Rheology of Fluids: Newtonian to Non Newtonian

Transcription

1 0/26 Rheology of Fluids: Newtonian to Non Newtonian Ali Najafi University of Zanjan, Zanjan Instituet for advanced Studies in Basic Sciences May 2015

2 1/26 Agenda: Fluid: Definition Rheology: Elementary concepts Navier-Stokes Equation Basic Solutions Dilute Suspension Non Newtonian OldRoyd Model

3 2/26 Fluid: Deborah number, a measure of fluidity Relaxation time in response to an externally applied force Observation time

4 3/26 Rheology: How does the system respond to an external force?

5 4/26 Viscosity: How does the system respond to an externally applied force? x Linear.and. isotropic Newtonian Nonlinear.or. anisotropic Non-Newtonian uy σ yz =η =η γ z

6 5/26 Some examples

7 6/26 Active Suspensions: Bacillus Subtilis Aranson, PRL 2009 Chlamydomonas Rafai, PRL 2010

8 7/26 Rheometry: Elementary tools Two cylinders, cone and plate Dimensional Analysis for water

9 7/26 Rheometry: Elementary tools Two cylinder, cone and plate Dimensional Analysis for water

10 7/26 Rheometry: Elementary tools Two cylinder, cone and plate Dimensional Analysis for water

11 8/26 Non-Newtonian

12 9/26 Toward the governing equations: Velocity field Acceleration field

13 9/26 Toward the governing equations: Velocity field Acceleration field a (x,t )=( u(x +δ x, t +δ t) u(x, t ))/ δ t =

14 9/26 Toward the governing equations: Velocity field Acceleration field a (x,t )=( u(x +δ x, t +δ t) u(x, t ))/ δ t = Stress Tensor: i'th component of the the force exerted on a surface that is pointed to j direction

15 10/26 Governing equations Continuity equation

16 11/26 Governing Equations: Phenomenological model for stress tensor uy σ yz =η =η γ z

17 12/26 Hydrodynamics:

18 13/26 Colloid's Universe: Low Reynolds Regime:

19 13/26 Colloid's Universe: Low Reynolds Regime: 3 3 L=0.1μ m, V =1μ m/s, η=10 Pa S, ρ=10 Kg/m 3

20 14/26 Hydrodynamics: Point force:

21 14/26 Hydrodynamics: Point force: Force Dipole:

22 15/26 Multipole Expansion:

23 9/24 Sink and Source another singular solutions:

24 16/26 Moving Sphere:

25 16/26 Moving Sphere:

26 17/26 Sphere in shear flow:

27 17/26 Sphere in shear flow:

28 17/26 Sphere in shear flow:

29 18/26 Passive suspension: Einstein's theory? In the presence of colloids, the applied force should do much work to establish the same velocity profile as in the bare fluid

30 18/26 Passive suspension: Einstein's theory? In the presence of colloids, the applied force should do much work to establish the same velocity profile as in the bare fluid Einstein 1906

31 18/26 Passive suspension: Einstein's theory? In the presence of colloids, the applied force should do much work to establish the same velocity profile as in the bare fluid Perrin+Einstein 1909 Einstein 1906

32 18/26 Passive suspension: Einstein's theory? In the presence of colloids, the applied force should do much work to establish the same velocity profile as in the bare fluid Perrin+Einstein 1909 Einstein 1906 Interactions: Batchelor 1972 Fluctuations: Batchelor 1977

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37 19/26 Force Dipole

38 20/26 Simple shear flow

39 20/26 Simple shear flow Simple shear flow + Sphere

40 21/26 Non Newtonian Behavior: Microscopic anisotropy I: Intrinsic Liquid Crystal II: Induced anisotropy Polymeric solutions

41 21/26 Non Newtonian Behavior: Microscopic anisotropy I: Intrinsic Liquid Crystal Molecular relaxation time II: Induced anisotropy Polymeric solutions Time scale for external force N-N Molecules will not have enough time to response and reach their isotropic structure Relaxation time for water= We expect to see N-N behavior in water for frequencies >

42 22/26 Phenomenological Description:

43 22/26 Phenomenological Description: Normal Stress Differences Macroscopic Manifestation: Weisenberg Effect

44 22/26 Phenomenological Description: Normal Stress Differences Macroscopic Manifestation: Weisenberg Effect Shear thickening: Viscosity increases by increasing external shear نشاسته ذرت و آب : برش وشکسان Shear thinning: Viscosity decreases by increasing external shear سس گوجه : برش روان

45 22/26 Phenomenological Description: Normal Stress Differences Macroscopic Manifestation: Weisenberg Effect Shear thickening: Viscosity increases by increasing external shear نشاسته ذرت و آب : برش وشکسان Shear thinning: Viscosity decreases by increasing external shear سس گوجه : برش روان Viscoelastic Behavior Kelvin Voigt Model

46 23/26 Molecular Anisotropy: Microscopic Model

47 23/26 Molecular Anisotropy: Microscopic Model

48 23/26 Molecular Anisotropy: Microscopic Model

49 23/26 Molecular Anisotropy: Microscopic Model

50 23/26 Molecular Anisotropy: Microscopic Model

51 23/26 Molecular Anisotropy: Microscopic Model

52 23/26 Molecular Anisotropy: Microscopic Model After averaging over noise, it is easy to show:

53 24/26 OldRoyd Model

54 24/26 OldRoyd Model Response to a simple shear flow:

55 24/26 OldRoyd Model Response to a simple shear flow:

56 24/26 OldRoyd Model Response to a simple shear flow:

57 25/26 Response to an harmonic shear flow: G(ω)=G r (ω)+ig i (ω) Gr (ω)=? Gi (ω)=?

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