PS9_Solutions - " c 6.1 The velocity in a certain...

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Unformatted text preview: " c./ 6.1 The velocity in a certain two-dimen- sional flow field is given by the equation ‘V = 2xti — 2ytj ‘ Where the velocity is in ft/s when x, y, and t are in feet and seconds, respectively. Determine expressions for the local and convective compo}: nents of acceleration in the x and y directions; What is the magnitude and direction of the veA; locity and the acceleration at the point x 4—r y = 2 ft at the time t = 0? ‘ Fm”? “Was/m fur velacl+5/ LL: zx-A 4nd 7;.- 4512 Since xx =' '3’? + a. 9.5 1— v3.3“ *fifln 4x {/007/)" %' 7" X and 1 a; (ea/w) = u ’3}: Mg; = (Zx£)(22£)+ (432%) = 43*," S/hi/ar/g/ : au- 9w Mr and f7 = lfflt’z' ’415 2C=°J=2sz and zf=~0 u= 2(2)/o)=c> 1r: -Z(z)[o)—=0 59 7714171: 7:0 WM 4 = 2x+ amt"? == 200+ 9(2Mo) = 4752/51 ~2j+6fi7iL~‘ “2(2) 1" ‘I‘fzJ/D) :”""Ci'/52 ’ 2;... L/f-saj‘ pie/sf M75, IZI=W”fiifi/§ 6.2 Repeat Problem 6.1 if the flow field is described by the equation = (3x2 + 1)? — 6x313 where the velocity is in ft/s when x and y are in feet. 4 IL 3 2 ” 36X (5’)! +/) = [9(1) [30) 2+1} #211! ~3-E/5 Z: n9»: 9 m. =Ixmm~im= 129% Thus : 25‘? 2+ 32:; #151 = VQV)1+(IZ)1 : 2L8 9575/51 G-z V6.3 ‘ The velocity in a certain flow field is given by the equa- tion V=xl~lfx223+yzl2 Determine the expressions for the three rectangular components of acceleration. 1 Fram apex/kn 4r tuba/{17) a = X 0-: {22 Since ax: 3-St- 5L5; +V§g ng"; five/7W __ 0 + r x‘ )(I) +3 06%)(0 ) [email protected],2){0> : X L S/mllar/g) ...__..._. fly: gig-F tt£+vggwr affix“ 43 2 0 7“ [ X )(zxz)+ (x32)[o) 143196?) : sza + x2327: 4/50) = Q}! as): 3w— 41!.“ a2! 22H“ “ ”+7753 “”22 50 flat l 2 4%: 0+ (X‘ We»); +(X?)(2)+f:rz)/s) Z = 52w 5% 6:"3 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 ezueélémj __ ——-“— -0 (I; 31, 1 $31? H Eguaiz'o/J (2) can be thifirymfiu/ th ”spec?! £0 2: -/o 0/174th [4“ _~_ _/.\5)(¢:/X +fX’[email protected]) OP __ 3 2x3 a = :Xz-f 5 + {(31) where 75(9) is an undekrmined fimeézéw 0:5 g. 6.10 For a certain incompressible flow field it is suggested that the velocity components are given by the equations ‘ u=2xy v=—x1y :WZO Is this a physically possible flow field? Explain. Any Physical/g PaSS/Z/é: I‘ncom/oress/Z/e, F/uu [fa/A Musi 541451.547 Majeraiélaaa 0/ mass as at‘pmsscq/ by 171e, VelaétéaS/np 9X 3 9 + 3-21 .2 p (I) Fir 7712 1/5 ADC/1‘7 c/ui 7‘r/Jmi1/fm 7I'um __ :0 fig; 2‘ > g)? = 23 M :' - X 2 aw Juémflim‘w» Mk EZ- 3(1) Jhaws 79:47! 29—X2+0 #0 Thus} 771/3 1.5 no?! a, pthIEal/y lows/We Hm; {fa/d, £3; 6.2‘l The radial velocity component in an incompressible; two-dimensional flow field (122 = O) is u, = 2r + 3r2 gin 0 Determine the corresponding tangential velocity component, 1),}, required to satisfy conservation of mass. ’90V/e’ / 9% 2V” f5 43:- l’ 3‘!» +FZH3+_é-2;0 7' J Smce ’V;—=d} aw? : _ 20m) , M &E_ "I and nut}; r‘?}-‘ 2P2+3r35m 6 [7‘ fo/x’ows 7714£ 1 2.4%): We + 47*‘25/11 a fil" Thus, 55.0) éecomes ‘ $94? = — Mr §+ W304 6) <1) EZuaré/bflfb (’4)? 5e lflréejfmefi/ why res/Sui zéo 9 “Ea déz’zr/f, deé‘ _— ‘fo/wf- W‘s/amate 1- 75M) 22;.— ~¥r9 « ?P2cosé +70%) 7Q in) IL: 4/1 MHz/e éer/I'ipt/ final-I49.» (9,5 k. 7.2 6 72 The velocity of a fluid particle moving along a horizontal streamline that coincides with the x axis in a plane, two- dimensional incom- pressible flow field was experimentally found to be described by the equation u = chz. Along this streamline determine an expression for: (a) the rate of change of the v—component of velocity with respect to y; (b) the acceleration of the particle; ‘ and (c) the pressure gradient 1n the x direction. The fluid 15 Newtonian. ' (a) From The Conimurfg 6514.119!“ an. US: 1 31¢”; 23 0 so 7714!: ail/771 u=xz 3V“_ __3u .. 3__.' 39 3x .. 1) ax— A180 58' (I) Can be inkymled («J/7‘7: VESFPC'Ii. fa 5 +13 air/11w fci'V’: f—i2xd5 or 1 0-: ~ij + Jim Since 171: 96-4st is a Shawn/Hie) 1r=o along, This axis amt 7‘719Ve14re 75160—10 oo 771417 72-: ~43 (A) “x: “3—3-2 +7; 2%: (ix‘)(2x)+ (~2263)(0) = 2x3 625: “335:... #23: = ixzjf—Zj) + {‘2x3){‘2’<) : 2X3 4/0/19 x-axv‘s/ gun/a and fhercbre 45:0.7/1145, __3 A QL=ZX§L CC) Flam 55 @1274 [w;17, 16:01; ..___ +/“ a2“ 036' fig-f Alf/3x175: 91:) 50 flat 2X3: “F§£+ i&{2+o: I and (I) 6. '13 Two horizontal, infinite, parallel plates are spaced w a distance (3 apart. A viscous liquid is contained between the T ’ ’ ‘ ' plates. The bottom plate is fixed and the upper plate moves parallel to the bottom plate with a velocity U.EE‘»eczause of "'V M. the no-slip boundary condition (see Video V6.5“), the liquid b motion is caused by the liquid being dragged along; by the i 3 moving boundary. There is no pressure gradient in the di— rection of flow Note that this 15 a so- called simple Couette * flow discussed in Section 6. 9 2 (a) Start with the Navier— Fixed Pl‘ie Stokes equations and determine the velocity distribution be- tween the plates (b) Determine an expression for the flowrate passing between the plates (for a unit lwidth) Ex- press your answer in terms of b and (I (a) Far sie'aa’y f/ou M7711 'Vztha ff {cl/tau: 7747f We Na’l’lflf’ J’é‘efS (544090-45 fen/ac; {—0 (I)? dlrcoilzm.’ off/av) &:—S—f +/L,L (52a) (£75612?) r425 f1: 0 LL— 0 and /' 7foflaws 7944: (’ :0 5/mI/ar/g) ’ U «LL (724: AL: U and C/ = E- Merefme U u=7§fl b i A -" I U U W - Ii]? (1,) g: “(1)497; 5‘13 = 1: a: a” z o o m where % /'-5 i’i’he. fi/owrai'e per Lin/'1' WId-fh. 6.75 Two fixed, hormonal, parallel plates are spaced 0.4 m. apart. A viscous liquid (5:. = 3 X 10—3 lb - s/ftz. $0 = 0.9) flows between the plates with a them) velocity of 0.5 ft/s. The flow is laminar. Determine the pressure drop per unit length in the direction of flow. What is tic maximum velocity in the channel? ’ iéqu Direction of flow 6.77 7 A viscous, incompressible fluid flows bet-f 7' tween the two infinite, vertical, parallelplates of Fig. P637. Determine, by use of the Navijeir- Stokes equations, an expression for the pressure gradient in the direction of flow. Express Your answer in terms of the mean velocity. Assume that the flow is laminar, steady, and uniform; l \\\\\\\\‘\\\\\\\\\\\ .\\\\\\\\\\\‘s\\\\\\'\t lbw FIGURE m1? W171! .7718. dwrd/iedéc Jqszéfniv Show}? “20)w-zo and 74mm 7718 (an‘tman‘g 654142615» 2315' ;p. 777115, fram The y—Mm/ommi of 77/6 Newer-Snakes fjmv‘ww (51 6-”-75)/’”’77’ €32.77?) --—i£..., _____ [I 0‘ as f’él‘V‘" w ) 5/1“: 7776: [unsure is ”oil at 7401425129” 071 x J 55, II) can be mum .45 ’ L24": :5. 0%” )4 [Where P -‘-' 5954703) ahf/ lh/fyra/vd 7‘2: dbl/20.1 dU" .. 31? (2) :72 ‘ g}; x t C' 5:907 ymmefiy ail-:0 ad} 35:0 Jo 774475 61:0- Ih'ltc’jmfiok 075 E512) Wye/.1, . 7/“: 7:1” * S/hfe af )Cri»£.)7/'=0 125‘ 71:0//owu 7744125 am! Wereére f g, 2 2, ‘Il‘: 3/7 WV #41 ) The flawrméc pagan/3} whit): n} The Z-direm‘mn Can be {KPNSSEA as 2/0. 3/4 777115) Inf/74 V [me/m Ve/ocHy ) yin/cm 197 s _ l_..L £12412 V: 2): " 3 /’- 1+ Mm 77m: . 6.80 An incompressible, viscous fluid 1 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~Stoli1es equations to derive an expression for the velocity 1 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 7714i 777‘ x'wmf’o’lpfi‘f 075 fire IVflV/é’k‘S'lkaJ‘ (Pi/”4151‘s,” (Eg‘é,/Z7£) reduces £01 J (/2 :1; d;: i0 ‘0 Inkyaémn o% E; m Weld" “: 6y 7" (z (7.) 5,, y: 0 ) 142—433“ mm’ fire/e5”: 79w» 57,622 610; E F5!» g: b) “=0, 501 7774f 15 = (jg —Z& 0" Eff); C, — W“ Thus) t) 14»e7 Jé,/ 6.81 Two immiscible, incompressible, viscous fluids having the same densities but different \fis- cosities are contained between two infinite, hbr- izontal, parallel plates (Fig. P6.8l . The bottom plate is fixed and the upper plate moves with a constant velocity U Determine the velocity at the interface. Express your answer in terms of U, yl,‘ and #2 The motion of the fluid' is caused entirely - ‘_ by the movement of the upper plate; that IS there 7 i. ‘ ' ’ -o is no pressure gradient in the x direction. The . FIGURE 1'6 3 l‘ fluid velocity and shearing stress is continuous 7 across the interface between the two fluids. As- sume laminar flow Far We spew/fled Canal/from: 21—0 arr: ~0 53? :0} and ai" 0’ so flmt 7712 X— wmponenf 0/; 7‘"): IVaV/erJ/v/(e: (iuaézéus {55,é,lz7a) 15w e/flver 7716 upper of Kate/er Myer Vedutas to 5,23% Im‘ejmfwn mt 55 (I) 9/5/55 LL~ Ag :48 “MiG/I 71-1495 77m I/e/acn’}? dish/214255;: I}: e/‘flvey layer. In file upper Alf/er mi 5221} a: U' .50 771,35 8:- U A (211) when 1719 sulscn‘pt I refer: 40 1k upper layer. F5? 7712 Lower Myer “‘23:“ 1 uzo .So 771415 B =16 2. when 771: saéscwpt z gwife/3 7‘0 five lower layer. Thug1 UL} :- A; (y—ZK)+U am _ i u = A1 3‘ 2. Ale 53%. I “I 26/2; .50 Wa’i' 4/ {’tf-ZA)+L( : Az’fla or U A. :2 ——,4 + ——- ' 2 I f; fan/1f) ire—~72 6.3] I [Cg/7'1) ! i Smce 17k Vet/00391 0/IJ.7Lr/.éq§£70'h L's fidem I): €4c/1 layer 7‘71: shear/r17 .3 frag: ~ i5 9:71: _~ .41: @x‘/ 391*5?) fidg and 791* 7512/0410 /47/py 1 429 f/ie m¥rné¢c 27:2; 50 7744i ”,4, =‘ LA; 01» All :4“: (3’ /4‘2. fl: Sués-[f'éuf/é; of 531;) Mic £3312) fife/Ls I41: “def/12 + 2 /a'/ flu or - 02% z I + f’z/M, 777$“) V€IOCI¥7 NJ: 771: Iii-ffrfice 13 “2 (9’3)? 142% = 4 ~77 p.13 6.88 A fluid is initially at rest; between two horizontal, infinite, parallel plates. A. constant pressure gradient in a direction parallel to the plates 18 suddenly applied and the? fluid starts to move. Determine the appropriate differential equation(s), initial condition and [Eoundary con- ditions that govern this type of flow. You need not solve the equation(s) DI fwzenwémY eguaz’wfim aErc 77/2 Mme as Ezs. 4127/ 5/30, 4/14 é/B/ E’xae/aé 7712:!- j“ 7':E 0 (5/9“, fire f/au Ii «hymn/7) 771115) 57 é./Z7 mu:: /IE1c/¢m/e 77¢ /oca/ acce/emfloi; £077; €21? J WM 7716 701/”me a’n‘érmfiq‘x eigmaiu are ,- (X— dfrfcfiah) J“ .=- —— Q]: +. Z“ &_ at /c 333- (Id/7% ax ‘ {y— dlkcfioia ) 0': "3-5 -- {’02 [21—— d/Vécfzéu .) 0 :7- -— g; mf— ' III/1947 Cong/Iflph: LL=E0 7%,» :0 Ar l// y Baum/my Came/”9.01153 Air-1:9 24911 3:134 Aw 7.430 * 1?: " la? 49115251011) 6.3 2‘ (a) Show that for Poiseuille flow in a tube of radius R the magnitude of the wall shear- ing stress, 1'”, can be obtained from the relation- ship i(Trz)walli = 4"uRQ E E for a Newtonian fluid of viscosity ,1; The volume rate of flow is Q. ([1) Determine the magnitude of the wall shearing stress for a fluid having a viscosity of 0.004 N-s/m2 flowing with an average velocity of 130 mm/s in a 2—mm-diameter tube. (a) T =,u(~b—1:’i’-§: (5;. 6.1257”) . . E 5r lpO/J'fMI//C 14an i”; 4 "Ii-ulna) 74:0] flnr/ flint/are. S/ivre, 2-: MM [I __ (7:)1] (E3 6./5“’U 7/1145) af 7kg axe/fl (fr-E); is _ fp/ (2":2')wa// [~— 5‘16 and 4/1771 49:: TRiV 1%) M— 7/23 Ida/I" (L) é~/23i ‘4 “W “7*“ u 4,473 6. 93 An incompressible, Newtonian fluid Fixed we“ flows steadily between two infinitely long, con- centric cylinders as shown in Fig P6315 The outer cylinder is fixed, but the inner cylinder moves with a longitudinal velocity V0 as shown. For what value of V0 will the drag on the inner cylinder be zero? Assume that the flow IS laminar, axisymmetric, and fully developed.‘ ©71/l/l/l/l/I/I/Il/I/II/Ill III/II/H/WIIIIIII Ill. Ezua‘ £10»: 4 /’f 7 which was gJamie/calcxd rér 1C/ow m Cl/‘Cu/ay fakes, app/[es m’ 711:: annular ”jig/7 Thus, ”5:2,“(3—5 )/“+Cnt—+( 1 U) W177: hum/dry cal/104799;“, [2:212) 1/2120) and Vz’Z') Ki-‘V0) If é/lows 7774i , 0~ sic/3f); l" + C In)“ (2 (2) I \ézé‘ .691. {§)n'z + 6'11” Eff-Ca. (3} Suéfiacf 55,”) from 53,:Y3) 71¢ oééa/L .1. 2 i / 2 V0 ‘” 6‘ {YQK ’2'2‘ ’3 )* “I f” W / fin... i” D The dmj (9/7 777: ”may Cy/I/‘vder w/// be 26% ['7’ 50 fl’lai r: 1 5mm , A (P2'=/a(13;‘+ 37) (52- 4124*) , [7‘ [Ia/liars 7921‘ ‘ Pi- a 1- (Con 't ) A- //L[ 4nd Ml‘fil 7/7,“ 0, 7 {Cap/1'2) 0,.F14renflézée 55,0) (@1771 reS/aecf {-9 k 154: 0/5797“; 3% _ _L ‘it? P + C: =0 V" PL./€H70- 0r 3 .. J. «al’i ‘ 2 V“ 2 2 Va“ ‘ 9%{3 [2’21"7: -(’2 ‘5) 1e; ’- 1/5“ i i i i i 1 ' 6.44 [ 6.414 An infinitely long, solid, vertical cylin- der of radius R is located in an infinite mass‘ of an incompressible fluid. Start with the Navier~ Stokes equation in the 6 direction and derive an expression for the velocity distribution for the steady flow case in which the cylinder 15 rotating about a fixed axis with a constant angular velodity a). You need not consider body forces Assume that the flow 15 axisymmetric and the fluid is? at rest at infinity.‘ (Ea. 4.35) 11‘ 7/0/0445 771;: (fee 44'7”“ Ar flaiaz‘zén) ”ms 7’77e Newer- 53/4/95 eguailou m 7712 <9 direct/on [£3 6/235) 4r 5£€4dy 7%“) ”Mace: 2.40 ’ au— v" o=—~ JEN/a 725(33qu Due; 15-0 7712 Jyrnmez‘r: ain/ 7716 Haw) 5114}: V; 1:5 6? 740457.410}, 6/ 011/5 1“] 53.”) (an 52 asp/essay as 4W ord/éywy a’:r%rmfié/ Fix/4119’”) 4/15/ VK- am 26’?» as c/ZV; d»?- ‘dkf )0 Egueézw'” 17-) (Ia/4 be M‘kgrm/et/ v4: glelz/ / [cw/Z) E5144 25/195 ‘3 ) C/m be expressed 4.5 dmg) (/1’ mm 4 seam/2’ ”flay/450% wield: 361’? I 2 ,45 fi—>ool Wig—~70, ~5Mce f/ufi/ 4'5 4'5 Wn‘ a?! //;24/;/}‘£7) Jo Mai C: . ThMS) V 0140/ J)HC€ az’ 2/? Va: Raj] /‘7L [ta/701.4“ 7714i sz/QQJ (214% 2 AWN? 6. ‘75 A viscous fluid is contained between two infinitely long vertical concentric cylinders. The outer cylinder has a radius r and rotates with an angular velocity co. The inner cylinder 13 fixed Eand has a radius r,-. Make use of the Navier—Stokes equations to obtain an exact solution for theE ve- locity distribution in the gap. Assume that the flow in the gap is axisyrnmetric (neither velocity nor pressure are functions of angular position 6 within gap) and that there are no velocity com— ponents other than the tangential component The only body force is the weight. E The Velociig disi-rlbubo'n In five annular Space is 7112614 by We 9161141:on =C‘" V9431 C2. r- (See 50/111919.» +0 Praia/em C's-‘14 ér (lynx-#1314.) WIT}: We. boundary Com/EEHMS I~=r- v;— ’“2’317’; ‘\ Z. 1- ...... ILE I? -' CH?» 4- ff; 1E]; 20.) 1—323 EZ- ’3 E 12 ‘En'fid E—:_ ’2' 7;: I'ME Y‘E':L__ _ I“ .1: ’44}!— r-a)‘ I 2 _._. __ (1— E.) r3:- '0 and l" a) (598E {1514M 1491* I:oi‘a.'élon)) 125 [OI/onus ...
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