CM465 Lectures -3: Intro, Mathematical //6 Chapter 4: Standard Flows CM465 Polymer Rheology Michigan Tech Newtonian fluids: vs. non-newtonian fluids: How can we investigate non-newtonian behavior? CONSTANT TORQUE MOTOR t fluid Chapter 4: Standard Flows for Rheology v ) ( H V H CM465 Polymer Rheology Michigan Tech H v ( x ) x constant shear x x 3 x, x elongation
CM465 Lectures -3: Intro, Mathematical //6 On to... Polymer Rheology... We now know how to model Newtonian fluid motion, v( x, t), p( x, t) : t v vv p g t v v v T Continuity equation Cauchy momentum equation Newtonian constitutive equation 3 Rheological Behavior of Fluids Non-Newtonian How do we model the motion of Non-Newtonian fluid fluids? t v v vv p g t Continuity equation Cauchy Momentum Equation f ( x, t) Non-Newtonian constitutive equation 4
CM465 Lectures -3: Intro, Mathematical //6 Rheological Behavior of Fluids Non-Newtonian How do we model the motion of Non-Newtonian fluid fluids? t v v vv p g t Continuity equation Cauchy Momentum Equation f ( x, t) Non-Newtonian constitutive equation This is the missing piece 5 Chapter 4: Standard Flows for Rheology Chapter 4: Standard flows Chapter 5: Material Functions Chapter 6: Experimental Data To get to constitutive equations, we must first quantify how non-newtonian fluids behave Chapter 7: GNF Chapter 8: GLVE Chapter 9: Advanced Constitutive equations 6 3
r z r z CM465 Lectures -3: Intro, Mathematical //6 What do we observe? Rheological Behavior of Fluids Newtonian. Strain response to imposed shear stress shear rate is constant x v ( x ) d =constant dt x t r. Pressure-driven flow in a tube (Poiseuille flow) z 3. Stress tensor in shear flow viscosity is constant 4 PR Q 8L Q 4 R =constant 8L P only two components are nonzero 3 7 What do we observe? Rheological Behavior of Fluids Non-Newtonian. Strain response to imposed shear stress shear rate is variable x v ( x ) Release stress x t r. Pressure-driven flow in a tube (Poiseuille flow) z 3. Stress tensor in shear flow viscosity is variable Q Q Q Q f P Normal stresses all 9 components are nonzero 3 3 3 3 33 3 P P P 8 4
CM465 Lectures -3: Intro, Mathematical //6 Non-Newtonian Constitutive Equations We have observations that some materials are not like Newtonian fluids. How can we be systematic about developing new, unknown models for these materials? Need measurements For Newtonian fluids, measurements were easy: shear flow one stress, one material constant, (viscosity) x 3 x x W v (x ) H V 9 Non-Newtonian Constitutive Equations Need measurements For non-newtonian fluids, measurements are not easy: shear flow (not the only choice) Four stresses in shear,,,, Unknown number of material constants in Unknown number of material functions in??? W V x v (x ) H x 3 x 5
CM465 Lectures -3: Intro, Mathematical //6 Non-Newtonian Constitutive Equations Need measurements For non-newtonian fluids, measurements are not easy: We know we need to make measurements to know more, shear flow (not the only choice) Four stresses in shear,,,, Unknown number of material constants in Unknown number of material functions in??? W V x v (x ) H x 3 x Non-Newtonian Constitutive Equations Need measurements For non-newtonian fluids, measurements are not easy: We know we need to make measurements to know more, shear flow (not the only choice) Four stresses in shear,,,, Unknown number of material constants in Unknown number of material functions in??? But, because we do not W V know the functional form of, we don t know what we x v (x ) H need to measure to know x x 3 more! 6
CM465 Lectures -3: Intro, Mathematical //6 Non-Newtonian Constitutive Equations What should we do? 3 Non-Newtonian Constitutive Equations What should we do?. Pick a small number of simple flows Standardize the flows Make them easy to calculate with Make them easy to produce in the lab Chapter 4: Standard flows 4 7
CM465 Lectures -3: Intro, Mathematical //6 Non-Newtonian Constitutive Equations What should we do?. Pick a small number of simple flows. Make calculations 3. Make measurements Standardize the flows Make them easy to calculate with Make them easy to produce in the lab Chapter 4: Standard flows Chapter 5: Material Functions Chapter 6: Experimental Data 5 Non-Newtonian Constitutive Equations What should we do?. Pick a small number of simple flows. Make calculations 3. Make measurements 4. Try to deduce Standardize the flows Make them easy to calculate with Make them easy to produce in the lab Chapter 5: Material Functions Chapter 6: Experimental Data Chapter 7: GNF Chapter 8: GLVE Chapter 9: Advanced Chapter 4: Standard flows 6 8
CM465 Lectures -3: Intro, Mathematical //6 Tactic: Divide the Problem in half Modeling Calculations Experiments Dream up models Calculate model predictions for stresses in standard flows Calculate material functions from model stresses Standard Flows Compare Pass judgment on models Build experimental apparatuses that allow measurements in standard flows Determine material functions from measured stresses Collect models and their report cards for future use 7 Standard flows choose a velocity field (not an apparatus or a procedure) For model predictions, calculations are straightforward For experiments, design can be optimized for accuracy and fluid variety Material functions choose a common vocabulary of stress and kinematics to report results Make it easier to compare model/experiment Record an inventory of fluid behavior (expertise) 8 9
CM465 Lectures -3: Intro, Mathematical //6 Newtonian fluids: vs. non-newtonian fluids: How can we investigate non-newtonian behavior? CONSTANT TORQUE MOTOR t fluid 9 Simple Shear Flow velocity field v ) ( H V H constant H v ( x ) x x x x ( ) x ( t ) t t V v ( t) x 3 path lines x
CM465 Lectures -3: Intro, Mathematical //6 Near solid surfaces, the flow is shear flow. Experimental Shear Geometries y (z-plane section) x y x (z-plane section) r (-plane section) H r (plane section) (z-plane section) (-plane section) (z-plane section) (-plane section) r
CM465 Lectures -3: Intro, Mathematical //6 Standard Nomenclature for Shear Flow x gradient direction neutral direction x 3 x flow direction 3 Why is shear a standard flow? simple velocity field represents all sliding flows simple stress tensor x x 4
CM465 Lectures -3: Intro, Mathematical //6 How do particles move apart in shear flow? t P, l, V l o Consider two particles in the same x -x plane, initially along the x axis. t v x x x P,, l, l lo ot P o l t,l, l, v x x x lo P o l t,ll,, l lo ot at long times 5 How do particles move apart in shear flow? x v 3 Each particle has a different velocity depending on its x position: v x Consider two particles in the same x -x plane, initially along the x axis (x =). P : P : v l v l The initial x position of each particle is x =. After t seconds, the two particles are at the following positions: P ( t) : P ( t) : x lt x l t length location initial time time 6 3
CM465 Lectures -3: Intro, Mathematical //6 What is the separation of the particles after time t? l t l t l l l l l l l l t t l l t t l l t negligible as l t t l l l t In shear the distance between points is directly proportional to time 7 Uniaxial Elongational Flow x 3 x x, x x velocity field ( t) ( x t) v ( x t) x3 3 ( t) 8 4
CM465 Lectures -3: Intro, Mathematical //6 Uniaxial Elongational Flow x x 3 x x, x path lines ( t) ( x t) v x ( t) x3 3 ( t) 9 Elongational flow occurs when there is stretching - die exit, flow through contractions fluid 3 5
CM465 Lectures -3: Intro, Mathematical //6 Experimental Elongational Geometries x x 3 air-bed to support sample fluid x x3 t o t o +t t o +t x 3 x h(t o ) R(t) h(t) thin, lubricating layer on each plate R(t o ) 3 Sentmanat Extension Rheometer (5) Originally developed for rubbers, good for melts Measures elongational viscosity, startup, other material functions Two counter-rotating drums Easy to load; reproducible www.xpansioninstruments.com http://www.xpansioninstruments.com/rheo-optics.htm 3 6
CM465 Lectures -3: Intro, Mathematical //6 Why is elongation a standard flow? simple velocity field represents all stretching flows simple stress tensor 33 How do particles move apart in elongational flow? x 3 Consider two particles in the same x -x 3 plane, initially along the x 3 axis. l o t P P l P,, o x l P,, o 34 7
CM465 Lectures -3: Intro, Mathematical //6 How do particles move apart in elongational flow? Consider two particles in the same x -x 3 plane, initially along the x 3 axis. varies ln o l loe t Particles move apart exponentially fast. x 3 P P lo P ot,, e x lo ot P,, e 35 A second type of shear-free flow: Biaxial Stretching before after P before air under pressure ( t) ( x t) v x ( t) x3 3 ( t) after 36 8
CM465 Lectures -3: Intro, Mathematical //6 How do uniaxial and biaxial deformations differ? Consider a uniaxial flow in which a particle is doubled in length in the flow direction. a a a fluid a a a 37 How do uniaxial and biaxial deformations differ? Consider a biaxial flow in which a particle is doubled in length in the flow direction. a a a a a a/4 38 9
CM465 Lectures -3: Intro, Mathematical //6 A third type of shear-free flow: Planar Elongational Flow ( t) x v ( ) t x3 3 ( t) a a a a a a/ 39 All three shear-free flows can be written together as: ( t)( b) x v ( t)( b) x ( t) x3 3 Elongational flow: b=, Biaxial stretching: b=, Planar elongation: b=, ( t) ( t) ( t) 4
CM465 Lectures -3: Intro, Mathematical //6 Why have we chosen these flows? ANSWER: Because these simple flows have symmetry. And symmetry allows us to draw conclusions about the stress tensor that is associated with these flows for any fluid subjected to that flow. 4 In general: 3 3 3 3 33 3 But the stress tensor is symmetric leaving 6 independent stress components. Can we choose a flow to use in which there are fewer than 6 independent stress components? Yes we can symmetric flows 4
CM465 Lectures -3: Intro, Mathematical //6 How does the stress tensor simplify for shear (and later, elongational) flow? ê P3,, 3 e P3,, 3 e ê 43 What would the velocity function be for a Newtonian fluid in this coordinate system? v v 3 H x V x V 44
CM465 Lectures -3: Intro, Mathematical //6 3 What would the velocity function be for a Newtonian fluid in this coordinate system? 3 v v V V H x x 45 Vectors are independent of coordinate system, but in general the coefficients will be different when the same vector is written in two different coordinate systems: 3 3 3 3 v v v v v v v For shear flow and the two particular coordinate systems we have just examined, however: 3 3 x H V x H V v 46
CM465 Lectures -3: Intro, Mathematical //6 V x H v 3 V x H 3 x x x x If we plug in the same number in for and, we will NOT be asking about the same point in space, but we WILL get the same exact velocity vector. Since stress is calculated from the velocity field, we will get the same exact stress components when we calculate them from either vector representation. This is an unusual circumstance only true for the particular coordinate systems chosen. 47 What do we learn if we formally transform v from one coordinate system to the other? 48 4
CM465 Lectures -3: Intro, Mathematical //6 What do we learn if we formally transform from one coordinate system to the other? 49 What do we learn if we formally transform v from one coordinate system to the other? (now, substitute from previous slide and simplify) You try. 5 5
CM465 Lectures -3: Intro, Mathematical //6 Conclusion: Because of symmetry, there are only 5 nonzero components of the extra stress tensor in shear flow. SHEAR: 33 3 This greatly simplifies the experimentalists tasks as only four stress components must be measured instead of 6 recall. 5 Summary: We have found a coordinate system (the shear coordinate system) in which there are only 5 non-zero coefficients of the stress tensor. In addition,. This leaves only four stress components to be measured for this flow, expressed in this coordinate system. 5 6
CM465 Lectures -3: Intro, Mathematical //6 How does the stress tensor simplify for elongational flow? x 3 x x, x x There is 8 o of symmetry around all three coordinate axes. 53 Because of symmetry, there are only 3 nonzero components of the extra stress tensor in elongational flows. ELONGATION: 33 3 This greatly simplifies the experimentalists tasks as only three stress components must be measured instead of 6. 54 7
CM465 Lectures -3: Intro, Mathematical //6 Standard Flows Summary Choose velocity field: ( t) x v 3 ( t)( b) x v ( t)( b) x ( t) x3 3 Symmetry alone implies: (no constitutive equation needed yet) 33 3 33 3 By choosing these symmetric flows, we have reduced the number of stress components that we need to measure. 55 Tactic: Divide the Problem in half Modeling Calculations Experiments Dream up models Calculate model predictions for stresses in standard flows Standard Flows Build experimental apparatuses that allow measurements in standard flows Calculate material functions from model stresses Compare Pass judgment on models Determine material functions from measured stresses Collect models and their report cards for future use 56 8
CM465 Lectures -3: Intro, Mathematical //6 Next, build and assume this Choose velocity field: Symmetry alone implies: (no constitutive equation needed yet) ( t) x v 3 ( t)( b) x v ( t)( b) x ( t) x3 3 33 3 33 3 Measure and predict this 57 One final comment on measuring stresses... What is measured is the total stress, : p 3 p 3 3 3 p 33 3 For the normal stresses we are faced with the difficulty of separating p from t ii. Compressible fluids: Incompressible fluids: Get p from measurements of T and V.? nrt p V 58 9
CM465 Lectures -3: Intro, Mathematical //6 Density does not vary (much) with pressure for polymeric fluids. 3 g / cm 4 3 gas density polymer density incompressible fluid M P RT 5 5 5 3 Pressure (MPa) 59 For incompressible fluids it is not possible to separate p from t ii. Luckily, this is not a problem since we only need p Equation of motion v v v g t P g We do not need t ii directly to solve for velocities Solution? Normal stress differences 6 3
CM465 Lectures -3: Intro, Mathematical //6 Normal Stress Differences First normal stress difference Second normal stress difference N N 33 33 In shear flow, three stress quantities are measured N,, N In elongational flow, two stress quantities are measured 33, 6 Normal Stress Differences First normal stress difference Second normal stress difference N N 33 33 In shear flow, three stress quantities are measured N,, N In elongational flow, two stress quantities are measured 33, Are shear normal stress differences real? 6 3
CM465 Lectures -3: Intro, Mathematical //6 First normal stress effects: rod climbing Extra tension in the -direction pulls azimuthally and upward (see DPL p65). Newtonian - glycerin Bird, et al., Dynamics of Polymeric Fluids, vol., Wiley, 987, Figure.3- page 63. (DPL) Viscoelastic - solution of polyacrylamide in glycerin 63 Second normal stress effects: inclined openchannel flow 33 Extra tension in the -direction pulls down the free surface where dv /dx is greatest (see DPL p65). Newtonian - glycerin R. I. Tanner, Engineering Rheology, Oxford 985, Figure 3.6 page 4 Viscoelastic - % soln of polyethylene oxide in water N ~-N / 64 3
CM465 Lectures -3: Intro, Mathematical //6 Example: Can the equation of motion predict rod climbing for typical values of N, N? z cross-section A: fluid v v r z R R r A www.chem.mtu.edu/~fmorriso/cm465/rod_climb.pdf What is d dr zz? Bird et al. p64 65 What s next? Shear ( t) x v 3 Even with just these (or 4) standard flows, we can still generate an infinite number of flows by varying ( t) and ( t ). Shear-free (elongational, extensional) ( t)( b) x v ( t)( b) x ( t) x 3 3 Elongational flow: b=, ( t) Biaxial stretching: b=, ( t) Planar elongation: b=, ( t) 66 33
CM465 Lectures -3: Intro, Mathematical //6 We seek to quantify the behavior of non- Newtonian fluids Procedure:. Choose a flow type (shear or a type of elongation).. Specify ( t) or ( t) as appropriate. 3. Impose the flow on a fluid of interest. 4. Measure stresses. shear, N, N elongation 33, 5. Report stresses in terms of material functions. Faith A. Morrison, Michigan Tech U. 6a. Compare measured material functions with predictions of these material functions (from proposed constitutive equations). 7a. Choose the most appropriate constitutive equation for use in numerical modeling. 6b. Compare measured material functions with those measured on other materials. 7a. Draw conclusions on the likely properties of the unknown material based on the comparison. 67 Done with Standard Flows. Let s move on to Material Functions 68 34
CM465 Lectures -3: Intro, Mathematical //6 Chapter 5: Material Functions Kinematics: ( t) x v 3 Steady Shear Flow Material Functions ( t ) constant CM465 Polymer Rheology Michigan Tech Material Functions: Viscosity First normal-stress coefficient Second normalstress coefficient 33 69 35