Rheology. How do we reach these goals? Polymer Rheology. What is rheology anyway? Phenomena/Continuum Modeling. What is rheology anyway?

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1 What is heology anyway? Rheology Ω To the scientist, enginee, o technician, heology is fluid Phenomena/Continuum Modeling Yield stesses coss-section A: A iscoelastic effects θ κr R Memoy effects γ Shea thickening and shea thinning Release stess t Pofesso Faith A. Moison Depatment of Chemical Engineeing Michigan Technological Univ. fmoiso@mtu.edu Fo both the laypeson and the technical peson, heology is a set of poblems o obsevations elated to how the stess in a mateial o foce applied to a mateial is elated to defomation (change of shape) of the mateial. CM465 Polyme Rheology Michigan Tech What is heology anyway? Rheology affects: Pocessing (design, costs, poduction ates) Rhe- ρει Geek fo flow What is heology anyway? pic/akon%extude.jpg Rheology the study of defomation and flow. What is Rheology Anyway? Faith A. Moison, The Industial Physicist, () 9-, Apil/May 4. End use (food textue, poduct pou, moto-oil function) Poma et al. JNNFM What is heology anyway? To the laypeson, heology is Mayonnaise does not flow even unde stess fo a long time; honey always flows Silly Putty bounces (is elastic) but also flows (is viscous) Dilute flou-wate solutions ae easy to wok with but doughs can be quite tempeamental Con stach and wate can display stange behavio poke it slowly and it defoms easily aound you finge; punch it apidly and you fist bounces off of the suface mpcm/aamp/examples.html Poduct quality (suface distotions, anisotopy, stength, stuctue development) Goal of the scientist, enginee, o technician: Undestand the kinds of flow and defomation effects exhibited by complex systems Apply qualitative heological knowledge to diagnostic, design, o optimiation poblems How do we each these goals? In diagnostic, design, o optimiation poblems, use o devise quantitative analytical tools that coectly captue heological effects

2 How? By making calculations with models in appopiate situations Undestand the kinds of flow and defomation effects exhibited by complex systems Apply qualitative heological knowledge to diagnostic, design, o optimiation poblems By obseving the behavio of diffeent systems In diagnostic, design, o optimiation poblems, Use o devise quantitative analytical tools that coectly captue heological effects CM 465 Newtonian fluids: (fluid mechanics) Non-Newtonian fluids: (heology) dv µ dx Newton s Law of iscosity Non-Newtonian Fluid Mechanics mateial paamete defomation This is an empiical law (measued o obseved) May be deived theoetically fo some systems Need a new law o new laws These laws will also eithe be empiical o will be deived theoetically Leaning Rheology (bibliogaphy) CM 465 Non-Newtonian Fluid Mechanics Desciptive Rheology Banes, H., J. Hutton, and K. Waltes, An Intoduction to Rheology (Elsevie, 989) Quantitative Rheology Moison, Faith, Undestanding Rheology (Oxfod, ) Bid, R., R. Amstong, and O. Hassage, Dynamics of Polymeic Liquids, olume (Wiley, 987) Industial Rheology Dealy, John and Kut Wissbun, Melt Rheology and Its Role in Plastics Pocessing (an Nostand Reinhold, 99) Polyme Behavio Lason, Ron, The Stuctue and Rheology of Complex Fluids (Oxfod, 999) Fey, John, iscoelastic Popeties of Polymes (Wiley, 98) Suspension Behavio Lason, Ron, The Stuctue and Rheology of Complex Fluids (Oxfod, 999) Macosko, Chis, Rheology: Pinciples, Measuements, and Applications (CH Publishes, 994) Newtonian fluids: (shea flow only) Non-Newtonian fluids: (all flows) stess tenso dv µ dx Constitutive Equation f ( γ ) By leaning which quantitative models apply in what cicumstances Rate-ofdefomation tenso non-linea function (in time and position) The Physics Behind Rheology:. Consevation laws mass Cauchy Momentum Equation momentum v ρ + v v p + ρ g enegy. Mathematics diffeential equations vectos tensos. Constitutive law law that elates stess to defomation fo a paticula fluid Intoduction to Non-Newtonian Behavio Rheological Behavio of Fluids, National Committee on Fluid Mechanics Films, 964 Type of fluid Inviscid (eo viscosity, µ) Newtonian (finite. constant viscosity, µ) Non-Newtonian (finite, vaiable viscosity η plus memoy effects) Momentum balance Eule equation (Navie- Stokes with eo viscosity) Navie-Stokes (Cauchy momentum equation with Newtonian constitutive equation) Cauchy momentum equation with memoy constitutive equation elocity gadient tenso γ Stess Defomation elationship (constitutive equation) Stess is isotopic Stess is a function of the instantaneous velocity gadient Stess is a nonlinea function of the histoy of the velocity gadient

3 Rheological Behavio of Fluids - Newtonian. Stain esponse to imposed shea stess x shea ate is constant. Pessue-diven flow in a tube (Poiseuille flow) viscosity is constant Q P v ( x ) 4 π PR Q 8µ L 4 πr constant 8µ L x γ γ t dγ constant dt. Stess tenso in shea flow only two components ae noneo Examples fom the film of.... Dependence on the histoy of the defomation gadient Polyme fluid pous, but spings back Elastic ball bounces, but flows if given enough time Steel ball dopped in polyme solution bounces Polyme solution in concentic cylindes has fading memoy Quantitative measuements in concentic cylindes show memoy and need a finite time to come to steady state Non-lineaity of the function f ( γ ) Polyme solution daining fom a tube is fist slowe, then faste than a Newtonian fluid Double the static head on a daining tube, and the flow ate does not necessaily double (as it does fo Newtonian fluids); sometimes moe than doubles, sometimes less Nomal stesses in shea flow Die swell Rheological Behavio of Fluids non-newtonian. Stain esponse to imposed shea stess x shea ate is vaiable. Pessue-diven flow in a tube (Poiseuille flow) viscosity is vaiable Q Q Q P P P Q f v ( x ) x ( P) γ. Stess tenso in shea flow Nomal stesses t all 9 components ae noneo Release stess Show NCFM Film on Rheological Behavio of Fluids Rheological Behavio of Fluids non-newtonian. Stain esponse to imposed shea stess x shea ate is vaiable. Pessue-diven flow in a tube (Poiseuille flow) viscosity is vaiable Q Q Q P P P Q f v ( x ) x ( P) γ Nomal stesses t all 9 components ae noneo Release stess Chapte : Chapte : Newtonian Fluid Mechanics Please eview Chapte and Chapte

4 How can we do actual calculations with vectos? Rule: any vecto may be expessed as the linea combination of thee, non-eo, non-coplana basis vectos any vecto a a + a a + a + a x x a j j j y y + a coefficient of a in the ê y diection ax ay a xy Einstein Notation (con t) To cay out a dot poduct of two abitay vectos... Detailed Notation Einstein Notation ( a e + a e + a e ) ( be + b e + b e ) a b ae be + ae be + ae b + a be + a be + a be + a be + ae be + ae be ab + a b + a b a b a b j j a δ b a b j jm m j j m m Einstein Notation a system of notation fo vectos and tensos that allows fo the calculation of esults in Catesian coodinate systems. a a + a + a j a j j j j a a m m the initial choice of subscipt lette is abitay the pesence of a pai of like subscipts implies a missing summation sign Tenso the indeteminate vecto poduct of two (o moe) vectos e.g.: stess velocity gadient γ tensos may be constant o may be vaiable Definitions dyad o dyadic poduct a tenso witten explicitly as the indeteminate vecto poduct of two vectos a d A dyad geneal epesentation of a tenso Einstein Notation (con t) The esult of the dot poducts of basis vectos can be summaied by the Konecke delta function Laws of Algeba fo Indeteminate Poduct of ectos: δip i p i p i p Konecke delta NO commutative yes associative yes distibutive a v v a ( a v) ( b a) v b a v a ( v + w) a v + a w b 4

5 How can we epesent tensos with espect to a chosen coodinate system? Just follow the ules of tenso algeba a m ( a + a + a )( m + m + m ) a m + a m + a m + a m + a m + a m + a m + a m + a m k w k k w a m k k w w a m w k w Any tenso may be witten as the sum of 9 dyadic poducts of basis vectos 4. Diffeential Opeations with ectos, Tensos To cayout the diffeentiation with espect to D spatial vaiation, use the del (nabla) opeato. This is witten in Catesian coodinates We can constuct a wide vaiety of quantities such as: v, A, ( αp) etc. Please eview this topic. Del Opeato x + + x x x x x p x x p p Einstein notation fo del p p Summay of Einstein Notation. Expess vectos, tensos, (late, vecto opeatos) in a Catesian coodinate system as the sums of coefficients multiplying basis vectos - each sepaate summation has a diffeent index. Dop the summation signs. Dot poducts between basis vectos esult in the Konecke delta function because the Catesian system is othonomal. Note: In Einstein notation, the pesence of epeated indices implies a missing summation sign The choice of initial index (i, m, p, etc.) is abitay - it meely indicates which indices change togethe 5. Cuvilinea Coodinates Cylindical Spheical Please eview this topic.,θ,,, θ,θ, φ, e e θ, These coodinate systems ae otho-nomal, but they ae not constant (they vay with position). This causes some non-intuitive effects when deivatives ae taken. e φ See figues. and. Tenso Invaiants I A tacea ta A A Fo the tenso witten in Catesian coodinates: III tace tacea App A + ( A A) Please eview this topic. II tace A : A A A ( A A A) Apj AjhAhp + A A Note: the definitions of invaiants witten in tems of coefficients ae only valid when the tenso is witten in Catesian coodinates. pk kp 5. Cuvilinea Coodinates Cuvilinea Coodinates (summay) The basis vectos ae otho-nomal The basis vectos ae non-constant (vay with position) These systems ae convenient when the flow system mimics the coodinate sufaces in cuvilinea coodinate systems. We cannot use Einstein notation on cuvilinea coodinates must use Tables in Appendix C (pp ). 5

6 Chapte : Newtonian Fluid Mechanics Please eview this topic. The solution of Non-Newtonian flow poblems will follow the same pocess as the solution of Newtonian flow poblems. Chapte : Newtonian Fluid Mechanics this is the tough one stess f at P on the foce on that suface choose a suface though P P We need an expession fo the state of stess at an abitay point P in a flow. Chapte : Newtonian Fluid Mechanics Mass Balance Think back to the molecula pictue fom chemisty: Continuity equation: micoscopic mass balance ρ + ( ρv) The specifics of these foces, connections, and inteactions must be captued by the molecula foces tem that we seek. Chapte : Newtonian Fluid Mechanics Momentum Balance Momentum is conseved. Conside an abitay volume enclosed by a suface S We will concentate on expessing the molecula foces mathematically; We leave to late the task of elating the esulting mathematical expession to expeimental obsevations. ate of incease net flux of sum of + of momentum in momentum into foces on esembles the ate tem in the mass balance esembles the flux tem in the mass balance Foces: body (gavity) molecula foces Fist, choose a suface: abitay shape small stess f at P on n What is f? f 6

7 Conside the foces on thee mutually pependicula sufaces though point P: x How can we wite f (the foce on an abitay suface ) in tems of the Π pk? n f ê x ê P x b ê a f is foce on in -diection f f e + f e + f e f is foce on in -diection f is foce on in -diection c Thee ae thee Π pk that elate to foces in the -diection: Π Π,, Π a b c is stess on a suface at P a suface with unit nomal ḙ is stess on a suface at P is stess on a suface at P How can we wite f (the foce on an abitay suface ) in tems of the quantities Π pk? f f e + f e + f e f, the foce on in -diection, can be boken into thee pats associated with the thee stess components:. n Π Π,, Π n f We can wite these vectos in a Catesian coodinate system: stess on a suface in the - diection a ae + a + a Π + Π + Π fist pat: pojection of ( Π) da onto the Πn suface foce ( aea) aea a ae + a + a Πe + Π + Π b be + be + be Π + Π + Π c ce + c + c Π + Π + Π So fa, this is nomenclatue; next we elate these expessions to foce on an abitay suface. a b c Stess on a p suface in the k-diection is stess on a suface at P is stess on a suface at P is stess on a suface at P Π pk f, the foce on in -diection, is composed of THREE pats: stess on a -suface in the - diection fist pat: second pat: thid pat: ( Π ) ( Π ) pojection of da onto the suface pojection of da onto the suface pojection of suface Π n ( Π ) da onto the Π n e Π n the sum of these thee f 7

8 Assembling the foce vecto: f f + f + f n + ( Πe + Π + Π ) n ( Π + Π + Π ) + n ( Π e + Π e + Π e ) [ n Πee + Π + Π + Πee + Π + Π + Π ee + Π + Π ] linea combination of dyadic poducts tenso Assembling the foce vecto: f, the foce in the -diection on an abitay suface is composed of THREE pats. Using the distibutive law: f Πn + Πn + Πn stess appopiate aea ( Π + Π + Π e ) f n Foce in the -diection on an abitay suface [ f n Πee + Π + Π + Πee + Π + Π + Π ee + Π + Π n n Π f n Π p m Π pm p m pm p m Total stess tenso (molecula stesses) ] Momentum Balance The same logic applies in the -diection and the -diection f n f n f n Assembling the foce vecto: ( Πe + Π + Π ) ( Π + Π + Π ) ( Π e + Π e + Π e ) f fe + f + f n + ( Πe + Π + Π ) n ( Π + Π + Π ) + n ( Π + Π + Π ) ate of incease net flux of sum of + of momentum in momentum into foces Please see the text fo the details of the deivation of the othe tems. ( ρv) d ( ρvv) molecula foces S d + molecula foces on S n Π Π d on molecula ρg d + foces Gauss Divegence Theoem 8

9 Momentum Balance ate of incease net flux of sum of + of momentum in momentum into foces ( ρv) d ( ρvv) Foces on vesus by cause a sign change, as does positive tension vesus positive compession molecula foces S d + molecula foces on S n Π Π d on molecula ρg d + foces Gauss Divegence Theoem Ou choice: positive compession (pessue is positive) Momentum Balance Final Assembly: ate of incease net flux of of momentum in momentum ( ρv) d ( ρvv) d + ρ g d ρv + Because is abitay, we may conclude: ρv + into sum of + foces on ( ρvv) ρ g + Π d ( ρvv) ρ g + Π Π Micoscopic momentum balance d Momentum Balance Micoscopic momentum ρv + ( ρvv) ρ g + Π balance Afte some eaangement: v ρ + v v Π + ρ g Dv ρ Π + ρ g Dt Equation of Motion Now, what to do with Π? 9