification of the re-stailization of vertical position for: * stale * a unstale c = a 9

30 Numerical verification of the re-stailization of vertical position for: * =6 e Z t =7 e Z t 30

31 Chaotic motion: plot of Poincaré sections (nutation,intrinsic rot ) 31

32 Thickness of the stochastic zone: visualization using Poincaré sections t n =n, n ℕ a=1, =1, c=0.05 t n t n 3

33 Thickness of the stochastic zone: visualization using Poincaré sections t n =n, n ℕ a=1, =1, c=0.05 t n a=1, =1., c=0.05 t n t n t n 33

34 Thickness of the stochastic zone: visualization using Poincaré sections t n =n, n ℕ a=1, =1, c=0.05 a=1, =1., c=0.05 t n t n a=1, =, c=0.05 t n t n t n 34 t n

35 Thickness of the stochastic zone: visualization using Poincaré sections t n =n, n ℕ e Z a=1, =, c=0.05 t n 35 t n

36 Thickness of the stochastic zone: visualization using Poincaré sections t n =n, n ℕ e Z a=1, =, c=0.05 t n 36 t n

37 Thickness of the stochastic zone: visualization using Poincaré sections t n =n, n ℕ e Z a=1, =, c=0.05 t n 37 t n

38 Thickness of the stochastic zone: visualization using Poincaré sections t n =n, n ℕ a=1, =, c=0.05 t n Analytical expression of the thickness of the stochastic zone? 38 t n

39 Very flat ojects: c = =0.01 a 39

40 Very flat ojects: c = =0.01 a stale a unstale c = a 40

41 Very flat ojects: c = =0.01 a Vertical rotation is less unstale, ut more likely to e unstale. stale a unstale c = a 41

42 Very flat ojects: =, c= = =0, 0 = 3, 0 =10 t e Z z t y x 4

43 Very flat ojects: =, c= = =0, 0 = t n 3, 0 =10 e Z z y x t n Thin stochastic zone near vertical position. 43

44 Time series: =, c= = =0, 0 = vertical flow-tangent position 3, 0 =10 y t vertical flow-tangent position Apparent random distriution, proaility density peaked at vertical flow-tangent position. NB: non-rownian 44 ojects.

45 CONCLUSION Non-axisymmetry is known to induce an apparent random orientation of flat ojects. How does the thickness of the stochastic zone scale with c/a and /a? What is the PDF of the orientation of a triaxial platelet? In a real dilute suspension, what is the contriution of non-axisymmetry and of rownian motion on the distriution of Euler angles? 45

46 Brownian axisymmetric ojects Brenner & Condiff 1974 Brenner 1974 Vlad Bezuggly (LEMTA, Nancy) C. Baravian, J.R. Angilella (LAEGO, Nancy) P. Adrian-Marie (LEMTA) 46

47 Brownian axisymmetric ojects Brenner & Condiff 1974 Brenner

48 Jeffery equations + noise: c 1 =, a=1= c 1 p = f n S n i t j t ' = D 0 ij t t ' d n = B n n n. B n n dt J n -> stochastic diff. equation: d ni =J i n Gij n j dt 0 n3 n G= n3 0 n1 n n1 0 48

49 Fokker-Planck equation (Stratonovitch) : P n, t = density proaility of n = r,, =r e r, : [ P 1 1 = J P sin J P t sin sin 1 Pe [ ] 1 P 1 P sin sin sin ] Simple shear flow: J = sin sin 4 1 J = cos 1 sin c 1 =, a=1= c 1 k Pe= D0 49

50 Numerical solutions of the Fokker-Planck equation 50

51 Numerical solutions of Fokker-Planck equation Pe = 1 Pe = 0.01 Uniform PDF for low Pe. 51

52 Numerical solutions of Fokker-Planck equation Pe = 500 Pe = 0 5

53 Numerical solutions of Fokker-Planck equation Pe = 500 Pe = 0 PDF peaked at vertical flow-tangent position 53

54 CONCLUSION Both non-axisymmetry and rownian motion induce a random orientation of flat ojects. How does the thickness of the stochastic zone (triaxial platelet) scale with c/a and /a? What is the PDF of orientation in the non-rownian triaxial case? In a real dilute suspension, what is the contriution of non-axisymmetry and of rownian motion on the distriution of Euler angles? 54

55 55

56 Thickness of the stochastic zone: z y x 0 =1.3 0 =1.4 0 = = 0 =1.3 0 =1.5 56

57 Numerical verification: =, c= = =0, 0 = 6, 0 =10 Chaotic scattering near vertical position: z t z e Z flow y x x 57

58 Solution analytique de l'équation de FP Ecoulement élongationnel 58

59 Fokker-Planck equation (Stratonovitch) : d ni =J i n Gij n j dt P n, t = density proaility of n [ : ] G P ik = J i D0 G jk P D0 Gik G jk P ] [ t ni nj ni n j Gik G jk = ni nj Gik G jk = n ij ni n j P =. J P D0 n. P D0 n P D0 n n : P t 59

60 Fokker-Planck equation (Stratonovitch) : Spherical coordinates: P n, t =0 r n = r,, =r e r, n. P=0 and n n : P=r e r e r : P=0 P =. J P D0 P t Adim. par k (1/sec) : 1 P k =. J P P, Pe= t Pe D0 60

61 Ecoulement élongationnel plan / âtonnets non-rowniens u = k x e x k y e y, =1 tan =e k t k 0 tan 0 ± 0 0 tan =e k t tan 0 ± 0 0 ± ± D'où : et : le âtonnet est // Oy61

62 Ecoulement cisaillé plan / disques non-rowniens u =k y e x ; k 0 = 1 tan = k t tan 0 0 tan =tan 0 cos 0 1 tan 0 kt ± 0 0 ± D'où : et : n Oy 6

63 Ecoulement élongationnel plan / disques non-rowniens u = k x e x k y e y, = 1 k 0 tan =e k t tan tan =e k t tan 0 ± 0 0 ± D'où : 0 et : n Ox 63

64 Solution analytique de l'équation de FP Ecoulement élongationnel phi theta sgn k Pe P, =N exp [ sin cos 1 ] Brenner 1974 Orientation préférentielle indépendante de Pe 64

65 Motion equation for the oject and the fluid. d ext G, t = G, t dt t G, t =I G, t ext G, t = S GM P n ds S GM. n ds V T = 0 V V P V = 0 t.v =0 with oundary conditions: M, t = V GM, if M S u x = A. x when x V 65

66 Thickness of the stochastic zone: z y x 0 =1.3 0 =1.4 t 0 = =1.5 t t t 66