0/26 Rheology of Fluids: Newtonian to Non Newtonian Ali Najafi University of Zanjan, Zanjan Instituet for advanced Studies in Basic Sciences May 2015
1/26 Agenda: Fluid: Definition Rheology: Elementary concepts Navier-Stokes Equation Basic Solutions Dilute Suspension Non Newtonian OldRoyd Model
2/26 Fluid: Deborah number, a measure of fluidity Relaxation time in response to an externally applied force Observation time
3/26 Rheology: How does the system respond to an external force?
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
5/26 Some examples
6/26 Active Suspensions: Bacillus Subtilis Aranson, PRL 2009 Chlamydomonas Rafai, PRL 2010
7/26 Rheometry: Elementary tools Two cylinders, cone and plate Dimensional Analysis for water
7/26 Rheometry: Elementary tools Two cylinder, cone and plate Dimensional Analysis for water
7/26 Rheometry: Elementary tools Two cylinder, cone and plate Dimensional Analysis for water
8/26 Non-Newtonian
9/26 Toward the governing equations: Velocity field Acceleration field
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 =
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
10/26 Governing equations Continuity equation
11/26 Governing Equations: Phenomenological model for stress tensor uy σ yz =η =η γ z
12/26 Hydrodynamics:
13/26 Colloid's Universe: Low Reynolds Regime:
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
14/26 Hydrodynamics: Point force:
14/26 Hydrodynamics: Point force: Force Dipole:
15/26 Multipole Expansion:
9/24 Sink and Source another singular solutions:
16/26 Moving Sphere:
16/26 Moving Sphere:
17/26 Sphere in shear flow:
17/26 Sphere in shear flow:
17/26 Sphere in shear flow:
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
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
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
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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19/26 Force Dipole
20/26 Simple shear flow
20/26 Simple shear flow Simple shear flow + Sphere
21/26 Non Newtonian Behavior: Microscopic anisotropy I: Intrinsic Liquid Crystal II: Induced anisotropy Polymeric solutions
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 >
22/26 Phenomenological Description:
22/26 Phenomenological Description: Normal Stress Differences Macroscopic Manifestation: Weisenberg Effect
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
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
23/26 Molecular Anisotropy: Microscopic Model
23/26 Molecular Anisotropy: Microscopic Model
23/26 Molecular Anisotropy: Microscopic Model
23/26 Molecular Anisotropy: Microscopic Model
23/26 Molecular Anisotropy: Microscopic Model
23/26 Molecular Anisotropy: Microscopic Model
23/26 Molecular Anisotropy: Microscopic Model After averaging over noise, it is easy to show:
24/26 OldRoyd Model
24/26 OldRoyd Model Response to a simple shear flow:
24/26 OldRoyd Model Response to a simple shear flow:
24/26 OldRoyd Model Response to a simple shear flow:
25/26 Response to an harmonic shear flow: G(ω)=G r (ω)+ig i (ω) Gr (ω)=? Gi (ω)=?
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