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# solutions2 - 4,5 4.5 The x and y components of velocity for...

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Unformatted text preview: 4,5 4.5 The x and y components of velocity for a two-dimen- sional ﬂow are u = 3 ft/s and v = 9x2 ft/s. where x is in feet. Determine the equation for the streamlines and graph repre- sentative streamlines in the upper half plane. a = 3 and Ar = 9x1 so 7%47‘ dream/Mes are 9M9!) by g? :: Afr..— ﬂ: :=.5’Xz 0r fdy =I3Xz” u 3 Thus) y = X3+C J WéE/"E C II: acaﬂJ/anf Rep resenfaiive dream/Mes corrupanJ/‘ny f0 differs/72’ Value: of c. are Show» be/ow. Streamlines y = x3 + C :50 150 F I I 0 g 100 ﬂeﬂf l > iC:-—J§0 l i 50 5—. l l l 4. 6 Show that the streamlines for a ﬂow whose velocity components are u = C(x2 -— yz) and v = j ~20xy, where c is a constant, are given by the equation xzy — y3/3 = constant. At which point (points) is the ﬂow parallel to the y axis? At which ' point (points) is the ﬂuid stationary? M=C(X"'-Y1) , v=-'26xy Stream/mes given by y=f(X) are sue/7 ﬁmi 5-} way; Consider Hie {Malian Xzyv glam/737‘. U) A/ofe‘ If is naf easy fa Wife #9119 exp/i017}; as yrf/X) However, we can differs/Waite 571/) 7‘0 gills zxydx +X2'a/y —yzc/y =0 1 or (x‘~yz)a/y +2Xy dx=0 77%; Me ﬁne: I}; 7% X—y p/ane give/7 by E; {/1 541/3 0 s/ope d , —2XY 6/ - 7% .. W or for any com/amt 6; 7%— - ya ale. Me {mafia/7 Xzy ~5- =cam7‘. represem’s Me straw/Mas of H19 given f/ow. 7719 How is para/le/ *0 Me x— axis We” %:0 J or y=0. 77an occurs whee/7 el'Mﬁ/‘IX=0 or y=0 J aﬁev 7%6 Wax/Le 0r 7%6’ y'dX/Ls The How is para/lei *0 the y'aXIlc when g§=ooj or am. This Occur: when X = i' >_/ 7716 {/w'c/ has zero valve/'7’} 47‘ X = y =0 4.7 The velocity ﬁeld of a ﬂow is given by u 1 —Voy/(x3 + yiz)“2 and v = Vex/(x2 + yz)“, where V0 is a cOnstant. Where in the ﬂow ﬁeld is the speed equal to V0? Determine equation of the streamlines and discuss the various char— acteristics of this ﬂow. V = Vo————(XL:)/")%‘ \$0 ll/M‘lt Stream/mes are given by :0};- = 7‘!- = TX);— or —ydy = de which can be I'n/egrafed 7L0 git/e X2+yl = and. 7721):, Me f/U/‘d How MW) circular sf/‘eamline: anal {/79 Speed is constant throughout 6.12 For each of the following stream functions, with units of mzls, detemiine the magnitude and the angle the velocity vector makes with the x-axis at x = 1 m, y = 2 m. Locate any stagnation points in the ﬂow ﬁeld. (a) ¢I=xy (b) ¢=-2x’+y H49»: 177: deﬁhlﬁﬂ‘ﬁ 0/ ’/71c afreﬂm Fahd-ten, 4+ X-‘J/zm/ g: lam/i Allows e/mé a: /-§v” and v-r-z. Thus) 5 5m “=0 aé x=° w v=o u‘ an, 0- slug/mm Pal/7f: 06400": at X=—'j:-0, (A) For L, 1.41M 9: T x / Suite @920, 7710/: are no Jéajod-xoh pawns. 6.15 The velocity components for an incom- pressible, plane ﬂow are u, = Ar‘1 + Br"2 cos 6 U0 = Br’3 sin 0 where A and B are constants. Determine the cor- responding stream function. 50m 771:? a’e/xb/ﬁéh 0/ {be sired/)7 740452915”, - 3-? (531.40.) __/_M _ ”PI/979 ”5‘ So 771415 74” “Me Ve/och’g club/hm?” 7/4/01] J. (2.“ .- ‘I F2 k 6 Ar + B Casé 41, —z | a. = — B!— 50749 3” Imleym/e 5;. 1/) w/ﬁr regard £0 6 £0 oéz‘a/h ﬂy = ﬁA+l3F_/cose)d9 + £(f‘) 0!" g; _— A9 + BF’J/my + 740) Sawﬂy/3, Avirqmz‘e £3.12) w'nh rag/acct A P L‘a 0624:“), [cm = -—/B/—'25/m9 dr + £09) 4; .— BPS/”a + 4(5) 7774(5) 710 \$4,295.79 5077) 535,6) 4/11/14) 175.— 49 + Br‘la‘lhﬁ +C where C Ls cm ark/ﬂaw consfmt. é-lﬁ' 6.16 For a certain two-dimensional ﬂow ﬁeld u = 0 v = V (a) What are the corresponding radial and tan- gential velocity components? (b) Determine the corresponding stream function expressed in Cartesian coordinates and in cylindrical polar co- ordinates. Ai‘ an aré}1Lmr7 [Do/ht P (See vcl'qure) (43%: 0 at? If' {cl/ow 777a'l 41/ 13 not 4 14096.19an 07! f7 and LP: - Vx + f 41’! 4ré/ﬁmy amném'z‘. FMSQ - l/l’cczsa +C é-lé mﬁr -—- _ mu: A '3.6 3.5 3.4 What pressure gradient along the streamline, dp/ds, is required to accelerate water in a horizontal pipe at a rate of30 m/sz? Qﬂ—g—J'S/ﬁG—PVQJEK 14/59/19 8=0 and 03 V33! :: as = 0% 761/5, 34?. = .. (pas -.: —977_§”3§630%) : —30,000-7,/,M1)/m or .3451 = — 30.0 kPa/m 3.5 At a given location the air speed is 20 m/s and the pressure gradient along the streamline is 100 N / m3. Estimate the air speed at a point 0.5 In further along the streamline. V i ’ 9 av av _ 12F neglect qravn‘y ) 71:: g “Pvt? 01“ T3— __ "ﬁg/9V 0r V ’25 :.- ’looﬁAg/(LB 13'3”” 3’) = “4.07:;- Thus, I JV: 5).}:- 53 = (—4.07%)(0Jm) = 4.03%) sol/ml V+6V=20-’§’--2'03—E9 0," Vz/é’ﬂga 3.6 What pressure gradient along the streamline, dp/ds, is required to accelerate water upward in a vertical pipe at a rate of 30 ft/sz? What is the answer if the ﬂow is downward? 3f; :: —Z's//76 ~9V§SZ Where 9 =90’ for 0,0 f/W , ()5 . a 9 = Jizvfor- down f/UW, and V’- =4, =30ﬁz T/lz/s, For (/pf/ou/ as 6 6 I}: a ,- 1 %‘§'= _62,¢(,)%% -—/.%< “3%;(30tﬁé) ; 5/20.6«g;)/H =—o.saa% and for clown ﬂow It; , __ lb ' 1 %§_ = -52,y(—/)7—{,% —/.74%%\$(30_§1l) - _4.zo 351-3)/H=Q0292 1g; 3-4 3J5 3.15 Water flows from a pop bottle that has holes in it as shown in Video V3.5 and Fig. P3.15. 'I‘wo streams com- ing from holes located distances h. and h; below the free surface intersect at a distance L from the side of the bottle. If viscous effects are negligible and the ﬂow is quasi-steady. show that L = 2(hlhz)l . Compare this result with experi- 2.0 L mental data measured from the paused video for which the _ ' ' holes are 2 inches apart. L—d’f W“; I FIGURE P3.15 For sfeady Mil/lead HUM/{#78 ve/ac/l’ies of {he horizanfa/ jails of wafer- ai poi/7*: (U and (1)0“? oln‘ained from {he Bernoul/i eyum‘ion as: J J 2 Thus, V, = MW), and V2423};1 (I) V 3 Once oufside ihe fan-k, graviiy is {he 00/), force on {he fluid pariicles. Hence {he horizam’al campaneml 0 of Veins/{y remains cansiant, bm‘ ihe par‘iic/es aces/smile afoul/mam! wH/i {/79 acceleraiian off grail/7y. Thus, fa,» a {/qu parh’éle Hmi exh‘ed {he funk L‘ sewn? ago 41‘ z=h if fol/ow: f/mf X=Vfandz=h-i'g£? . hz= By alibi/hailing z" #713 give: {/76 pupil's/e pa”) (cl-e. ﬁle shape of ihe Minder jail as 2 = h - 2%., X T/WSJ #73 shapesof {he iwo Waier jefs are as shown in 7‘69 figure. 2I=(h2—/71)—E% xia) [co/7'11) 7756 Sfr‘eams infersec?‘ when X=/_ and Z,=Z,_. Tim, from E'yns.(2)and (3; (/12" I71) " 2%]? 1-2 -'= " 5E? 12 w/7/‘c/7 Cd” be fell/ranged /a give _ / 201er .1. _ .1. (4) L” ______._.__? / v12 V22. From Eqn. (1)) l l _,___L-._L..=.LJ—,.L ___ (bar/)1) (5) _——_——— vﬁ V11 " 23/2, 23/72 25 ’7: ’71 2;A,/11 Thug by comb/Why Ems. (Ha/101(5) W6 057!th __ 2072347!) {/Iz’hl) _ L—]/ g /]/2___—WL “zyhﬂaz. Nofe #m‘ a/Maoy/I Wand V2 am a {anal/m My] f/ze al/lr/ance Z 139 00/. Two 7’4”ka one 0/) far/4 7% offer M Met/1r Way/d a/m/‘II 47‘0/1'7’7’6/‘0/7/ [”0wa Am‘ 7%9 x’nfenred/on dink/70g 1.7 of 1% sired/77: Would be 7% awe. 3-l3 ...
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