Rheology of Fluids: Newtonian to Non Newtonian
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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
33 19/26
34 19/26
35 19/26
36 19/26
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)=?
58 26/26 از توجه شما ممنونم
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