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set12-answers - 6.7 For a certain incompressible,...

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Unformatted text preview: 6.7 For a certain incompressible, two—dimensional flow field the velocity component in the y direction is given by the equation ' U=3xy—x2y% Determine the velocity component in theix direction so that the continuity equation is satisfied. ‘ 72> Sal/37% 771? cvnélifliaifly 9550:2955 __ ——-“— (I; 3-,, 1 $31? ll Eguaiz'o/J (2) can be Mflyméw/ th raped £0 2: -/o ain‘th [4“ _~_ _/.\5)(¢:/X +fX’dx+f-(5) or x3 u = ‘i—xzdr 44(3) where 75(9) is an undekrmined fimeézéw of g. 6J2 For each of the following stream ‘fufnctions, with units of mzls, detemtine the magnitude and the janglc the velocity vector makes with the x-axis at x = l m, y 2 m. Locate any stagnation points in the flow field. ’ _(a) ¢=xy (b) ¢=~2xz+y H défihl‘flo'h 0/ ’fih: afreém , 1 _; ,L‘I’ .._;.<>‘{’ : (35437)? “" 2M ’ w * “(a—J X3) ‘ , f , V 4+ Xenia/7: fi/Iowig fiat r=~2L4f§€ 5’7‘.‘ “‘0 5” “9’9"?” “*0? “"5 “97“?”34‘mh .é/Izi'WWZW: ms raw-10.x, = - ’ » ‘ f” j E 3 :Ai- x: ) “j if Agl/aws #31411. *émd 11‘5'5% mus) ; i V S/nce Lt¢ol fine/ci are Ina Jéajfl¢£70h [Jo/975;. +0 6.22 The stream function for an incompres- y, my sible flow field is given by the equation ‘ II! = 3ng — Y3 1.0: where the stream function has the units of nil/s with x and y in meters. (at) Sketch the stream— line(s) passing through the origin. (b) Determine the rate of flow across the straight path AB shown * in Fig, P622. A 1.0' m FIGURE P62 2 ( a») lines of wad-401‘ ¢ are slra’m/Mes. V 5r % —‘ 3X2j ~73 72': 3:5/7f4/n/l/“76 Fax/)7;- ffimufi 777: 0/2709 :0) yzo) has a Vd/m: ¢=0. 7714’s) 7718 9344:4143” for fire skew/biz: fit/wilt The air/th 115 0 = 3x 23 ~ ya or :1: i’l/Ex 4 sl’eéc/z 0/ 771356 sfrevmi/hes 13 Mow/1 in 7746 flit/re. (A) _, 6»— sM 45 [5’ 96:0) 9:1,». 17mi- 2 9g smug — m3= «Inns/5 (P9, “my Wm) 41‘ ,4 15:14") 3:5 .59 7710.1! 46, = 30070) — (0)3 i: 0 ll Hus ) 49: 9% = ‘JWVS (per am't Low/“I14 The‘ nejm‘xk 5/7.” fitn‘ 77/: flaw is From [Vi/It kt lifl‘ as £90k 14mm 24 -/o 5. fiéC—u23 l 6376; A layer of viscous liquid of consthnti thickness (no velocity perpendicular to plate)" flows steadily down an infinite, inclined plalne; Determine, by means of the Navier—Stokes eqiuai tions, the relationship between the thickness of, the layer and the discharge per unit width. The flow is laminar, and assume air resistance is neg- ligible so that the shearing stress at the free sur- face is zero. led/.774 771a word/mug: 5115145”: shown /;'1 five figure : = I n p - .— 7/ OJ w— ol and from 1112 amt/mutt, egawttcw a? 40. //Ml$, From 7776 x— (ImM/Wffif 07‘ :Me A’m/m'r -5¥a/(e.s elude»; [53», 4,1274.) , J 2—21: ‘0! Effl- ax quid; s/n +/u. d3; (I) 14/30) _5;'h(c 77mm is 4‘. free sunéce/ 771m Cannot! be a. Pmmre jmd/eflf I}: 7712 x—d/recy‘wiv So ‘fha-é :0 4,44! 5 [U can? be. tat/«1‘79?» 4: a” 3- 0/24" = *fl3 who! djz 77 $111457” :49}; y/é/KZS %5-;- m @3 What): 1!- C, (2) Slhce The Sheflrlhj shes—S i Q} 2.”. 73x 7‘; at: T M Gian/5, 7ero ai‘ 77m free Jar/ace. (3:4,) [£- [o/lams 7‘7mi g5;- -'-" (Li: {7 = ’K .50 771a2§ 722 60/617011 53,12) 1:5 C ,& .Slhol I /" In 147/7796” 0/ 1.55/2) y/é/is .i .97“ - ‘ _ Jar S/nte (4:0 at 3:0] v[o//aw.s fiué CZ :0) 0115/ Wereére : a _ 612 LC {Cg—d 5mg“ "2) "L 77“ f/owmfe ’0” “my: width can A: expressed as j. =/ rule; 60 flat fl 3 o _- A? S/nol {1635,— ffi/ : p376 ma 5 we a j 3/“ 0 l i l l l l . 6.80 An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates as is shown in Fig“ P680. The two plates move in opposite directions with constant veloc— ities, U, and U2, as shown. The pressure gradient in the x direction is zero and the only body forice is due to the fluid weight. Use the Navier~Stolies equations to derive an expression for the velocity ‘ distribution between the plates. Assume lamiriar flow. 1 FIGURE P630 l x i i E Far five. Sfecx'f/éc/ Candy/7M5) ’V’=0) “3% 35:91 4”"! f2: =0 ‘50 flat 777‘ x'wmf’o’lpfi‘f 075 fire IVflV/é’k‘S'lkaJ‘ (744154,” (Eg‘é,/Z7£) reduces £0; J (/2 d;: i0 ‘0 Inkyaémn o% E; m yL/e/tls a: 7" (z (Z) 5,, y: o ) Aer—JD;3 ana’ fire/e5”: 79w» 57,62) 1:»; E F5!» g: b) “=0, 50E 7774f 15 = (jg —Z& or C, — Thus) i4»e7 E” $9 $53 rum...“ . i: _ { g fifiéfifi 5% mews flag: {ggmgifis migh; : 3%} ibfifi Wages» “ === 33.633 1% ~ 3&2} i3 fianmmaé @meefi mm éfi‘fifiiifi, 3mm“ ' Emmi pamfiai giazass 33 Shawn fizz 'igig. P2183; The flask? maws : bgtwem 3313 3313335 unda me action of a pmgsum gmaiiéem; anti {has uppm‘ piaic movag whiz a wallmciiy if whfle this bamm mam ' is mm? A Limbs mafiamsfigr maximized beiwsm {gainm akmg 3% Emma; iindicams a giiffemnfiafi mafing 0? {2.7: ism. if {Em : apper- pfiam mums with a vcicmiiy @f {162 Iii/5? a: win: dis'zmcc i {mm {ha batman plate $053 me;- maximum valagity in {he gap - haw-3m me mg piams assist? Agsums lamina: flow” i E é s? g i g , 3 § 2‘ E g §?§Q§RE Pfiaéfii‘s A 4 d “rm i f a": {‘M «g» m #5; f a .a a “j £33 ubgfijégwéflg éJiféf {WQ’XEMHM Wimffgg WW flag:sz €22? fféSEQfiEQ $2M wfiéwe g5“: Ziéwsg ! : é 3 w ‘? £553“? ’9 m man w. “M “g9 m “é” ‘ w d5; 4% 3% 53% Jig/23f M (swag? wag? £45? £3,613 ‘ Egg ,M; 64’? £9 g g%%% “fig-ff “55% 1 l & WM 3 NW mmmgéflw {we ifigmm 3325 Kiwi}; Uf‘fi ‘flzj . . ..€=‘_ «$ng “=0 '3 g E E E E E E E i E g E g E g E E” g MW“? f7 . 1m 3"“ i Q ...
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set12-answers - 6.7 For a certain incompressible,...

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