( ) ( ) CHAPTER 8. Lai et al, Introduction to Continuum Mechanics. Copyright 2010, Elsevier Inc 8-1

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1 CHAPER 8 81 Shw ha f an incmpessible Newnian fluid in Cuee flw he pessue a he ue cylinde ( R ) is always lage han ha a he inne cylinde ha is bain R i Ri ρ ω R R d B Ans In Cuee flw v vz and vθ A+ [see Eq (6154) and (6157)] hus dvθ vθ θθ zz p z θ z and θ μ funcin f nly d 1 z hus he -equain f min θ θθ θ z ρ ω becmes: R R ρω Nw d ρ ω d Ri hus Ri ρ ω R R d R i he igh hand side f his las equain is always R i psiive 8 Shw ha he cnsiuive equain τ τ1+ τ + τ wih τn + n τ n / μnd n 1 is equivalen τ + a1 τ / + a τ / + a τ / bd+ b1 D/ + b D/ whee a + + a + + a b b ( μ μ μ ) b μ ( ) μ ( ) μ ( ) ( μ μ μ ) Ans τ τ j τ j i i j i i i 1 i 1 τ j 1 i 1 j 1 i 1 j 1 τ τ i j τ j i + i ( μid τ i) + i i 1 i 1 j 1 i 1 i 1 j 1 j i j i τ j D μi τ i i D μi τ + i 1 i 1 i 1 j 1 i 1 i 1 j 1 j i j i i + ha is τ j 8-1 Cpyigh 1 Elsevie Inc

2 τ τ τ τ1 τ τ1 τ i D μi i 1 τ (i) i 1 Nex we have + + τ / + + τ / τ / ( 1 1) ( 1 1) 1 ( 1 1) ( ) ( ) ( ) + τ / + ( + ) τ / + τ / 1 1 τ / 1 1 τ1/ τ1/ 1 τ / ie ( ) τ / μ1( + ) + μ( 1+ ) + μ( + 1) D / + τ / + τ / + τ / + τ / + τ / τ / + Finally we have τ τ1 τ τ D τ1 D τ D τ μ1 + 1 μ + 1 μ D D D τ τ τ 1 μ + μ + μ 1 ha is 1 τ μ D 1 μ D 1 1 μ D τ τ 1 τ (iii) hus (i) + (ii)+ (iii) gives τ τ / τ / + τ / ( 1 ) ( 1 1) 1 D( μ1 μ μ) μ1( ) μ( 1 ) μ( 1) ( μ μ μ ) D D/ / ha is τ τ τ D D τ + a1 + a + a b D + b1 + b whee a + + a + + a b b ( μ μ μ ) b μ ( ) μ ( ) μ ( ) ( μ μ μ ) Obain he fce-displacemen elainship f he Kelvin-Vig slid which cnsiss f a dashp (wih damping cefficien η ) and a sping (wih sping cnsan G ) cnneced in paallel Als bain is elaxain funcin Ans Since he sping and he dashp ae cnneced in paallel heefe he al fce is given by: S Ssp + Sdash and he al displacemen ε is given by ε εsp εdash Nw (ii) 8- Cpyigh 1 Elsevie Inc

3 dε dε Ssp Gε and Sdash η heefe S Gε + η find he elaxain funcin we le d d ε ε H () whee H is he Heaviside funcin hen S GεH() + ηεδ () hus he elaxain funcin is S / ε GH + ηδ 84 (a) Obain he fce-displacemen elainship f a dashp (damping cefficien η ) and a Kelvin-Vig slid (damping cefficien η and sping cnsan G see he pevius pblem) cnneced in seies (b) Obain is elaxain funcin Ans (a) Le Sv and Sd be he fce ansmied by he Kelvin-Vigh elemen and he dashp especively and le εv and εd be he elngain f he Kelvin-Vigh elemen and he dashp especively hen we have he al fce is given by S Sd Sv (i) and he al displacemen dεd dεv is given by ε εd + εv(ii) whee Sd η S Sv Gεv + η (iii) Fm (ii) and d d dε dεd dεv S 1 S S G (iii) we have + + ( S Gε v ) + ( ε εd ) (iv) hus d d d η η η η η d ε η η + ds G dε G dεd ηη d ε ( η + η ) ds dε + η + S d ηη d η d η d G d G d d hus he fce-displacemen elainship is given by: ( η + η ) ds dε ηη d ε S + η + (v) G d d G d (b) Le ε ε H () whee H is Heaviside funcin hen Eq (v) gives ds G G () d + S ε η δ + ε ηη δ (vi) d η + η η + η η + η d whee () hus δ is he Diac funcin he inegain fac f his ODE is exp G / ( η + η ) G G G d η G d Se + η εη η η εηη η η δ e δ () e d ( η η) + + ( η η) + and + + d G G G η + η G η η η η () + δ + η η ( η η) εη εηη dδ d Se e d e d εη G εηη G/( η η) / () G G η η e δ δe d ( η η) ( η η) η + η ha is G G G G G G Se η + η εη εηη e () e () η + η εη εηη δ η + η + + δ η + η η + η η + η η + η η + η G + S η G η η ηη hus he elaxain funcin is e + δ () ε ( η + η ) ( η + η) 8- Cpyigh 1 Elsevie Inc

4 85 A linea Maxwell fluid defined by Eq (81) is beween w paallel plaes which ae ne uni apa Saing fm es a ime he p plae is given a displacemen u v while he bm plae emains fixed Neglec ineia effecs bain he shea sess hisy Ans he velciy field f he fluid in his min is given by (ineia negleced) v1 vh() x v v whee H is he Heaviside Funcin he nly nn-ze ae f defmain cmpnen is D1 v H / hus fm he cnsiuive equain f he linea ds1 d Maxwell fluid we bain S1 + μvh() hus / / 1 v S e μ e H () ha is d d / v / v / v / / S1e μ e μ H() d e μ d e μv ( e 1) hus he shea sess hisy is: 1 ( 1 / S μv e ) 86 Obain Eq (81) ie ' ' ' / φ( ) ( ) d whee φ ( μ / ) e S D by slving ds he linea nn-hmgeneus dinay diffeenial equain S+ μd d exp / ha is he equain can be wien as; Ans he inegain fac f his ODE is [ ] d d ( e ) μ / / / / S e D hus S ( / ) D / / ( μ / ) ( ' )/ S e e D d ha is e μ e d φ S μ e ' d' ' ' d' D D 87 Shw ha f he linea Maxwell fluid defined by Eq (81) φ whee φ () is he elaxain funcin and J is he ceep cmpliance funcin ' J ' d' S S H be applied he p plae f a channel f uni deph in which is he linea Ans Le 1 Maxwell fluid [ H is he uni sep funcin ie Heaviside funcin] Neglecing ineia he velciy field is v( x ) v x v is he velciy f he p plae hen fm he whee cnsiuive equain d / d μ S+ S D we bain S + S δ μd μ v / μdu / d 1 du S whee u is he displacemen f he p plae Fm Sδ () d μ + μ we bain du S S S d d Sδ () u S ( ) d μ μ hus he ceep cmpliance μ μ μ funcin is: J () u / S ( + )/ μ / Since he elaxain funcin is φ() ( μ / ) e heefe 8-4 Cpyigh 1 Elsevie Inc

5 ( ) ( ) ( )/ ( )/ e d + (1 / ) e ( ) d ( )/ ( ) ' / Nw ' / / φ J d ( μ / ) e ( + )/ μd (1/ ) e ( + ) d e d [ e ] and 1 ' ' ' φ J d + + ( )/ / / / ' e d ( de )( ) d e e d hus 88 Obain he sage mdulus and lss mdulus f he linea Maxwell fluid wih a H ( ) / cninuus elaxain specum defined by Eq (841) ie φ () e d Ans Le he shea sain be: γ1 γe iω F his sain hisy he ae f defmain hisy is dγ1 given by D1 i e i ω ωγ hus fm he cnsiuive equain d φ d S D d i e i ω φ ωγ φ d S D we have 1 1 / Wih φ () H / e d we have / H ( )/ iω H e / iω S1 i e e d d i e e dd ' Nw / iω ( 1 + i ω) / e e d e d e ( 1 + i ω ) / hus 1+ iω 1 i S ω H i e d G e i ω ω ( ) γ whee G iω d + iω ( + iω) ωγ ωγ H 1 1 is he cmplex mdulus Nw H iω H G iω d iω d 1+ iω 1+ iω 1iω ( 1 ) ( iω+ ω ) H ω H ωh d d + i d (1 + ω) (1 + ω) (1 + ω) ωh ωh hus G d G d (1 + ω ) (1 + ω ) 89 Shw ha he viscsiy μ f a linea Maxwell fluid define by φ ( ) is elaed he elaxain funcin φ and he memy funcin f ( s) by he elain φ( s) ds sf ( s) ds μ S D d 8-5 Cpyigh 1 Elsevie Inc

6 S φ D d φ s D s ds φ s D s ds Ans s s F simple sheaing flw v 1 x v v D1 s ha S 1 φ( s ) ds μ S 1 / φ( s ) ds s Nw he memy funcin s elaxain funcin φ ( s) by he elain dφ ( s) / ds f( s) hus dφ() s μ φ( s) ds sφ( s) s ds sf ( s) ds ds f s is elaed he 81 Shw ha he elaxain funcin f he Jeffey mdel [Eq (87)] wih a is given by [ne: Refeence Eq(87) is missing in he pblem saemen in he ex] S1 b b1 a / b 1 1 φ() 1 e + δ() δ() Diac Funcin γ a1 ba1 b Ans Le he shea sain γ 1 be given by γ1 γ H () hen D1 dγ1 / d γδ whee δ () is Diac funcin Fm he cnsiuive equain we have S1 γ γ δ S1 1 γ b b1 δ S1 + a1 b δ + b1 + S1 δ + a1 a1 a1 a / 1 a / b 1 b1 δ ( S1e ) e γ δ + a1 a1 S1 a / b 1 a / b 1 1 a / dδ () e e δ d e 1 d γ a + 1 a 1 d b b1 a / b a / b 1 b1 a / b e δ() δ() e d + e δ() a 1 a 1 a 1 a1 a1 a1 a1 hus he elaxain funcin is: S1 b b () / a b1 b b 1 () / a b 1 φ 1 e + δ 1 e + δ () γ a1 ab 1 a1 a1 ab 1 b v1 v v x1 v Obain (a) he paicle pahline equains using he cuen ime as he efeence ime (b) he elaive igh Cauchy- Geen defmain ens and (c) he Rivlin-Eicsen enss using he equain C I+ τ - A1+ τ - A / + (d) he Rivlin-Eicsen ens A using he ecusive 811 Given he fllwing velciy field: equain [ ] [ D / D] + [ ][ ] + [ ] [ ] A A1 A1 v v A 1 ec Ans (a) Le x xi e ibe he psiin a ime τ f he paicle which is a x xieia ime hen x x x x x τ gives he pahline equain hus i i 1 dx1 dx dx v1 (i) v v( x1 ) (ii) (iii) dτ dτ dτ 8-6 Cpyigh 1 Elsevie Inc

7 wih he iniial cndiins: xi xi ( x1 x x ) Eq (i) gives x1 f ( x1 x x) x1 dx gives x g( x x x ) x Eq (ii) becmes v( x ) x v( x ) τ h( x x x ) Eq (iii) dτ x ' v x1 + h x1 x x h x1 x x x v x1 x x + v( x1)( τ ) hus x1 x1 x x + v( x1)( τ ) x x 1 1 (b) [ F] [ x ] ( dv / dx1)( τ ) 1 ( τ ) 1 dv / dx ( τ ) 1 1+ ( τ ) ( τ ) [ C] [ F] [ F] 1 ( τ ) 1 ( τ ) ( τ ) 1 + ( τ ) + 1 dv (c) [ A1] [ A ] dx1 DA1 (d) [ A] + [ 1][ ] + [ ] [ 1] D A v v A whee [ A 1] hus DA ( 1) 1 A1 DA1 A1 A + [ 1][ ] + v + D A v D x [ A1][ v] [ v] [ A 1] hus [ A] [ A1][ v] + [ v] [ A 1] 81 Given he fllwing velciy field: v 1 x 1 v x v Obain (a) he paicle pahline equains using he cuen ime as he efeence ime (b) he elaive igh Cauchy- Geen defmain ens and (c) he Rivlin-Eicsen enss using he equain C I+ τ - A1+ τ - A / + (d) he Rivlin-Eicsen ens A and A using he ecusive equain [ ] [ D / D] + [ ][ ] + [ ] [ ] A A1 A1 v v A 1 ec 8-7 Cpyigh 1 Elsevie Inc

8 Ans (a) (a) Le x hen x x ( x x x τ ) i i x ie ibe he psiin a ime τ f he paicle which is a i i gives he pahline equain hus 1 dx1 dx dx v1 x1 (i) v x (ii) (iii) dτ dτ dτ wih he iniial cndiins: xi xi ( x1 x x ) Nw dx1 x ' 1 ln x1 τ + g ( x1 x x) g ( x1 x x) ln x1+ dτ ln ln ln ln ( τ x τ + x + x x x x e ) ( τ ) dx Similaly ( τ ) dτ hus ( τ) ( τ x xe x x e ) x x x x xe and x f ( x1 x x) x 1 1 ( τ) ( e τ e ) ( τ) ( τ F ) x e [ C] [ F] [ F ] e 1 1 (b) [ ] [ ] x x e a ime m Since ( τ ) 4 8 e 1m ( τ ) + ( τ ) m ( τ ) + heefe! 4 8 ( τ ) ( τ ) [ C ] [] I + ( τ ) ! (d) wih v 1 x 1 v x v [ v] [ 1] A DA ( 1) 1 A1 DA1 A1 A + [ 1][ ] + v + D A v D x (c) [ A ] [ A ] [ A ] [ A ] [ A ][ v] + [ v] [ A ] Nex 8-8 Cpyigh 1 Elsevie Inc

9 4 4 [ A ][ v ] hus [ A] [ ] + [ A][ v] + [ v] [ A ] 8 81 Given he fllwing velciy field: v 1 x 1 v x v x Obain (a) he paicle pahline equains using he cuen ime as he efeence ime (b) he elaive igh Cauchy-Geen defmain ens and (c) he Rivlin-Eicsen enss using he equain C I+ τ - A1+ τ - A / + (d) he Rivlin-Eicsen ens A and A using he ecusive equain [ ] [ D / D] + [ ][ ] + [ ] [ ] A A1 A1 v v A 1 ec Ans (a) Le x xi e ibe he psiin a ime τ f he paicle which is a x xieia ime hen x x x x τ gives he pahline equain hus i 1 dx1 dx dx v1 x1 (i) v x (ii) x (iii) dτ dτ dτ wih he iniial cndiins: xi xi ( x1 x x ) Nw dx1 x1 ln x1 τ + g ( x1 x x) g ( x1 x x) lnx1 dτ ln ln ln ln ( x τ x x x x xe τ + ) ( τ ) dx Similaly ( τ x ) x xe and ( τ x ) xe hus dτ ( x ) 1 xe τ 1 x xe τ x xe τ (b) ( τ) ( e τ e ) [ ] ' τ τ F x e [ C] [ F] [ F ] e ( τ) 4 ( τ e e ) Since ( τ ) 4 8 e 1+ ( τ ) + ( τ ) + ( τ ) +! 4( τ ) e 14 ( τ ) + ( τ ) ( τ ) +! C I heefe [ ] [] 8-9 Cpyigh 1 Elsevie Inc

10 4 8 ( τ ) ( τ ) + ( τ ) ! (c) [ A1] [ A] 4 [ A ] (d) wih v 1 x 1 v x v x [ v] [ 1] A 4 DA ( 1) 1 A1 DA1 A1 A + [ 1][ ] + v + D A v D x [ A ] [ A ][ v] + [ v] [ A ] Nex 4 4 [ A ][ v ] hus [ A] [ ] + [ A][ v] + [ v] [ A ] 8 Ec Given he fllwing velciy field: v 1 x v x 1 v Obain (a) he paicle pahline equains using he cuen ime as he efeence ime (b) he elaive igh Cauchy- Geen defmain ens and (c) he Rivlin-Eicsen enss using he equain C I+ τ - A1+ τ - A / + (d) he Rivlin-Eicsen ens A and A using he ecusive equain [ ] [ D / D] + [ ][ ] + [ ] [ ] A A1 A1 v v A 1 ec Ans (a) Le x xi e ibe he psiin a ime τ f he paicle which is a x xieia ime hen x x x x x τ gives he pahline equain hus i i 1 dx1 dx dx v1 x (i) v x1 (ii) (iii) dτ dτ dτ 8-1 Cpyigh 1 Elsevie Inc

11 wih he iniial cndiins: x x ( x x x ) i i 1 Nw dx1 d x1 dx d x1 (i) x x 1 x 1 dτ dτ dτ dτ x1 Asinh τ + Bcsh τ x1 Asinh + Bcsh (iv) 1 dx1 (ii) x Acsh τ + Bsinh τ x Acsh + Bsinh (v) dτ (iv) and (v) gives A x1sinh + xcsh B x1csh xsinh x x csh csh τ sinh sinh τ + x csh sinh τ sinh csh τ ( csh sinh τ sinh csh τ) ( csh csh τ sinh sinh τ) 1 1 x x1 + x ha is x1 x1csh ( τ ) + xsinh ( τ ) x x1sinh ( τ ) + xcsh ( τ ) x x (b) Since [ F ] [ x ] ( τ ) ( τ ) ( τ ) ( τ ) csh sinh sinh csh 1 csh x+ sinh x csh xsinh x [ C ] [ F ] [ F ] csh xsinh x sinh x+ csh x x ( τ ) 1 4 x 5 x cshx 1+ + O( x ) sinhx x+ + O( x ) x 5 csh x 1+ x + O( x ) sinh x x + O( x ) sinh xcsh x x+ + O( x ) 4 1+ x + x+ x + 4 [ C ] x + x + 1+ x + [] I + ( τ ) ( τ ) ( τ ) hus (c) [ A ] [ A ] [ A ] 4 8 ec 8-11 Cpyigh 1 Elsevie Inc

12 (d) wih v 1 x v x 1 v [ v] [ 1] A DA ( 1) 1 A1 DA1 A1 A + [ 1][ ] + v + D A v D x [ A ] [ A ][ v] + [ v] [ A ] [ A ][ v] [ A ] Nex [ A ][ v ] hus [ A] [ ] + [ A][ v] + [ v] [ A ] 8 θ z bain he secnd Rivlin-Eicsen enss A N N using he ecusive fmula dv Ans [ v] [ A ] [ ] [ ] 1 v + v d 815 Given he velciy field in cylindical cdinaes: v v v v Since ( A 1) cnsan independen f ime and space heefe DA A A v [ ] D Cpyigh 1 Elsevie Inc

13 [ ] ( ) + ( ) A A1 v v A1 + [ A ] A v + v A + hus A N N Using he equains given in Appendix 81 f cylindical cdinaes veify ha he θ cmpnen f he hid de ens is given by: 1 θ + θ ( ) θ θ Ans Fm he equains ( ) hm + qjγ qmi + iqγqmj n sum n m sum n q m xm and h 1 h h 1; Γ 1 Γ 1 all he Γ θ z θθ θθ we have ( ) h q q q q θ + Γ θ + Γ θ + θ Γ θθ + θ θγθθ hus θ θ 1 θ + θ ( ) + ( 1) ( 1 ) ( ) θ θ + θ θ θ θ 817 Using he equains given in Appendix 81 f cylindical cdinaes veify ha he θθ cmpnen f he hid de ens is given by: 1 θ θθ ( ) + θθ θ Ans Fm he equains ( ) hm + qjγ qmi + iqγqmj n sum n m sum n q m xm and h 1 h h 1; Γ 1 Γ 1 all he Γ we have θ z θθ θθ 8-1 Cpyigh 1 Elsevie Inc

14 θ θ ( ) h + Γ + Γ ( ) h + Γ + Γ θθ θ θθ θ θ 1 θ θθ ( ) + ( 1) ( 1 ) ( ) θθ θθ + + θ θθ θ θ qθ qθ q qθθ θ θθ θθ θθ 818 Using he equains given in Appendix 81 f spheical cdinaes veify ha he φ cmpnen f he hid de ens is given by: ( φ + φ ) 1 ( ) φ sinθ φ Ans Fm he equains ( ) hm + qjγ qmi + iqγqmj n sum n m sum n q m xm and h 1 hθ hφ sin θ; Γ θθ 1 Γ φφ sin θ Γ φφ sin θ Γ φφθ cs θ Γ θθ 1 Γ θφφ cs θ all he Γ we have ( ) h + Γ + Γ ( ) h + φ φ φ φ Γ + Γ ( ) ( sin θ) + ( sin ) ( sin ) φ φ θ + φ θ φ 1 φ + φ ( ) φ sinθ φ φ q qφ q qφ φ φ φφ φ φφ 819 Using he equains given in Appendix 81 f spheical cdinaes veify ha he φφφ cmpnen f he hid de ens is given by: ( φ + φ) ( θφ + φθ )c 1 φφ θ + + sinθ φ Ans Fm he equains ( ) hm + qjγ qmi + iqγqmj n sum n m sum n q m xm h 1 hθ hφ sin θ; Γ θθ 1 Γ φφ sin θ Γ sin θ Γ cs θ Γ 1 Γ cs θ all he Γ φφ φφθ θθ θφφ we have 8-14 Cpyigh 1 Elsevie Inc

15 φφ ( ) hφ + qφγ qφφ + φqγ φφφ qφφ φ φφ ( ) hφ + φ Γ φφ + θφ Γ θφφ + φ Γ φφ + φθ Γ φφφ θφφ φ φφ ( ) hφ + ( φ + φ) Γ φφ + ( θφ + φθ ) Γ φφφ θφφ φ φφ + ( φ + φ)( sinθ) + ( θφ + φθ ) csθ φ 1 φφ ( φ + φ) ( θφ + φθ ) cθ ( ) + + φφφ sin θ φ v vθ v vz bain (a) he fis Rivlin-Eicsen ens A 1 (b) A 1 (c) he secnd Rivlin-Eicsen enss A using he ecusive fmula v 1 v v v v () θ θ z v θ 1 vθ v θ dv Ans [ v] v + θ z d v 1 z vz v z θ z dv v() [ A1] [ v] + [ v] d DA1 A 1 ( 1) v D + A ( A1) v ( 1) v ( 1) v θ θ A θθ θ A zθ θ ( A1) v ( 1) v ( 1) v ( 1) v A θ θ θ A θθθ θ A θ zθ θ ( A1) v z θ ( A1) v z θ ( A1) v θ θθ zzθ θ A can be bained fm Appendix 81 as: 8 Given he velciy field in cylindical cdinaes: he cmpnens f he hid de ens 1 1 A A + A 1 Aθ A Aθθ 1 θ θ 1 + θθ θ 1 Az Aθ z A 1 zθ θ 1 Aθ A Aθθ 1 Aθ z Az A 1 + ( A ) θθ 1 ( A ) θ θθθ 1 + θzθ θ θ θ ( A ) ( A ) 8-15 Cpyigh 1 Elsevie Inc

16 1 Az Azθ 1 Azθ Az 1 Azz ( A 1) zθ ( A 1) + θ zθθ ( A 1) θ zzθ θ / v/ DA1 hus v / θ v/ D v/ dv / d [ A1][ v] dv / d v / [ v] [ A ] 1 dv / d dv / d v ()/ v/ ( ) dv/ d v/ DA1 A [ 1][ ] [ ] [ 1] D + A v + v A D N 81 Deive Eq (811) ie N+ 1 A A N N D + A v + v A Ans We had {see Eq (8117)} N ( N+ 1 D D ) ( Ddx DAN Ddx ds dx A N Ndx ds N 1 ) ANdx+ dx dx+ dx A + N ha is D D D D D N + 1 DAN ( ) N N ( ) D ds v dx A dx+ dx dx+ dx A v dx N + 1 D D DAN dx ( v) ANdx+ dx AN ( v) dx+ dx dx D DAN dx ( v) A N +AN ( v ) + d d N+ 1d D x x A x D N hus N+ 1 A A N N D + A v + v A 8 Le S D/ D+ WW whee is an bjecive ens and W is he spin ens shw ha S is bjecive ie () () S Q SQ Ans Since is bjecive heefe () () W dq/ d Q () + Q() WQ () heefe Q Q and fm Eq (811) 8-16 Cpyigh 1 Elsevie Inc

17 S D + W -W D dq D dq dq Q + Q Q + Q + QQ Q + QQ QWQ d D d d dq Q QQ QWQ Q () Q () d dq dq Nw Q() Q () I Q () Q () heefe he abve equain becmes d d dq D dq dq S Q + Q Q + Q Q + QWQ d D d d dq () Q d QWQ ha is D S Q() + W W () () () D Q Q SQ 8 Obain he viscsiy funcin and he w nmal sess funcin f he nnlinea 1 viscelasic fluid defined by S f () s s I C ds Ans F v 1 x v v we have [see Secin 89 Eq(891)] 1 ( τ ) 1+ ( τ ) ( τ ) [ ] 1 C τ τ τ + 1 C ( τ) ( τ ) s s s s 1 ( s) 1 C s 1 hus IC ( s) s 1 S S sf s ds sf s ds 1 μ S11 S s f( s) ds σ S S S s f s ds S S σ 84 Deive he fllwing ansfmain laws [Eqs(818) and Eq (811)] unde a change f fame ( τ) ( τ) and ( τ) V Q VQ R Q R Q 8-17 Cpyigh 1 Elsevie Inc

18 Ans Since FVR and F VR heefe fm ( τ) ( τ) ( τ) V R Q( τ) VR Q Q( τ) VQ ( τ) Q( τ) RQ whee ( τ ) ( τ ) symmey ens and ( τ ) F Q F Q we ge Q VQ is a Q R Q is an hgnal ens heefe he uniqueness f he pla decmpsiin leads V Q( τ) VQ ( τ) and R Q( τ) RQ L DJ τ 85 Fm ( DF τ and v shw ha Dτ Dτ τ τ ( + D+D [ne mispin in he pblem in ex] Ans Fm JL ( τ ) F ( τ) ( τ) F ( τ) we have D J L( τ ) D ( τ) D ( τ) D ( τ) F ( τ) F( τ) + F ( τ) F( τ) + F ( τ) ( τ) F Dτ Dτ Dτ Dτ DJL( τ ) D hus ( v) + + () v [Ne () () F F I] Dτ D τ Nw vd+w heefe DJL( τ ) D D + D + W + D + W + D + D + W + W Dτ D D τ D + D + D + ( W W ) + D + D D 86 Cnside 1 J τ F ( τ) ( τ) F 1 ( τ) Shw ha (a) J ( τ) U D U / Dτ τ and (b) D D U ( τ) / Dτ τ Dτ J v v D+D is bjecive Ans (a) Given J ( τ ) F ( τ) ( τ)( F ) ( τ) and ( τ ) ( τ) ( τ)( ) ( τ) U J F F U In a change f fame (see Secin 81 Eq(816) ( τ) ( τ) ( τ) F Q F Q s ha ( τ) ( τ) ( τ) and ( ) ( τ) ( τ)( ) ( τ) F Q F Q F Q F Q Als () Q() () Q () hus 1 1 ( τ) ( τ) ( τ) τ τ τ ( τ) ( τ) JU Q F Q Q Q Q F Q 1 1 () ( τ) τ ( τ) Q F F Q ha is J ( τ) Q J ( τ) Q and U U ( U ) () N N N N D JU τ / Dτ Q D J τ / Dτ Q hus 8-18 Cpyigh 1 Elsevie Inc

19 N U τ D J τ U () () N D J Q Q N N Dτ Dτ τ 1 1 (b) + [ ] + () τ DJU τ / Dτ DF / Dτ D/ Dτ D / Dτ τ τ τ F F τ F τ I DF / Dτ F τ + F τ DF / Dτ DF ( τ) / Dτ F [ DF / Dτ] F () [ DF / Dτ] ( ) ( τ ) τ v F ( v) hus D D D U ( τ) / Dτ τ D J v v D D W D+W ha is he uppe cnveced deivaive f can be wien: ˆ DJU ( τ ) D + W W ( D + D) ( D + D ) Dτ D τ Nw [ ] τ τ τ τ 87 Given he velciy field f a plane Cuee flw: v 1 v x 1 (a) F a Newnian fluid find he sess field [ ] and he c-ainal sess ae (b) Cnside a change f fame (change f bseve) descibed by: x 1 csω sinω x1 csω sinω [ ] x sinω csω x Q sinω csω Find and v v D W (c) Find he c-ainal sess ae f he saed fame (d) Veify ha he w sess aes ae elaed by he bjecive ensial elain Ans / / (a) [ v ] [ ] [ ] D / W / hus sess ens is [ p / p μ ] + μ p / μ p p μ / / p μ μ [ W] [ W ] μ p / / μ p μ D μ μ C-ainal sess ae is: D + μ μ (b) Fm Eq (5561) f Chape 5 we have [ v( x) ] [ Qv ] + ( d / d) [ ] ( d / d) Q x Qv + Q Q x hus v 1 csω sinω sinω csω csω sinω x 1 ω sin cs v v ω ω + csω sinω sinω csω Since x 8-19 Cpyigh 1 Elsevie Inc

20 x1 csω sinω x1 x 1 csωx1 sinωx v ( csωx1 sinωx) x + + sinω csω x heefe v 1 vsin 1 x csωsinωx 1 1 sin ωx ω + x ω ω v v csω x ( cs ωx1 + sinω csωx) x1 fm which we ge ( sin ω) / sin ω 1 ( v ) + ω cs ω ( sin ω) / 1 ( sin ω) / ( cs ω) / [ D ] [ ( cs ω) / ( sin ω) / W 1/ 1 ] + ω 1/ 1 (c) F he Newnian fluid he sess field in he saed-fame is: pμ( sin ω) μcs ω [ ] μ( cs ω) p+ μ( sin ω) whee he indeeminae pessue p is ime independen hus D ( cs ω) ( sin ω) D μω and ( sin ω) ( cs ω) μcs ω p+ μsin ω μcs ω p+ μsin ω W + ω p+ μ( sin ω) μcs ω p + μsin ω μcsω μcs ω pμsin ω μcs ω pμsin ω W + ω p μsin ω μcsω pμsin ω μcs ω hus μcs ω μsin ω cs ω sin ω W W + μω μsin ω μcs ω sin ω csω cs ω sin ω hus D / D+ W W μ sin ω csω (d) [ Q] [ Q ] csω sinω μ csω sinω csω sin ω μ sinω csω μ sinω csω sin ω csω hus we have [ Q] [ Q ] 88 Given he velciy field: v 1 x 1 v x v Obain (a) he sess field f a secnd-de fluid (b) he c-ainal deivaive f he sess ens Cpyigh 1 Elsevie Inc

21 v D W Ans (a) [ ] [ ] [ ] [ ] 4 4 [ A ] [ D] [ A ] 1 1 [ ] D / D+ ( ) + ( ) ( ) + ( ) A A1 A1 v v A1 A1 v v A he secnd-de fluid is defined by Eq(8186): pi+ μ A + μ A + μ A [ ] [ I] p + μ 1 + μ 4 + μ 4 11 μ1 4( μ μ) μ1 4( μ μ) p + + p+ + + p bain he pessue p we fis calculae he acceleain: x1 x 1 [ a] [ v/ ] + [ v][ v ] x x Equains f min ρai hen give x ρ x1 ρ x 1 ρ ( 1 )/ j p p p hus x x x p x + x +C (b) he c-ainal deivaive f : D/ D+ W-W Since W 1 D p p v1 v v1 v 1 D + x1 x x1 x p p x1 x 1 ρ x1 x 1 ρ v v1 x1 x + I 1 1 [] 8-1 Cpyigh 1 Elsevie Inc

22 ( 89 Shw ha he Lwe Cnveced deivaive f A 1 is A ie A1 A Ans Fm Eq(819) ( A A + A D+ DA DA / D+ A W WA + A D+ DA DA1 D A1 W D D W A1 DA1 D A1 v v A 1 A / / he Reine-Rivlin fluid is defined by he cnsiuive equain: φ pi+s S φ1 I I D+ I I D whee Ii ae he scala invaians f D Obain he sess cmpnens f his fluid in a simple sheaing flw Ans In a simple sheaing flw v 1 x v v / /4 [ ] / D D /4 I I 4 / /4 [ ] p[ I ] + φ1( /4 ) / + φ( /4) /4 81 he expnenial f a ens A is defined as: exp [ ] ens is exp [ A ] als bjecive? Ans Yes Because () () () () () () () () N N ( A ) Q() A Q () A Q AQ A Q AQ Q AQ Q A Q N 1 n A I+ A If A is an bjecive n! ha is ( A ) N is bjecive f all N As a cnsequence exp [ A] is bjecive 8 Why is i ha he fllwing cnsiuive equain is n accepable: pi+s S α v whee v is velciy and α is a cnsan Ans Because v is n bjecive 1 8- Cpyigh 1 Elsevie Inc

23 8 Le da and da dene he diffeenial aea vecs a ime τ and ime especively F an incmpessible fluid shw ha N N N 1 N D da / Dτ da D C / Dτ da da MNdA τ τ whee da is he magniude f da and he enss MN ae nwn as he Whie-Mezne enss - Ans Fm Eq (71) we have [ne hee da is he efeence aea and da is he aea a he 1 unning ime τ ] d ( de ) d a F F da F an incmpessible fluid ( de F ) 1 s ha a F da ha is 1 A C 1 da da F da F da da F F da da F F da da d da hus N N 1 N 1 D da D C D C d d d Nd whee N A N A A M A M N N Dτ Dτ Dτ τ τ τ 84 (a) Veify ha Oldyd's lwe cnveced deivaives f he ideniy ens I ae he Rivlin-Eicsen ens A N (b) Veify ha Oldyd uppe deivaives f he ideniy ens ae he negaive Whie-Mezne enss [see Pb 8 f he definiin f Whie-Mezne ens] Ans (a) he Nh lwe cnveced deivaive f is given by N N D JL / Dτ whee JL( τ) F ( τ) ( τ) F ( τ ) F I τ ( τ ) N N D J L D C JL τ F ( τ) F( τ) C ( τ) hus N A N Dτ Dτ τ τ (b) he Nh uppe cnveced deivaive f is given by N N 1 1 D JU / Dτ whee JU ( τ) F ( τ) ( τ)( F ) ( τ ) F I τ N N D J U D C JU ( τ ) F ( τ)( F ) ( τ) C ( τ) hus N M N N Dτ Dτ τ τ N 1 ( τ ) ( 85 Obain he equain D/ D+ v+ ( v) whee ( is he lwe cnveced deivaive f Ans By definiin he lwe cnveced deivaive is DJ L ( τ) / Dτ whee τ JL τ F ( τ) ( τ) F ( τ) hus D L( τ) / D τ D / D τ J F ( ) F ( ) τ τ + F ()[ τ] F () + F () () F Nw F F ( F ) ( v ) [see Eq(81)] and D / D D / D τ τ τ D τ / Dτ D / D D / D τ heefe F I 8- Cpyigh 1 Elsevie Inc

24 ( τ ) ( DJL D + v+ ( v) Dτ D τ 86 Cnside he fllwing cnsiuive equain: ( D / D) whee ( D / D) + S+ S μd S S α DS+SD and S is c-ainal deivaive f S Obain he shea sess funcin and he w nmal sess funcins f his fluid Ans Wih v 1 x v v he ae f defmain ens and spin ens ae: / / [ ] / [ ] / S D W Since he flw is seady he c-ainal deivaive is f symmeic S: + [ ] [ ] S SW WS SW SW Nw S1 S11 S1 S S SW S S 1 S11 S1 S 1 SW S S1 S1 S11 S S S S S 11 S S 1 1 [ ] [ ] S1 S11 + S S SD + DS S S 11 S S S S1 S S1 hus D S/ D S + α ( DS+SD ) gives ( α 1) ( 1+ α) + ( α 1) ( α 1) D S ( 1+ α) + ( α 1) ( 1+ α) ( 1+ α) D ( α 1) S ( 1+ α) S S1 S11 S S S11 S S1 S1 1 D S heefe S+ μd D S11 + ( α 1) S1 (i) S1 + ( 1 + α) S11 + ( α 1 ) S μ (ii) S1 + ( α 1) S (iii) S + ( 1+ α) S1 (iv) S + ( 1+ α ) S1 (v) S (vi) Nw (iii) (v) and (vi) give S1 S S S 1 α S Eq(iv) gives ( 1 α) ( α )( ) S + S hus wih 1 A 1+ 1 we have Eq (i) gives ( ) 11 1 S1 μ / A( ) S11 μ 1 α / A( ) S μ 1 + α / A( ) he shea sess funcin is S1 μ / A( ) he nmal sess funcins ae: 1 11 σ S S μ / A( ) σ S S μ 1 + α / A( ) 8-4 Cpyigh 1 Elsevie Inc

25 87 Obain he appaen viscsiy and he nmal sess funcins f he Oldyd -cnsan fluid [see (C) f Secin 8] Ans F he simple sheaing flw / / [ D] / [ ] / W S1 S11 S S S [ ] + [ SW] [ WS ] ( / ) S11 S S1 S 1 S S1 S1 S11 + S S [ SD ] + [ DS ] ( / ) S11 S S1 S + 1 S S1 4S1 S S SS ˆ ( SD+DS ) ( / ) S S / 1 D [ ] + [ DW] [ WD ] / DD 1 4 μ μ ˆ DD D ( ˆ μ D+ D ) μ S11 1S1 S1 1S S1 1S S+ ˆ 1S S1 1S S S S1 1S S S S+ Sˆ μ D+ Dˆ 1 ( ) S11 1S1 S1 1S S1 1S μ μ S S S S μ 1 1 S1 1S S S hus S S S S S μ S S μ s ha we have μ 11 μ ( 1 ) all he S S S he appaen viscsiy is η S1 / μ σ1 11 μ 1 σ1 8-5 Cpyigh 1 Elsevie Inc

26 88 Obain he appaen viscsiy and he nmal sess funcins f he Oldyd 4-cnsan fluid [see (D) f Secin 8] Ans F he simple sheaing flw / / S1 S11 S S [ ] / [ ] / D W S S11 S S1 S 1 S S1 4S1 S S SS ˆ ( SD+DS ) ( / ) S S / 1 D [ ] + [ DW] [ WD ] / DD ˆ DD D 1 1 / μ ( S)[ D ] μ ( S11 + S + S ) / 1 ( ) ( ˆ ) S+ Sˆ + μ S D μ D+ D S S S S + μ S + S + S / S S S1 1S + μ( S11 + S + S )/ S S S1 1S S S μ μ μ hus S S S S1 S11 1S1 μ S1 + μs11 / μ Fm which we ge wih B( ) (1 + μ ) 1 ( ) ( + ) 11 μ 1 / 1 μ 1 μ / S B S B hus he appaen viscsiy is: η μ μ S / (1 + )/ B( ) ( 1 )/ B( ) Nmal sess funcins ae: σ μ σ 8-6 Cpyigh 1 Elsevie Inc

27 1 1 { n } 89 Given [ Q] 1 and [ N] 1 and i 1 and { n } A N+N A N N (a) Veify ha QA Q A and QA Q A (b) Fm i 1 1 pi+f( A1 A) and ( Qf A1 A Q f QA1Q QAQ ) shw ha Q( ) Q ( ) and (c) Fm he esuls f pa (b) shw ha he viscmeic funcins have he ppeies: S( ) S ( ) σ1( ) σ1( ) σ( ) σ( ) 1 Ans (a) [ A1] ( N+N ) and 1 A 1 1 QA1Q 1 1 [ A 1] QAQ 1 1 [ A ] 1 1 (b) Q( ) Q Q pi+f( A1 A) Q pi+qf( A1 A) Q pi+f( QAQ 1 QAQ ) pi+f( A1( ) A( ) ) A A and A A hus Nw 1 1 Q( ) Q pi + f ( A1( ) A( ) ) and ( ) p 1( ) ( ) ha is Q( ) Q ( ) and Q( ) Q ( ) (c) [ ] Q Q I + f A A Q Q ( ) ( ) 11 ( ) 11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) S( ) S( ) Q Q hus σ1 σ1 σ σ [Ne in viscmeic flw 1 ] 84 F he velciy field given in example 81 ie v vθ vz v he sess cmpnens in ems f he shea sess funcin S( ) and he nmal sess funcins σ ( ) and σ ( ) whee dv/ d (b) bain he fllwing velciy disibuin f he 1 (a) bain 8-7 Cpyigh 1 Elsevie Inc

28 Piseuille flw unde a pessue gadien f ( f ): v γ ( f /) d whee γ is he invese shea sess funcin and (c)bain he elain γ π Rf / 1/( R f ) f Q / f Ans (a) In example 81 we see ha he velciy field v vθ vz v descibes a viscmeic flw wih he nnze Rivlin-Eicsen enss given by [ A1] [ A ] n i whee n1 ez n e n e θ and dv / d (see Example 81 bu ne he diffeences in he de f bases) hus he sess cmpnens wih espec he basis { n i } ae given by (See secin 8): Sz τ( ) Szz S σ1( ) S Sθθ σ( ) Szθ Sθ (b) Wih S depending nly n he equains f min becme: S S Sθθ p p 1 p + ( i) (ii) ( Sz ) ( iii) θ z p p p Eq (i) gives z Eq (ii) gives θ z and Eq (iii) gives z z hus p / z a cnsan f Eq (iii) becmes 1 f C ( Sz ) f ( Sz ) f Sz + Since S z mus be finie a hus C and S f / Nw S τ ( ) whee τ ( ) is he shea sess funcin and dv/ d z z hus τ ( ) f / Inveing his equain we have τ ( f /) γ ( f /) n i R 1 Since τ ( ) is an dd funcin f heefe γ is als an dd funcin f s ha γ f / dv/ d γ f / dv γ f / d R R hus v( R) v γ ( f /) d Since v( R ) heefe γ ( /) R (c) he vlume dischage is given by Q v π d heefe v f d R R R dv R dv R Q v d v d d f d π π π π γ ( /) d d R hus Q/ π γ ( f /) d Le f / s d ds / f and 4 s / f hen R R f / R f / s s Q / π γ f / d 8 s / f γ sds f Q / π 8 sγ sds Diffeeniaing he las equain wih espec f we bain 1 f Q Rf Rf Rf R Rf Rf Rf 8 γ 8 γ R f γ π f f Rf 1 ( f Q) hus γ π R f f 8-8 Cpyigh 1 Elsevie Inc

29 8-9 Cpyigh 1 Elsevie Inc