Chapter03

Mechanics of Fluids

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
34 CHAPTER 3 Introduction to Fluids in Motion 3.1 3.2 Pathlines: Release several at an instant in time and take a time exposure of the subsequent motions of the bulbs. Sreakline: Continue to release the devises at a given location and after the last one is released, take a snapshot of the “line” of bulbs. Repeat this for several different release locations for additional streaklines. 3.3 3.4 streakline pathline streamline streakline pathline hose boy time t t = 0 streamlines t = 2 hr pathline t = 2 hr streakline at t = 3 hr y x
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
35 3.5 a) u dx dt t v dy dt t = = + = = 2 2 2 x t t c y t c = + + = + 2 1 2 2 2 = + y y 2 - + = x xy y y 2 2 2 4 parabola. b) x t t c c c = + + = - = - 2 1 1 2 2 8 4 . , . and = + + ± + - y y 4 2 4 8 ( ) - + + - = x xy y x y 2 2 2 8 12 0. parabola. 3.6 ˆ ˆˆ () ˆ ˆ ˆ using , . z V d r ud y vdx V u i v j wk d r dx i dy j dzk ij kj ik ×=- =++ × = × =- v v v v 3.7 Lagrangian: Several college students would be hired to ride bikes around the various roads, making notes of quantities of interest. Eulerian: Several college students would be positioned at each intersection and quantities would be recorded as a function of time. 3.8 a) At t V = = = 2 0 0 0 2 2 2 and m / s ( , , ) . At t V = - = + = 2 1 2 0 3 2 3 606 2 2 m / s ( , , ) . . b) At t V = = 2 0 0 0 0 ( , , ) . t V = - = - + - = 2 1 2 0 2 8 8 246 2 2 and m/s ( , , ) ( ) ( ) . . c) At t V = = - = 2 0 0 0 4 4 2 m / s ( , , ) ( ) . t V = - = + - + - = 2 1 2 0 2 4 4 6 2 2 2 m / s ( , , ) ( ) ( ) . 3.9 ( D ) 5 ˆ ( 51.4 1 0) j - A simultaneous solution yields 4/5 and 3/5. nn == (They must both have the same sign. 3.10 a) cos $ / ( )/ . . . a a = = + + = = v o V i V 1 2 3 2 0 832 33 69 2 2 v V n i j n i n j n n n n n n n n x y x y x y y x x x = + + = + = + = = - + = $ . ( $ $ ) ( $ $ ) . 0 3 2 0 3 2 0 1 3 2 9 4 1 2 2 2 2 = = - = - n n n i j x y 2 13 3 13 1 13 2 3 , $ ( $ $ ). or 39.8 o y x streamlines t = 5 s (27, 21) (35, 25)
Background image of page 2
36 b) cos $ / / ( ) ( ) . . a a = = - - + - = - = v o V i V 2 2 8 0 2425 104 2 2 v V n i j n i n j n n n n n n n n x y x y x y x y y y = - - + = - - = + = = - + = $ . ( $ $ ) ( $ $ ) . 0 2 8 0 2 8 0 1 4 16 1 2 2 2 2 = = - = - + n n n i j y x 1 17 4 17 1 17 4 , $ ( $ $ ). or c) $ / / ( ) . . . a a = = + - = = - v o V i V 5 5 8 0 6202 51 67 2 2 v V n i j n i n j n n n n n n n n x y x y x y x y y y = - + = - = + = = + = $ . ( $ $ ) ( $ $ ) . 0 5 8 0 5 8 0 1 8 5 64 25 1 2 2 2 2 = = = + n n n i j y x 5 89 8 89 1 89 8 5 , $ ( $ $ or 3.11 a) [ ] v v V dr x i xtj dxi dyj × = + + × + = 0 2 0 . ( ) $ $ ( $ $ ) . + - = + = ( ) . x dy xtdx t xdx x dy 2 0 2 Integrate: [ ] t x dy t x n x y C + = - + = + 2 2 2 . . l 2 1 2 3 2 ( ) . - = - + l n C C = 0.8028. [ ] t x n x y - + = + 2 2 0 8028 l . b) [ ] v v V dr xyi y j dxi dyj × = - × + = 0 2 0 2 . $ $ ( $ $ ) . + = = - xydy y dx dx x dy y 2 0 2 2 . Integrate: 2 2 1 2 l l l l nx n y C n n C = - = - - ( / ( ) ( / ). = - = - - = - C n y x y 2 2 2 2 2 . ( / ). . l l c) [ ] v v V dr x i y tj dxi dyj × = + - × + = 0 4 0 2 2 . ( ) $ $ ( $ $ ) . ( ) . x dy y tdx tdx x dy y 2 2 2 2 4 0 4 + + = + = - Integrate: t x C y C 2 2 1 2 2 1 2 1 2 1 1 tan . . - - +  = +  = - = - -  = - C yt x 0 9636 2 0 9636 2 1 . . .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
37 3.12 ( C ) 2 ˆ ˆ ˆ ˆ ˆˆ 2 ( 2 ) ( 2 2 ) 1 68 16. VVVV a uvw x y y iy x i y j i ij txyz ∂∂∂∂ = +++ = - - = - -+ vvvv v 22 8 1 6 17.89 m/s a =+= 3.13 a) DV Dt u V x v V y w V z V t v v v v v = + + + =0 . b) u V x v V y w V z V t x i y j xi yj v v v v + + + = + = + 2 2 2 2 4 4 ( $ ) ( $ ) $ $ = 8 4 $ $ i j - c) u V x v V y w V z V t x t xti ytj xyt xtj ztk x i xyj v v v v + + + = + + + + + 2 2 2 2 2 2 2 2 ( $ $ ) ( $ $ ) $ + = - - 2 68 100 54 yzk i j k $ $ $ d) u V x v V y w V z V t x i yzj xyz xzj tz xyj tk zk v v v v + + + = - - - + - + + ( $ $ ) ( $ ) ( $ $ ) $ 2 2 2 2 = xi yz x yz xyzt j zt z k $ ( ) $ ( ) $ - - + + + 2 4 2 2 2 2 = 2 114 15 $ $ $ i j k - + 3.14
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 16

Chapter03 - CHAPTER 3 Introduction to Fluids in Motion 3.1...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online