( t) Steady Shear Flow Material Functions. Material function definitions. How do we predict material functions?
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1 Rle f aeial Funins in Rhelgial Analysis Rle f aeial Funins in Rhelgial Analysis QUALIY CONROL QUALIAIVE ANALYSIS QUALIY CONROL QUALIAIVE ANALYSIS mpae wih he in-huse daa n qualiaive basis unknwn maeial mpae daa wih lieaue eps n vaius fluids mpae wih he in-huse daa n qualiaive basis unknwn maeial mpae daa wih lieaue eps n vaius fluids nlude whehe n a maeial is apppiae f a speifi appliain measue maeial funins, e.g., G'(ω, G"(w, G( nlude n he pbable physial behavi f he fluid based n mpaisn wih knwn fluid behavi nlude whehe n a maeial is apppiae f a speifi appliain measue maeial funins, e.g., G'(ω, G"(w, G( nlude n he pbable physial behavi f he fluid based n mpaisn wih knwn fluid behavi ODELING WORK mpae measued wih pedied nlude whih nsiuive equain is bes f fuhe mdeling alulains alulae pediins f maeial funins fm vaius nsiuive equains We will fus hee fis ODELING WORK mpae measued wih pedied nlude whih nsiuive equain is bes f fuhe mdeling alulains alulae pediins f maeial funins fm vaius nsiuive equains aeial funin definiins kinemais. Chie f flw (shea elngain ( x v 3 ε ( ( b x v ε( ( b x ε( x 3. Chie f deails f ( ε(. 3 Elngainal flw: b, ε ( > Biaxial sehing: b, ε ( < Plana elngain: b, ε ( > 3. aeial funins definiins: will be based n in shea, N, N 33, in elngainal flws. Kinemais: ( x v 3 aeial Funins: Vissiy Seady Shea Flw aeial Funins ( nsan Fis nmal-sess effiien Send nmalsess effiien ( Ψ ( 33 Ψ Hw d we pedi maeial funins? ANSWER: Fm he nsiuive equain. f (v Wha d we measue f hese maeial funins? Wha des he Newnian Fluid mdel pedi in seady sheaing? µ µ [ v ( v ]
2 Seady shea vissiy and fis nmal sess effiien Seady shea vissiy and fis nmal sess effiien.e5 Ψ.E6.E5.E9.E8, Pise Ψ, (dyn/m s.e4.e3 Ψ ( ( 83 kg/ml 57 kg/ml 35 kg/ml kg/ml, Pise.E4.E3.E.E Ψ.E7.E6.E5.E4 Ψ, dynes/m.e.e3.e.. Figue 6., p. 7 enzes and, s Gaessley n. PB sluin SOR Sh Cuse Beginning Rhelgy Figue 6., p. 7 enzes and Gaessley n. PB sluin;.676 g/m 3 SOR Sh Cuse Beginning Rhelgy.E-.., s.e Seady shea vissiy f linea and banhed PDS linea 3 kg/mle banhed 56 kg/mle linea 48 kg/ml banhed 48 kg/ml Wha have maeial funins augh us s fa? Newnian nsiuive equain is inadequae. Pedis nsan shea vissiy (n always ue. Pedis n shea nmal sesses (hese sesses ae geneaed f many fluids Behavi depends n he maeial (hemial suue, mleula weigh, nenain Figue 6.3, p. 7 Piau e al., linea and banhed PDS SOR Sh Cuse Beginning Rhelgy Can we fix he Newnian Cnsiuive Equain? µ [ v ( v ] Wha des his mdel pedi f seady shea vissiy? ( v ( v Le s eplae µ wih a funin f shea ae beause we wan pedi a nn-nsan vissiy in shea ( v ( v Answe: (
3 If we hse: ( n m < Bu wha abu he nmal sesses? ( v ( v lg slpe (n- v 3 3 I appeas ha shuld n be simply ppinal Pblem slved! lg lg y smehing else... µ I f ( v f ( v v ( v A v ( v B v C v ( Bu whih nes? s u hw fix he Newnian equain, we need me bsevains ( give us ideas. Le s y anhe maeial funin ha s n a seady flw (bu sik shea. Sa-up f Seady Shea Flw aeial Funins Kinemais: ( x v 3 aeial Funins: ( Shea sess gwh funin ( Fis nmal-sess gwh funin Send nmalsess gwh funin Ψ Ψ < ( ( 33 Wha des he Newnian Fluid mdel pedi in sa-up f seady sheaing? µ µ [ v ( v ] aeial funins pedied f sa-up f seady sheaing f a Newnian fluid < ( µ ( µ Again, sine we knw v, we an jus plug i in and alulae he sesses. ( Ψ ( 33 Ψ D hese pediins mah bsevains? 3
4 Saup f Seady Sheaing ( x v 3 ( < Wha abu he nn-seady flws? ( Ψ ( Figues 6.49, 6.5, p. 8 enezes and Gaessley, PB sln SOR Sh Cuse Beginning Rhelgy Cessain f Seady Shea Flw aeial Funins Kinemais: Cessain f Seady Sheaing ( x v 3 ( < ( x v 3 ( < ( aeial Funins: ( Shea sess deay funin Fis nmal-sess deay funin Send nmalsess deay funin Ψ Ψ ( ( 33 Figues 6.5, 6.5, p. 9 enezes and Gaessley, PB sln Ψ ( SOR Sh Cuse Beginning Rhelgy Wha des he mdel we guessed a pedi f sa-up and essain f shea? ( ( v ( v n m < Obsevains he mdel pedis an insananeus sess espnse, and his is n wha is bseved f plymes he pedied unseady maeial funins depend n he shea ae, whih is bseved f plymes (, N nmal sesses ae pedied ( n m ( v ( v < Pgess hee 4
5 ( v ( v Obsevains ( n m < he mdel pedis an insananeus sess espnse, and his is n wha is bseved f plymes Laks memy he pedied unseady maeial funins depend n he shea ae, whih is bseved f plymes (, Pgess hee N nmal sesses ae pedied Relaed nnlineaiies peed bee-designed nsiuive equains, we need knw me abu maeial behavi, i.e. we need me maeial funins pedi, and we need measuemens f hese maeial funins. e nn-seady maeial funins (maeial funins ha ell us abu memy aeial funins ha ell us abu nnlineaiy (sain Summay f shea ae kinemais (pa a. Seady. ( (, ( he nex hee families f maeial funins inpae he nep f sain.. ( (, ( b. Sess Gwh. ( (, (. Sess Relaxain Summay f shea ae kinemais (pa Shea Ceep Flw d. Ceep. ( (, ( Cnsan shea sess impsed. ( (, ( samples e. Sep Sain ε f. SAOS. ( ε sω ε (, sinω ( ε sin( ω δ δ ven ASS 5
6 Beause shea ae is n pesibed, i bemes smehing we mus measue. Ceep Shea Flw aeial Funins Kinemais: ( x v ( aeial Funins: 3 I is unusual pesibe sess ahe han. < Sine we se he sess in his expeimen (ahe han measuing i, he maeial funins ae elaed he defmain f he sample. We need disuss measuemens f defmain befe peeding. ( Defmain (sain ( x ( x( x3( x( ( x( 3( x 3 3 ( u( flw paile pah, ( ( x 3 u, x ( P( u(, ( Shea sain x Displaemen funin P( x Physial inepeain f sain in shea fluid paile a ime v ( x P u (, x P u ( P x x he sain is elaed he hange f shape f he defmed paile. u ( P x u fluid paile a ime he sain is he invese f he slpe f he side f he defmed paile. Defmain in shea flw (sain ( x ( x( x3( x ( ( x( 3( x u( 3 3, ( ( x ( 3 ( x ( x ( (, u x x ( x 3 3 Shea sain Displaemen funin In sain (ninued F seady shea, wih : (sh ime ineval u (, x F a lng ime ineval, we add up he sains ve sh ime inevals. (, p p N, ( p, p p ( N ( Same, beause flw is seady. F unseady shea: u ( p, p ( p x (sh ime ineval F a lng ime ineval, we add up he sains ve sh ime inevals., ( ( p p p N N (, ( p, p ( p p p aking he limi as ges ze, N, lim ( p p ( ( d Sain a wih espe fluid nfiguain a in unseady shea flw. 6
7 Kinemais: ( x v aeial Funins: Ceep Shea Flw aeial Funins ( 3 (, J (, Shea eep mpliane < ( J (, Reveable eep mpliane aeial funins pedied f eep f a Newnian fluid J ( emve sess µ Shea eep maeial funins Shea Ceep ( x v 3 ( < Seady-sae mpliane nsan slpe ( J(, J ( R( ', R( s Ulimae eil funin J ( ρ J p ρ Daa have been eed f veial shif. (, J (,, eep ', evey Figue 6.53, p. Plazek; PS mel Shea Ceep - Reveable Cmpliane J ( R( eveable sain ( ( al sain nneveable sain A lng imes he eep mpliane J(, bemes a saigh line. dj d d seady d sae ( he slpe a seady sae is he invese f he seady vissiy Figues 6.54, 6.55, p. Plazek; PS mel dj d seady sae J ( ( C seady sae ( Seady-sae mpliane J s ( 7
8 Ceep Revey - afe eep, sp pulling fwad and allw he flw evese Reveable sain Reil sain ( (, (, Sain a he end f he fwad min ( J (, Reveable eep mpliane Sain a he end f he evey Linea Viselasi Ceep ( ( al sain eveable sain nneveable sain J ( R( his is a way ge R( wihu measuing i Shea eep maeial funins nsan slpe Linea-viselasi limi Seady-sae mpliane J s J ( R(' R J s Ulimae eil funin, eep 8
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