Review of Vector Algebra and Vector Calculus Operations

Transcription

1 Revew of Vecto Algeba and Vecto Calculus Opeatons Tpes of vaables n Flud Mechancs Repesentaton of vectos Dffeent coodnate sstems Base vecto elatons Scala and vecto poducts Stess Newton s law of vscost Stess at a pont Stess epesentaton Foce on a suface Pessue and vscous components Eamples Vecto calculus Gadent opeato Gadents of scalas & vectos Dvegence of vecto and 2 nd ode tensos Othe vecto calculus opeatons

2 Impotant Vaable Tpes n Flud Mechancs Scalas A value (1 component); no decton Speed, vscost, pessue, othe Vectos Magntude and decton 3 component epesentaton Veloct, foce, momentum, toque, gavt, othe Second ode quanttes (tensos) 2 dectons, 9 components Stess, veloct gadents, ate of defomaton, othe

3 Geneal Repesentaton of a Vecto 3 v 3 Unt base vectos: v v v 2 2,, v v v v v v vv + v + v v v + v + v v v v (mpled summaton conventon) Summaton conventon: Repeated ndces mpl summaton fom 1 to 3

4 Vecto Repesentaton n Rectangula Coodnates v k j,,,, 1 2 3,,, j, k , j 1, k 1 Othonomal base vectos Fo eample, the veloct vecto s gven b: 3 v v v v + v + v whee o v v + v j+ v k v v(,,, t) v v (,,, t) v v (,,, t)

5 k Clndcal Coodnates Repesentaton e e,,,, 1 2 3,, e, e, k e 1, e 1, k 1 Othonomal base vectos The veloct s then epesented b: v and 3 v v v e + v e + v 1 v v (,,, t) v v (,,, t) v v (,,, t) k

6 2D Clndcal Coodnates e j cos e e sn j cos e sn Relatons: e cos + sn j e sn + cos j cos e sn e j sn e + cos e

7 Sphecal Coodnates e,,,, φ 1 2 3,, e, e, e Othonomal base vectos φ φ e e φ e e φ e In decton of coodnate n plane and pependcula to e In decton pependcula to e and e φ

8 Rectangula and Sphecal Base Vecto Relatons sn cosφ e + cos cosφ e sn φ e j sn snφ e + cos snφ e + cos φ e k cos e sn e e sn cosφ + sn snφ j + cos k e cos cosφ + cos snφ j sn k e snφ + cosφ j φ φ φ

9 Impotant Vecto Opeatons Scala Poduct (o Dot Poduct) a b a b cos a Note: 1, j j 1, k k 1 j j j k k j k k 0 ( ) Fnd n tems of components: ab ( a + a j+ ak) ( b+ b j+ bk) ab + ab + ab a a ( a a) [( a + a j+ a k) ( a + a j+ a k)] 1/2 1/2 ( a + a + a ) /2 a ( a + a j+ ak) a b a a + a j+ a k a

10 The Dot Poduct Geneal Catesan Coodnates But Then 3 3 ab a b j j 1 j 1 a b ab j j j j δ 1 f j j 0 f j j j 1 3 ab abδ ab ab ab + a b + ab j

11 Scala (Dot) Poduct (cont) Dot Poduct useful n man was, e.g., detemnng elatons between vecto components n dffeent coodnate sstems. In the eample below we detemne the component of the veloct n tems of the clndcal and components v v ( ve+ v e+ vk) v cos v sn e j e e cos + sn j e sn + cos j cos e sn e j sn e + cos e

12 Unt Nomal to a Suface 2D: n n + n j n n n cos, n j cos(90 ) sn n cos + sn j Eample of use: Detemnng flow ate acoss a suface j n [ ] ( ) ( ) ( cos sn) Q v n A v+ vj ( n+ nj) A v + v j (cos + sn j) A v + v A v A n

13 Dffeental Aea: Dffeental Aea Vecto da Dffeental Aea Vecto: dsd n n + n k sn + cos k da nda sn da + cos da k da sn dsd + cos dsd k d sn, cos ds d ds da dd + dd k da n n d ds d ds d 2D: da da + da k, da dd, da dd 3D: da da + da j+ da k, da dd, da dd, da dd

14 The Vecto Poduct (o Coss Poduct) Defnton a b a b sn e Hee e s a unt vecto pependcula to a and b and n the decton of a ght hand ule. Useful n detemnng moments, angula momentum Τ F and H m v What s ( j)? ( e )?

15 Local Inteactons and Stesses n Flow Felds V Smple Shea Flow: Fed Flud fcton between laes A aea between laes and to F Δv Δ ~ A o F A dv μ d o τ dv μ d Smple fom of Newton s Law of Vscost τ s the shea stess (fst nde- aea oentaton; second, decton of foce)

16 Inteactons n a Convegng Flow v v (, ), v v (, ) decton: Elements ae beng stetched: Elements ae beng sheaed: decton: Elements ae beng squeeed: Elements ae beng sheaed: τ?? τ Geneal 3 D Flow How man components?

17 Newton s Law of Vscost Movng Plate V B F A τ V μ B dv μ d Hgh μ : Thck flud (e.g., molasses) Low μ : Thn flud (e.g., a) Fo a geneal flow, we would have 9 stess components: τ, τ, τ, τ, τ, τ, τ, τ, τ

18 Stess at a Pont n a Flud n ΔA ΔF ΔF T lm n n ΔA 0 ΔA df da Two dectons: n (suface oentaton) ΔF (decton of foce actng on suface)

19 Calculaton of Foce on a Suface Fom T n df da F n o df n T da Suface Hee T can be dvded nto a pessue pat and a pat due to the defomaton of the flud,.e., T pnn+ τ o nt pn+ nτ Then F pnda+ da A n τ A If the flud s statc, o n non-defomng moton, then τ 0.

20 Rect. Cood: Repesentaton of the Stess τ τ τ + τ j + τ k + τ j + τ jj + τ jk+ τ k+ τ kj + τ kk Note: j j (ode counts) Cl. Cood: τ τ ee + τ ee + τ ek+ τ ee... + τ ke + τ kk How do we fnd τ fom τ, τ,... τ? τ e τ e Mess, but smple eecse (HW)...

21 Smple Eample: Calculaton of Foce on Suface Smple shea flow wth v V B Detemne F and F on bottom plate. Assume pessue on bottom plate to be p. At bottom plate: n j Then F ( pj) da+ ( j τ) da A 0 0 A 0 V B Fed dv V and F F ( jτ ) da τ 0 da μ da μ A 0 d B A A A 0 and F Fj p da p A A 0 0

22 Vecto Calculus Opeatons 1. The gadent opeato: 3 + j + k (ectangula cood.) 1 1 e + e + k (clndcal cood.) 2. Gadent of a scala funton: p p p p p + j + k Late n couse we shall use the Eule equaton of moton: v ρ + v v p + τ + ρg t What does v epesent? v v? τ?

23 3. Gadent of a vecto Vecto Calculus Opeatons (cont) v j v ( v j j) j? (9 components) 4. Dvegence of a vecto v v v v ( v j j) ( j) δj v v v + + j j 5. Dvegence of 2nd ode quantt ( τ) τ jk τ jk τ ( τ jk jk) ( j) k δjk τ τ k τ τ τ τ τ τ k j+ τ τ τ k

24 6. The tem v v Vecto Calculus Opeatons (cont) v v ( v) j ( vkk) vδj ( vkk) j j v v k v v v v v vj k v + v + v + v + v + v + j j + v v v + v + v v k