HW#9%20solution - ‘ 7(1 Water sloshes back and forth in a...

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Unformatted text preview: ‘ 7. (1 Water sloshes back and forth in a tank as shown in Fig. P7. 6. The frequency of sloshing w, is assumed to be a function . of the acceleration of gravity, 3, the average depth of the wa- _. _ ,. ter, h, and the length of the tank, e Develop a suitable set of ' 1 ' dimensionless parameters for this problem using g and 6 as re- _ L e _____1 j ._ _ Wm. o . . I F g c; u R E p 7 , 5 peating variables. wT OPLT‘thLgL From 77“ P' 1.1180,”, L}_- 2 Z d1menswn/ess‘ Parameier: regal/184’ Mae 3. 4nd 4 4.5 V¢P€44mi var/ewes 77111.5, ‘ ‘ 501714.16: (Li-5:0 [Arl)y“ ‘ «124—0 V (flu—7') I4 fi/laws ’Ma'l' 52::— ‘/2 am! #7912149” 64.» I.) . ¥ , . , /§ #1611; mi ’ 2-09521 and 111m ére ? 'i" “ ::: iv :- ... 3 ' : ‘ :i i i "‘ f ‘ ; 77" ~ E ~— 1 i ‘ l 1 1 z I' ' f ‘ 1 ' ' s ' 3 , 7.12 The velocity, V, of a spherical particle falling slowly in a viscous liquid can be expressed ' as V = f(d, ll. 1', 7:) where d is the particle diameter, ,u the liquid vis- cosity, and y and y, the specific weight oil the liquid and particle, respectively. Develop a set of di- mensionless parameters that can be used to in- vestigate this problem. Vz' LT-I 45L #5 FL“? 3”: FL'3 33-;- F153 From 7718 pf Theorem I 5~3= :2 pi firms refill/Perl. [/53 03/9 and 3’ 4.5 reflmf/ég Varmé/es. Thas, 77’,= Vdfiu‘a" and _ 1, _3 c. a a a CC f”’)(L)/FL'ZT) [PL ) = Fl 7' ‘50 777415 éfc :0 {941* F) /-ra.—2..b—3€==o (5/1.) " l'f‘é:0 (6r T) If él/ows #512? Cir-2) b= I) C =—// and flare/are . ‘ 77’ : 4223’ Check d/mt’nslans aslhg MLT 51151116771 -' m W 2-. We,» ,10k 423’ ‘ (LY-(Mrzr‘z) 77,2: 3; da/ubalc fibril/L)“ {FL-=7} boxer-v“: #2 ‘7" /+b-.rC=o. Gng) ‘3 {HQ-#251522 =0 (fir L) .520 (fir T) If 741/014: 77741 a-‘o,$=0, C-‘*/, and 77/6/279/2 7,. ... I; 2 3w Ld/w‘ch 1.3 oéV/DHS/g dimensxm/tss. 77Iu5 sew-5:3) ——._—_. ———_.—._ ) 7—15 7.19 One type of viscometer consists of an open reservoir with a small diameter tube at the bottom as illustrated in Fig. P7.l‘t. To measure viscosity the system is filled with the liquid of interest and the time required for the liquid level to fall from level H,- to H, is determined. Use dimensional analysis to obtain a relationship be:- tween the viscosity, y, and the draining time, 1'. Assume that the other variables involved are the initial head, 11,-, the final head, [1,, the tube di- ameter, D, and the specific weight of the liq- uid, y. a. ‘ #f‘ _. — 7r: 13:9 I /0. Cheek “5/95 M173 Fér 77; [Cam/012”}; f4); -H‘ ”3‘ 7f -— HP 773‘ ”5 Thus, 73.9:/( /u. 7vza an- FE3 5'0”: 77)! PI“ Theorem, 4—3“:- 3, 1"“ firms rezuu‘rpcl. By Inspeczfzéh/ a?” 77; fc‘anp‘alhlég '2') .' (7‘)./FZ'3)/U 4 FL‘37' F°L°T° m .4 (7'__)______{_MA‘Z__T"___Zjfl) =2 M‘L‘T" ', 0K /" M 4" 7" which 1.5 Obviously drinensmn/ess, Sinai-lady) TH) _ #c’ Bf [a — 54 (—3; D) and 15k a 14294 yeam‘eztrg { ”Educ/n}; ”1' and gr) (I) Where /< 13 a (basic-”25 J depend/a7 on 59/0 4m! léc/D. 579m 53 (I) ‘ ' - KD , /“‘ [(3,2- where [4,: D/k and K, 1:5 4 catfish/12‘ 76;» a flied yea/newly. ”31ft? The drag, ‘22), on a sphere located in a pipe through which a fluid is flowing is to be determined experimentally (see Fig. P750). Assume that the drag is a function of the sphere diameter, d, the pipe diameter, D, the fluid velocity, V, and the fluid density, p. (a) What dimensionless parameters Would you use for this problem? (b) Some experiments using Water indi- cate that for d = 0.2 in., D = 0.5 in., and V = 2 ft/sj, the drag is LS X 10‘3 lb. If possible, estimate the drag on a sphere located in a 2—ft—diameter pipe through which water is flowing with a velocity of 6 ft/s. The sphere diameter is such that ge— ometric similarity is maintained. If it is not possible, explain why not. (a) 08:76 [91,0] Vl/o) '95": 45:1 13:1 1/:‘LT" fl: Ill-“7'2 Fifam We ff flew/em, 5-3 : 2 pi vhf/rim raga/Md, and a. dimenfix'amr/ 4/44/7513 9/22/49 ”it we) (A) The Shin/”[717 regal/Emmi A} at” a! _—-—— 0,“ mo 5" ”Vi 0.21;}. _.__ d 446) age/n. 2 +2- and A: 0. 3 74:5 (raga/it’d 074ml”). Thus, 7716 PY€JIC51L1031 geium/z/aiv LI: ,8 g .29 ,‘ 444 M201 xii/4:04: 7. 47‘ i in? A viscous fluid is contained between wide, parallel plates spaced a distance it apart as shown in Fig.1 P7.67. The upper plate is fixed, and the bottom plate oscillates harmonically with a velocity amplitude U and frequency w. The differential equation for the velocity distribution between the plates is Bu - 32” p _ [‘1' ayz Fixed plate at where u is the velocity, t is time, and p and ,u. are fluid density and viscosity, respectively. Rewrite this equation in a suitable nondimensional form using h, (J, and a) as reference parameters. l a My" 1: are" & 9‘ g, if ,- 20.1f- &_‘j 33/ a3 51 IL: 5 or a: 3 flaw] W ~ 2W" '17. 63 The deflection Of the cantilever beam of \ ‘ Fig. P7. la? is governed by the differential equation dy dx- where E 15 the modulus of elasticity and I IS the; moment of inertia of the beam cross section. The‘ boundary conditions are y= 0 at x =0 and dy/(ix= 0 at x =0 (3) Rewrite the equationé and boundary conditions 111 dimensionless form using the beam length l’, as the reference length '(b) Based On the results Of part (a) what are the similarity requirements and the prediction equa- 1: tion for a model to predict deflections? - ‘ E1“ P=(x — L') (a) Lei j*= [-3,- and x =f: 5o "Thai 6’44 _. dafhm" 1 alt)“ (I g is: (1)6 d)<.* ”2:4 2:" I) ax" and (12 )4? 4d?" 21;: 422* dx¥ déx X 7*?— Thus) +71: (2:129;th cll'Fferénézkzl egueém‘n beeame; [15:13]"? i when) dx*z or dag; {-2-}? (w!) and The boundary cam/Hicks are *- ‘jr=0 a‘é xto and 0’9 =0 £12) The SIMIIqu+y regulmmeni’ We 94,: _;¢" 0, 25.x: = f and B». :12,” EMIM The midtown ...
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