Chapter 2. Review of Hydrodynamics and Vector Analysis
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1 her. Ree o Hdrodmcs d Vecor Alss. Tlor seres L L L L ' ' L L " " " M L L! " ' L " ' I s o he c e romed he Tlor seres.
2 O he oher hd ' " L
3 . osero o mss -dreco: L L IN ] OUT [mss l [mss l] mss ccmled h me ce -dreco Ne ccmlo 3 drecos or me O he oher hd mss crese ce drg s ccmlo 3 drecos drg Ec cosero o mss eqo
4 D D here D D ol dere or merl dere Bl modls d d E d d d d E 9 N/m.7 E or er comressle ld For comressle ld o eqo
5 .3 Hdrosc ressre Neo s d l: o lo Q m F s l F θ cos g g g l F θ As he rsm ecomes smller d smller or W e he Pressre s sclr. B he ressre orce cg o smerged od cs he dreco orml o he srce.
6 F L L Smlrl F From d 3 I -dreco g F g 3 g g 4 From 3 d 4. Becse 3 cos e he. Fll 3 g
7 .4 Eqo o moo Srce orces: σ σ σ orml sresses ec sher sresses The s sscr dces he le hle he d sscr dces he dreco. Sg coeo: Sresses re ose ose dreco o ose le or ege dreco o ege le.
8 Bod orce: Z Y X here X Y d Z re od orces er mss d drecos resecel. Ne orces cg o he ce o mss re : X F σ : Y F σ : Z F σ O he oher hd Neo s d l ges D D F D D F D D F Noe: Neo s d l les o ld rcle cossg o he sme molecles or ssem so he ol ccelero or locl ccelero ls coece cceleros shold e sed. X D D σ Y D D σ
9 D σ Z D Irodce ressre sclr: σ σ σ ose s comresso 3 Dee σ σ σ The D X D D Y D D Z D For d L X Y Z I s ose ercll rds / /. I gr s he ol od
10 orce X Y d Z g. The g g hdrosc ressre
11 Assme ll he sher sresses re ero scd ld h s ld or mos er e rolems. The D D D D D g D Eler eqo Eqo o moo or scd ld osder glr momem: M Iω& here M mome I mome o er d ω& glr ccelero. ω & ω & O ω& As he ce ecomes smller d smller ω& creses h mles s sg o er rcle. Thereore ms e ero. I geerl
12 For lmr lo sher sress s roorol o re o sr so h μ μ μ here μ coece o scos. Ssg o -dreco momem eqo X X D D μ μ μ μ μ μ Usg he co eqo e o he Ner-Soes eqo eqo o moo or comressle Neo ld: Z D D Y D D X D D μ μ μ
13 .5 Ree o ecor lss here ecors drecos. The legh or mgde o he ecor s ge.5. Do rodc cos θ comme here θ gle eee he o ecors. For ecors e he cos cos9 Usg he oe relo Noe: Do rodc s sclr. I he Q cos θ 3 Proeco o oo /
14 .5. ross rodc θ s o comme ross rodc o ecors s ecor mgde dreco. The mgde s ge he oe eresso d he dreco s ge he rgh-hd rle. For ecors s θ Q Noe: I he // s θ Q
15 .5.3 Vecor derel oeror sclr grde o Grde dces he sl re o chge o sclr. Grde s ecor hose dreco dces he mmm re o chge. dergece o I co eqo ecor oo
16 Llc oeror: o crl Usel ecor dees: Dergece o crl s ero rl o grde s ero
17 .6 Roo o ld rcle A me A me α α α glr eloc or re o roo β The me glr eloc or orc s ge β α ω For rrool lo ω. Noe h rrool lo he oerll she o he ld rcle c e dsored he me glr eloc or orc ms e ero.
18 .7 Veloc oel Deo 3-D: I ω / / / / / / he he lo s rrool. Theorem: ω d ol here ess sclr eld sch h. Proo Greeerg 978. Fodos o Aled Mechcs Prece-Hll 7-7: Assme h here s sclr sch h. The ω Q crl o grde 3 We ms sho h mles he esece o sclr sch h. B Soes heorem dr dσ Q S
19 Hoeer dr dr dr dr dr Thereore dr d r deede o h o egro Dee d r The he eloc oel s qel deed he o P o he h o egro. Usg d r d d d e he d r d d d d d d
20 Usg he dmel heorem o egrl clcls d d ξ dξ e he d d d I he sme Thereore Ths s he ed o he roo. osderg he oe dgrm hscll he lo ms e ose dreco. B ccordg o he deo o he eloc oel e he < or he lo s ege dreco. Thereore e redee he eloc oel s
21 so h he lo occrs he dreco rom hgh oel o lo oel..8 Srem co osder D le Ψ dr lo re cross he le coecg P d d d d d P For cosero o mss egrd ms e ec derel Ψ s deede o h o egro. For hs he d Ψ. Ths reqres h d Ψ Ψ Ψ d d d d or
22 Ψ Ψ Usg Ψ d Ψ e he hch s -D co eqo. Thereore srem co ess or -D comressle lo. : le egrl log Ψ : le egrl cross d Ψ re orhogol Fll d Ψ sses he ch-rem codos: Ψ Ψ
23 .9 Beroll eqo Eler eqo -dreco s Usg -D rrool lo codo / / Smlrl - dreco e he g Irodcg d / / / / : / / : g Iegro ges / / : / / : g The oe o eqos ge
24 g Thereore / / g hch s he Beroll eqo or sed lo gg relosh eee ressre eld d lo emcs. Usg he Beroll eqo ecomes [ ] g Deg ' e he ' ' ' g The Beroll erm s clded '. Hoeer he lo emcs re he sme or ' d : ' '
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