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Chapter 7: Generalized Newtonian Fluids

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Chapter 7: Generalized Newtonian Fluids Generalized Newtonian Fluid CM4650 3/21/2018 Chapter 7: Generalized Newtonian fluids CM4650 Polymer Rheology Carreau- Michigan Tech Yassuda GNF log positionposition of break of break on onscale is determinedscale is determined by by o slope is determined by n curvaturecurvature here here is is determineddetermined byby a log 1 1 © Faith A. Morrison, Michigan Tech U. Develop Non-Newtonian Constitutive Equations So Far: • We seek constitutive equations for non- Newtonian fluids. • We start with the Newtonian constitutive equation • We “modify it” and make the Fake-O model. • We checked out the model to see how we did. 2 © Faith A. Morrison, Michigan Tech U. 1 Generalized Newtonian Fluid CM4650 3/21/2018 ≡ constant When we tried the model out with elongational flow material functions, something odd happened … 3 © Faith A. Morrison, Michigan Tech U. Steady Elongational Flow Material Functions Imposed Kinematics: 0, ≡ 0 0 constant Material Stress Response: ̃ ̃ 0 Material Functions: Elongational ̃ ̃ ≡ Viscosity Alternatively, ̅ 4 © Faith A. Morrison, Michigan Tech U. 2 Generalized Newtonian Fluid CM4650 3/21/2018 2) Predict what Newtonian fluids would do. 1. Choose a material function 2. Predict what Newtonian fluids would do 3. See what non-Newtonian fluids do 4. Hypothesize a 5. Predict the material function 6. Compare with what non-Newtonian fluids do 7. Reflect, learn, revise model, repeat. 2) Predict what Newtonian fluids would do. ? 5 © Faith A. Morrison, Michigan Tech U. 2) Predict what Newtonian fluids would do. 1. Choose a material function 2. Predict what Newtonian fluids would do 3. See what non-Newtonian fluids do 4. Hypothesize a 5. Predict the material function 6. Compare with what non-Newtonian fluids do 7. Reflect, learn, revise model, repeat. 2) Predict what Newtonian fluids would do. 3 Trouton viscosity (Newtonian fluids) • Constant • Three times shear viscosity 6 © Faith A. Morrison, Michigan Tech U. 3 Generalized Newtonian Fluid CM4650 3/21/2018 Investigating Stress/Deformation Relationships (Rheology) 1. Choose a material function 2. Predict what Newtonian fluids would do 3. See what non-Newtonian fluids do 4. Hypothesize a 5. Predict the material function 6. Compare with what non-Newtonian fluids do 7. Reflect, learn, revise model, repeat. 5) Predict the material function (with new ) What does the Fake-O-model predict for steady elongational viscosity? 7 © Faith A. Morrison, Michigan Tech U. 5) Predict the material function What does the Fake-O-model predict for steady elongational viscosity? Garbage. is tied to shear flow only. 8 © Faith A. Morrison, Michigan Tech U. 4 Generalized Newtonian Fluid CM4650 3/21/2018 Investigating Stress/Deformation Relationships (Rheology) 1. Choose a material function 2. Predict what Newtonian fluids would do 3. See what non-Newtonian fluids do 4. Hypothesize a 5. Predict the material function 6. Compare with what non-Newtonian fluids do 7. Reflect, learn, revise model, repeat. What if we make the following replacement? → Maybe this “guess” will work in more types of This at least can be written for flows. any flow and it is equal to the shear rate in shear flow. But it assumes a coordinate system. 9 © Faith A. Morrison, Michigan Tech U. 4) Hypothesize a constitutive equation Fake-O-model Observations •The model contains parameters that are specific to shear flow – makes it impossible to adapt for elongational or mixed flows •Also, the model should only contain quantities that are independent of coordinate system (i.e. invariant) We are identifying constraints on our freedom to “make up” constitutive models. 10 © Faith A. Morrison, Michigan Tech U. 5 Generalized Newtonian Fluid CM4650 3/21/2018 4) Hypothesize a constitutive equation In Chapter 7, we find a solution: • take out the shear rate and • replace with the magnitude of the rate-of- deformation tensor (which is related to the second invariant of that tensor). This is going to work. 11 © Faith A. Morrison, Michigan Tech U. Chapter 7: Generalized Newtonian fluids CM4650 Polymer Rheology Carreau- Michigan Tech Yassuda GNF log positionposition of break of break on onscale is determinedscale is determined by by o slope is determined by n curvaturecurvature here here is is determineddetermined byby a log 1 12 © Faith A. Morrison, Michigan Tech U. 6 Generalized Newtonian Fluid CM4650 3/21/2018 Current Goal: Develop new constitutive models by trial and error: • Account for rate-dependence • Accommodate known constraints Known • no parameters specific to flow • only use quantities that are constraints: independent of coordinate system 13 © Faith A. Morrison, Michigan Tech U. Develop new constitutive models by trial and error: Rate-Dependence Back to our main goal: Constitutive Equation – an accounting for all stresses, all flows stress tensor Rate-of- Newtonian fluids: (all deformation tensor flows) In general: In the general case, f needs to be a non-linear function (in time and position) What should we choose Answer: something that for the function f? matches the data. Let’s start with the steady shear data. 14 © Faith A. Morrison, Michigan Tech U. 7 Generalized Newtonian Fluid CM4650 3/21/2018 Develop new constitutive models by trial and error: Rate-Dependence Non-Newtonian, Inelastic Fluids log First, we concentrate on the observation that shear viscosity depends on shear rate. o ≡ loglog Non-Newtonian ≡ viscosity, We will design a constitutive equation that shear rate predicts this behavior in shear flow 15 © Faith A. Morrison, Michigan Tech U. Develop new constitutive models by trial and error: Rate-Dependence Newtonian Constitutive Equation v 1 For Newton’s experiment (shear flow): v 0 0 123 v 0 1 0 x2 11 12 13 v1 21 22 23 0 0 x 2 31 32 33 123 0 0 0 We could make this equation give 123 the right answer (shear thinning) in steady shear flow if we substituted a function of shear rate for the constant viscosity. 16 © Faith A. Morrison, Michigan Tech U. 8 Generalized Newtonian Fluid CM4650 3/21/2018 Develop new constitutive models by trial and error: Rate-Dependence Generalized Newtonian Fluid (GNF) constitutive equation v1 FLOW SHEAR v 0 0 0 0 123 00 000 GNF v v v v v v 2 1 1 2 1 3 1 v v FLOWS ALL x1 x2 x1 x3 x1 2 GNF is v v v v v v3 designed in 1 2 2 2 3 2 123 shear flow; x2 x1 x2 x2 x3 fingers v1 v3 v3 v2 v3 crossed it’ll 2 x x x x x work in all 3 1 2 3 3 123 flows. 17 © Faith A. Morrison, Michigan Tech U. Develop new constitutive models by trial and error: Rate-Dependence Constitutive Equation – an accounting for all stresses, all flows A simple choice for : Generalized Newtonian Fluids (GNF) GNF is designed in 1 shear flow; ≡ : fingers 2 crossed it’ll work in all flows. 18 © Faith A. Morrison, Michigan Tech U. 9 Generalized Newtonian Fluid CM4650 3/21/2018 Develop new constitutive models by trial and error: Rate-Dependence Generalized Newtonian Fluids (GNF) What do we pick for ? •Something that matches the data; •Something simple, so that the calculations are easy 19 © Faith A. Morrison, Michigan Tech U. Develop new constitutive models by trial and error: Rate-Dependence In processing, the high-shear-rate behavior is the most important. + linear 131 kg/mole branched 156 kg/mole linear 418 kg/mol branched 428 kg/mol Figure 6.3, p. 172 Piau et al., linear and branched PDMS 20 © Faith A. Morrison, Michigan Tech U. 10 Generalized Newtonian Fluid CM4650 3/21/2018 Develop new constitutive models by trial and error: Rate-Dependence Power-law In shear flow ≡ model for viscosity (in shear flow) On a log-log plot, this would give a straight line: dv log logmn1 log 1 dx2 Y B M X 21 © Faith A. Morrison, Michigan Tech U. Develop new constitutive models by trial and error: Rate-Dependence steady Power-law shear Non-Newtonian shear flow viscosity model for viscosity ≡ (GNF) ≡ log Newtonian , 1 slope = n-1 shear thinning , 1 log 22 © Faith A. Morrison, Michigan Tech U. 11 Generalized Newtonian Fluid CM4650 3/21/2018 Develop new constitutive models by trial and error: Rate-Dependence Power-Law Generalized Newtonian Fluid (PL-GNF) m or K = consistency index (for Newtonian) n = power-law index (n = 1 for Newtonian) ≡ (Usually 0.5< n <1) 23 © Faith A. Morrison, Michigan Tech U. Calculate flow fields with GNF constitutive equations Let’s use it! EXAMPLE: r Pressure-driven flow z A of a Power-Law cross-sectioncross-section A: A: Generalized Newtonian fluid in a r tube P0 z •steady state •well developed •long tube L vz(r) fluid R g PL 24 © Faith A. Morrison, Michigan Tech U. 12 Generalized Newtonian Fluid CM4650 3/21/2018 Calculate flow fields with GNF constitutive equations Velocity field Poiseuille flow of a power-law fluid: 1 1 1 RLg P P n R r n v r o L 1 z 2Lm 1 R 1 n 25 © Faith A. Morrison, Michigan Tech U. Calculate flow fields with GNF constitutive equations Solution to Poiseuille flow in a tube incompressible, power-law fluid n=1.0 2.0 0.8 0.6 1.5 0.4 z,av 0.2 /v z v 1.0 0.0 0.5 0.0 0 0.2 0.4 0.6 0.8 1 1.2 r / R 26 © Faith A. Morrison, Michigan Tech U. 13 Generalized Newtonian Fluid CM4650 3/21/2018 Calculate flow fields with GNF constitutive equations Solution to Poiseuille flow in a tube incompressible, power-law fluid 1 0.8 0.6 max /v z n= 0.2 v 0.4 n= 0.4 n= 0.6 0.2 n= 0.8 n= 1 0 0 0.2 0.4 0.6 0.8 1 1.2 r/R 27 © Faith A.
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