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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: (Li5:0 [Arl)y“ ‘ «124—0 V (ﬂu—7')
I4 ﬁ/laws ’Ma'l' 52::— ‘/2 am! #7912149” 64.» I.) . ¥ , . , /§ #1611; mi ’ 209521 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' LTI 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? Cir2) 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} boxerv“: #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
sew5:3) ——._—_.
———_.—._ ) 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, Sinailady) 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, 53 : 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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 Spring '08
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