v T Pressure Extra Molecular Stresses Constitutive equations for Stress v t Observation: the stress tensor is symmetric


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1 Momenum Blnce (coninued Momenum Blnce (coninued Now, wh o do wih Π? Pessue is p of i. bck o ou quesion, Now, wh o do wih? Π Pessue is p of i. Thee e ohe, nonisoopic sesses Pessue E Molecul Sesses definiion: An isoopic foce/e of molecul oigin. Pessue is he sme on ny sufce dwn hough poin nd cs nomlly o he chosen sufce. p pessue = p I = p ee ˆ ˆ + p eˆ eˆ + p eˆ eˆ = Tes: wh is he foce on sufce wih uni noml? ṋ p p definiion: The e sesses e he molecul sesses h e no isoopic E sess enso, Now, wh o do wih? Π p I i.e. eveyhing compliced in molecul defomion This becomes he cenl quesion of heologicl sudy Momenum Blnce (coninued Momenum Blnce (coninued Consiuive equions fo Sess e enso equions ele he velociy field o he sesses geneed by molecul foces e bsed on obsevions (empiicl o e bsed on molecul models (heoeicl e ypiclly found by ilndeo e jusified by how well hey wok fo sysem of inees e obseved o be symmeic = f ( v, meil popeies Obsevion: he sess enso is symmeic Micoscopic momenum blnce ρ + v v = Π + ρ g In ems of he e sess enso: ρ + v v = p + ρ g Equion of Moion Equion of Moion Momenum Blnce (coninued Momenum Blnce (coninued Newonin Consiuive equion = µ ( v + ( v T fo incompessible s (see e fo compessible s is empiicl my be jusified fo some sysems wih molecul modeling clculions How is he Newonin Consiuive equion eled o Newon s Lw of iscosiy? = µ ( v + ( v T = µ incompessible s incompessible s eciline flow (sigh lines no viion in diecion
2 Momenum Blnce (coninued Momenum Blnce (coninued Bck o he momenum blnce... ρ + v v = p + ρ g We cn incopoe he Newonin consiuive equion ino he momenum blnce o obin momenumblnce equion h is specific o incompessible, Newonin s = µ Equion of Moion ( v + ( v T NvieSokes Equion v ρ + v v = p + µ v + ρ g incompessible s Newonin s EXAMPLE: Dg flow beween infinie pllel ples sedy se incompessible infiniely wide, long W EXAMPLE: Poiseuille flow beween infinie pllel ples sedy se incompessible infiniely wide, long W v ( H H = v ( =L p=p o p=p L EXAMPLE: Poiseuille flow in ube Sedy se incompessible long ube A z cosssecion A: z EXAMPLE: Tosionl flow beween pllel ples Sedy se incompessible v = zf ( cosssecionl view: Ω Ω z v z ( L H R R
3 Newonin s: = µγ& vs. nonnewonin s: µγ& Simple She Flow velociy field v ( H = = γ& H γ& = consn How cn we invesige nonnewonin behvio? CONSTANT TORQUE MOTOR H v ( ( ( + ς& ( v ph lines Ne solid sufces, he flow is she flow. Epeimenl She Geomeies (zplne y secion (zplne secion y H (plne secion φ ο (φ plne secion (zplne secion (plne secion (zplne secion (plne secion Sndd Nomenclue fo She Flow gdien diecion Why is she sndd flow? simple velociy field epesens ll sliding flows simple sess enso neul diecion flow diecion
4 How do picles move p in she flow? Conside wo picles in he sme  plne, iniilly long he is. = > ( l o v ( P (, l,, P (,, l, l v lo lo γ& o P ( γ& oll,l,, l P ( γ& ol,l,, l l loγ& o long imes How do picles move p in she flow? Conside wo picles in he sme  plne, iniilly long he is ( =. γ& v = Ech picle hs diffeen velociy depending on is posiion: v = γ& P : P : v = & γ l v = & γ l The iniil posiion of ech picle is =. Afe seconds, he wo picles e he following posiions: P ( : P ( : = & γ l = & γ l lengh locion = iniil + ime ( ime Wh is he sepion of he picles fe ime? γ& l Uniil Elongionl Flow l γ& ( l l l = l + = l l = l [& γ ( l l ] = l + & γ l ( + & γ + & γ l & γ negligible s γ& l l lγ& l In she he disnce beween poins is diecly popoionl o ime velociy field, & ε( ( & ε v ( & ε ε& ( > Uniil Elongionl Flow Elongionl flow occus when hee is seching  die ei, flow hough concions, ph lines & ε( ( & ε v ( & ε ε& ( > 4
5 Epeimenl Elongionl Geomeies ibed o suppo smple Why is elongion sndd flow? simple velociy field epesens ll seching flows simple sess enso o o+ o+ h(o R( h( hin, lubicing lye on ech ple R(o How do picles move p in elongionl flow? Conside wo picles in he sme  plne, iniilly long he is. l o o l = loe ε& = > P P P P l P,, o l P,, o lo ε& P o,, e lo ε& o P,, e A second ype of shefee flow: Biil Seching befoe fe & ε( ( & ε v ( & ε P i unde pessue ε& ( < fe befoe How do uniil nd biil defomions diffe? How do uniil nd biil defomions diffe? Conside uniil flow in which picle is doubled in lengh in he flow diecion. Conside biil flow in which picle is doubled in lengh in he flow diecion. /4 5
6 & ε ( ε& ( > v ( & ε A hid ype of shefee flow: Pln Elongionl Flow All hee shefee flows cn be wien ogehe s: & ε( ( + b v = & ε( ( b & ε( / Elongionl flow: b=, ε& ( > Biil seching: b=, ε& ( < Pln elongion: b=, ε& ( > Why hve we chosen hese flows? ANSWER: Becuse hese simple flows hve symmey. And symmey llows us o dw conclusions bou he sess enso h is ssocied wih hese flows fo ny subjeced o h flow. In genel: = Bu he sess enso is symmeic leving 6 independen sess componens. Cn we choose flow o use in which hee e fewe hn 6 independen sess componens? Yes we cn symmeic flows How does he sess enso simplify fo she (nd le, elongionl flow? P (,, e ê e ê P(,, Wh would he velociy funcion be fo Newonin in his coodine sysem? H v v = 6
7 Wh would he velociy funcion be fo Newonin in his coodine sysem? H v v = ecos e independen of coodine sysem, bu in genel he coefficiens will be diffeen when he sme veco is wien in wo diffeen coodine sysems: v v = v v v = v v Fo she flow nd he wo picul coodine sysems we hve jus emined, H v = H = H v = H = Wh do we len if we fomlly nsfom v fom one coodine sysem o he ohe? If we plug in he sme numbe in fo nd, we will NOT be sking bou he sme poin in spce, bu we WILL ge he sme ec velociy veco. Since sess is clculed fom he velociy field, we will ge he sme ec sess enso when we clcule i fom eihe veco epesenion lso. v p = v p This is n unusul cicumsnce only ue fo pk = he picul coodine pk sysems chosen. eˆ eˆ eˆ = = e = e e = = = Becuse of symmey, hee e only 5 nonzeo componens of he e sess enso in she flow. Conclusion: SHEAR: = This gely simplifies he epeimenliss sks s only fou sess componens ( = mus be mesued insed of 6. We hve found coodine sysem (he she coodine sysem in which hee e only 5 nonzeo coefficiens of he sess enso. In ddiion, =. This leves only fou sess componens o be mesued fo his flow, epessed in his coodine sysem. 7
8 How does he sess enso simplify fo elongionl flow? Becuse of symmey, hee e only nonzeo componens of he e sess enso in elongionl flows. ELONGATION:, = Thee is 8 o of symmey ound ll hee coodine es. This gely simplifies he epeimenliss sks s only hee sess componens mus be mesued insed of 6. One finl commen on mesuing sesses... Densiy does no vy (much wih pessue fo polymeic s. Wh is mesued is he ol sess, Π : p + Π = p + p + 4 ρ g / cm gs densiy polyme densiy incompessible M ρ = P RT Fo he noml sesses we e fced wih he difficuly of seping p fom ii. Compessible s: Incompessible s: Ge p fom nrt p = mesuemens of T nd.? Pessue (MP Fo incompessible s i is no possible o sepe p fom ii. Noml Sess Diffeences Luckily, his is no poblem since we only need Π = p + Equion of moion + v v = Π + ρ g = P + ρ g Soluion? Noml sess diffeences We do no need ii diecly o solve fo velociies Fis noml sess diffeence Second noml sess diffeence N N Π Π In she flow, hee sess quniies e mesued Π Π N, =, N = In elongionl flow, wo sess quniies e mesued, 8
9 Fis noml sess effecs: od climbing < Second noml sess effecs: inclined openchnnel flow > E ension in he diecion pulls down he fee sufce whee dv /d is gees (see DPL p65. E ension in he diecion pulls zimuhlly nd upwd (see DPL p65. Newonin  glycein iscoelsic  % soln of polyehylene oide in we Newonin  glycein iscoelsic  soluion of N ~ N / polycylmide in glycein Bid, e l., Dynmics of Polymeic Fluids, vol., Wiley, 987, Figue. pge 6. (DPL Wh s ne? Emple: Cn he equion of moion pedic od climbing fo ypicl vlues of N, N? z cosssecion A: Ω v = v z κr R A dπ zz Wh is? d Bid e l. p64 We seek o qunify he behvio of nonnewonin s R. I. Tnne, Engineeing Rheology, Ofod 985, Figue.6 pge 4 She ς&( v Even wih jus hese (o 4 sndd flows, we cn sill genee n infinie numbe of flows by vying ς&( nd ε&(. Shefee (elongionl, eensionl & ε ( ( + b v = ε& ( ( b ε&( Elongionl flow: b=, Biil seching: b=, Pln elongion: b=, ε&( > ε& ( < ε&( > Pocedue:. Choose flow ype (she o ype of elongion.. Specify ς& ( o ε&( s ppopie.. Impose he flow on of inees. 4. Mesue sesses. she, N, N elongion, 5. Repo sesses in ems of meil funcions. 6. Compe mesued meil funcions wih pedicions of hese meil funcions (fom poposed consiuive equions. Fih A. Moison, Michign Tech U. 7. Choose he mos ppopie consiuive equion fo use in numeicl modeling. 6b. Compe mesued meil funcions wih hose mesued on ohe meils. 7. Dw conclusions on he likely popeies of he unknown meil bsed on he compison. 9
ME 141. Engineering Mechanics
ME 141 Engineeing Mechnics Lecue 13: Kinemics of igid bodies hmd Shhedi Shkil Lecue, ep. of Mechnicl Engg, UET Emil: sshkil@me.bue.c.bd, shkil6791@gmil.com Websie: eche.bue.c.bd/sshkil Couesy: Veco Mechnics
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