Ch.1. Group Work Units. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

Transcription

1 Ch.. Group Work Unis Coninuum Mechanics Course (MMC) - ETSECCPB - UPC

2 Uni 2 Jusify wheher he following saemens are rue or false: a) Two sreamlines, corresponding o a same insan of ime, can never inersec each oher, unless he velociy a he inersecion poin is null. b) Two differen rajecories can never inersec each oher.

3 Uni - Soluion 3 a) The saemen is TRUE because if wo sreamlines would inersec each oher, he inersecion poin would be assigned wo differen velociy direcions corresponding o he angens of boh sreamlines. This would be physically impossible unless he velociy a such poin is null. v v 2 b) The saemen is FALSE because differen paricles may pass hrough a given spaial poin a differen ime insans, herefore wo rajecories may indeed inersec

4 Uni 2 4 Consider he following velociy field in maerial descripion: A V( X,) Ae X BX CX3 T Wih A, B and C consans. Obain is spaial descripion and he condiions ha consans A, B and C mus fulfill, so ha he moion is feasible for 0 < <. Consider 0 as he reference ime where x X.

5 Uni 2 - Soluion The maerial descripion of he velociy field is : V( X,) xx (,) Through he inegraion of he velociy we can obain he equaions of moion: x x x 2 3 Ae X dx Ae X d x e X + C A A A BX dx BX d x B X + C CX dx CX d x CX + C Applying he condiion xx (,0) X C C C 0 X 2 2 X 3 3 x e X A x BX + X 2 x CX + X

6 Uni 2 - Soluion From he equaions of moion, inverse equaions of moion can be easily obained: x e X A x BX + X 2 x CX + X X e x A X x B e x 2 x3 X3 C + 2 A 2 2 The spaial descripion of he velociy is: v( x,) V( X( x,),) So we jus have o subsiue: T A A A x3 V( X( x,),) Ae e x, Be x, C C + T A Cx3 v( x,) Ax, Be x, C +

7 Uni 2 - Soluion For he moion o be feasible J F > 0 A e A F B 0 J F e ( c + ) > C f () e A F Jacobian marix F : x x x X X2 X3 x 2 x2 x2 X X2 X3 x 3 x3 x3 X X2 X3 A e > 0, A, ( C + ) > 0 C >

8 Uni 2 - Soluion f () For 0 < < : C > C > 0

9 Uni 3 For he moion defined by he velociy field: z vx 2 ax, vy by, vz + c a) Find he equaions of he rajecories in canonical from and he equaion of he sreamlines. b) Deermine he possible values for a, b and c such ha he moion has a physical sense for 0,. [ ) c) Consider now a b c. Obain he expression of he densiy knowing ha is value a he reference ime 0 is ρ cons. Consider ρ0 ρ F o 9

10 Uni 3 - Soluion dx() a) Trajecories: v( x( ), ) d dx d dy d 2ax dx 2ad ln( x) 2a k x Ce x + by dy bd ln( y) b + k2 y C2e y dz z C3 dz d ln( z) ln( + c) + k3 z d + c z + c + c 2a b Applying he consisency condiion: xx (,0) X C C C 2 3 X Y cz x y z Xe Ye 2a b Zc + c

11 Uni 3 - Soluion Sreamlines: dx( λ) λ λ v( x( ), ) d dx 2ax dx 2adλ ln( x) 2aλ + k x Ce dλ x dy by dy bdλ ln( y) bλ + k2 y C2e dλ y dz z λ dz dλ ln( z) + k3 z C3e dλ + c z + c + c 2aλ bλ λ + c x y Ce Ce 2 z Ce 3 2aλ bλ λ + c

12 Uni 3 - Soluion b) For he moion o be feasible J F > 0 F e b 0 e 0 2a c + c J 2a b > F e c + c 0 f () (2 a b ) e e (2 a b) > 0; ab, c + c > 0

13 Uni 3 - Soluion C f () + C c + c > 0 C > 0 c) ρ0 ρ F a b c J 2 e 0 0 e F 0 e ρ ρ O + e

14 Uni 4 Given he following moion: x Xe y Ye a b z Z + cx Calculae he acceleraion as a funcion of ime of : ( ) a) An observer locaed a he spaial poin,,. b) An observer ravelling wih he paricle ha a 0 was locaed a he spaial poin,,. ( ) ( ) c) An observer locaed a he spaial poin,, measuring he acceleraion as he difference, per uni of ime, of he velociies of he paricles ha pass hrough ha spaial poin.

15 Uni 4 - Soluion a) Firs mehod: The spaial descripion of acceleraion is : ax (,) AX ( ( x,),) Through derivaion, we can obain he maerial descripion of velociy from he equaions of moion: x a Vx Vx axe xx (,) y b V( X,) Vy Vy bye z Vz Vz cx V( X,) The maerial descripion of acceleraion is : AX (,) A Through derivaion, we can obain he maerial descripion of acceleraion from he maerial A descripion of velociy: A x y z 0 2 a a Xe 2 b b Ye

16 Also from he equaions of moion, we can obain he inverse equaions of moion: x Xe y Ye a b z Z + cx X xe Y ye a b a Z z cxe Now we jus have o subsiue: ax (,) AX ( ( x,),) AX x 2 a a 2 b b T ( (,),) ( a xe e, b ye e,0) 2 2 ax (, ) ( axby,,0) T * * 2 2 For x (,,) ax (, ) ( a, b,0) T

17 Uni 4 - Soluion Second mehod: The spaial descripion of acceleraion can also be obained as: dv( x,) v( x,) ax (,) + v v d We can find he spaial descripion of velociy jus by subsiuing: vx (,) VXx ( (,),) a a b b a VXx ( (,),) ( axe e, bye e, cxe ) a (,) (,, ) v x ax by cxe T a 2 0 a 0 ce a x v( x,) a 2 ax (,) + v v 0 + ax, by, cxe 0 b 0 b y a acxe * For x (,,) * 2 2 ax (, ) ( a, b,0) T

18 b) 2 a 2 b (, ) (,,0) AX a Xe b Ye We mus deermine he maerial poin which a 0 was locaed a he spaial poin x* (,,). This can be done by subsiuing he poin ino he equaions of moion. * a 0 * Xe X (, 0) (,,) Ye Y (,,) * * * Z + cx 0 Z * * * b 0 * * x X X A X * (,,) AX * 2 a 2 b T (, ) ( ae, be,0) c) v lim v 0 * For x (,,) * vx (,) (0,0, ace ) a T

19 Uni 5 The spaial descripion of a velociy field in a fluid is: v ye ze 0 A ime insan a colored ink is poured in he plane y 0. Obain he spaial equaion of he sain along ime.

20 Uni 5 - Soluion Firs mehod: The equaions of he rajecories are given by: dx() v( x( ), ) d Taking he componens of he velociy field and inegraing hem, he equaions of he rajecory are obained: dz 0 dz 0 z C3 d dy d dx d ze dy ze d C e d y C e + C ye dx ye ( Ce + C) ed ( C + Ce) d x C Ce + C So he equaions of he rajecory are: x C Ce + C y Ce + C z C 3 Remark: The equaions of he rajecory are he equaions of moion paricularized for a given paricle. So, hese are, in fac, he equaions of moion.

21 Uni 5 - Soluion Applying he consisency condiion: xx (,0) X C X + Y Z C Y Z C 2 3 Z ( )( ) ( ) x X + Y Z e + Z y Y + Z e z Z From he equaions of moion, inverse equaions of moion can be easily obained: ( )( ) ( ) x X + Y Z e + Z y Y + Z e z Z ( )( ) ( ) X x y ze e + z Y y z e Z z

22 Uni 5 - Soluion Inroducing and y * 0 ino he y-equaion of moion, he maerial equaion of he surface in which he ink is poured ( sreak-surface ) is obained: * y Y Z e ( ) + 0 The spaial equaion of his surface can be obained by subsiuing: ( y z( e )) + z( e ) 0 y + z( e e ) 0

23 Uni 5 - Soluion Second mehod: The equaions of he rajecories are obained in he same way as in he previous mehod: x C Ce + C 3 2 y Ce + C z C Then, he values of he consans are obained aking as he reference ime: X C Ce + C C Ce + C Y Ce 3 + C2 Ce 3 + C 2 Z C ( ) C X Z + Y Ze e C2 Y Ze C3 Z

24 Uni 5 - Soluion Then, he equaions of moion aking as he reference ime are: ( ) ( ) ( ) ( ) x Z Y Ze e + X Z + Y Ze e y Ze + Y Ze z Z ( 2) ( ) ( ) ( ) x X + Y e e + Z + e y Z e e + Y z Z Then, for, y* Y 0 and z Z, so he spaial equaion of his surface is obained: ( ) + 0 y + z( e e ) 0 y z e e

25 Uni 6 Exam quesion A coninuous medium keeps a uniform densiy, which varies wih ime bu no in space, hrough al spaial poins. Indicae which of he following saemens are TRUE: a ρ 0 c dρ d ρ( x,) b dρ d 0 d ρ 0

26 Uni 6 - Soluion Densiy varies wih ime bu no in space: ρ( x,) ρ() ρ a) 0 Saemen A is FALSE. ρ 0 ρ 0 dρ ρ ρ b) + v ρ 0 Saemen B is TRUE. d c) Saemen C is TRUE. d) ρ 0 Saemen D is TRUE.

27 Uni 6 Exam quesion A coninuous medium keeps a uniform densiy, which varies wih ime bu no in space, hrough al spaial poins. Indicae which of he following saemens are TRUE: a ρ 0 c dρ d ρ( x,) b dρ d 0 d ρ 0

28 Uni 7 Exam quesion The equaions of moion of a coninuous medium are given by: x Xe, y Y( + ), z Z( + ) Indicae which of he following saemens are TRUE a The acceleraion a ime measured by an observer locaed a he spaial poin is c (,, ) [ ],, T The acceleraion a ime measured by an observer locaed a he spaial poin (,,), who would be measuring he pseudo-acceleraion as he incremen of velociy per uni of ime a ha poin, is. [ ],0,0 T b The acceleraion a ime measured by an observer who is ravelling wih he paricle and a 0 ( ) was locaed a he spaial poin is. e,0,0 T,, [ ] d The velociy measured a ime by and observer locaed a he (,,),, 2 2 spaial poin is. T

29 Uni 7 - Soluion From he equaions of moion, inverse equaions of moion can be easily obained: x Xe X xe y Y( + ) y Y z Z( + ) + z Z + The maerial descripion of velociy is : xx (,) V( X,) V( X,) Xe, Y, Z The spaial descripion of velociy is : v( x,) V( X( x,),) y z v( x,) x,, + + T T

30 Uni 7 - Soluion The maerial descripion of acceleraion is : V( X,) AX (,) AX (, ) Xe,0,0 T The spaial descripion of acceleraion can be obained as: dv( x,) v( x,) (,) + d ax v v ax [ x ] (, ),0,0 T 0 * a) For ax (, ) Saemen A is TRUE. * [,0,0] T x (,,) b) We mus deermine he maerial poin which a 0 was locaed a he spaial poin x (,,) : * xx (, 0) (,,) X (,,)

31 Uni 7 - Soluion For * X (,,) * AX (, ),0,0 T [ e ] Saemen B is TRUE. c) v v y z 0,, ( + ) ( + ) 2 2 T For * x (,,) * v( x,) 0,, 4 4 T Saemen C is FALSE. d) For * x (,,) * v( x, ),, 2 2 T Saemen D is TRUE.

32 Uni 7 Exam quesion The equaions of moion of a coninuous medium are given by: x Xe, y Y( + ), z Z( + ) Indicae which of he following saemens are TRUE a The acceleraion a ime measured by an observer locaed a he spaial poin is c (,, ) [ ],, T The acceleraion a ime measured by an observer locaed a he spaial poin (,,), who would be measuring he pseudo-acceleraion as he incremen of velociy per uni of ime a ha poin, is. [ ],0,0 T b The acceleraion a ime measured by an observer who is ravelling wih he paricle and a 0 ( ) was locaed a he spaial poin is. e,0,0 T,, [ ] d The velociy measured a ime by and observer locaed a he (,,),, 2 2 spaial poin is. T