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mt1soln

# mt1soln - ale*5#145467 FOR NO CREDIT FOR 7 ANY MATHEMch...

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Unformatted text preview: ale *5 #145467 FOR NO CREDIT FOR 7 ANY MATHEMch ERRORS D\MENS‘OMAULH’ iNCoRREQT IN unﬁt: . uxbmc‘ - UNIVERSITY OF CALIFORNIA, BERKELEY MECHANICAL ENGINEERING _ .r. ME106 . Fluid Mechanics - NAME QEizELLlEES lst Test, SOB Prof S. MOMS 1.(65) (a) Find the ﬂuid acceleration a for plane stagnation point ﬂow in which the velocity is given by the expression V = cmi — cyj (the Constant c > 0). - - Exam/x stats «uh—«MA Ewuaa if: z :92. text-1 __ :23; .2 ‘ Mean '. \QH . ‘5 5200 E: Q J at rt" Cw “to . 8r. ‘Dev . 37. ) : at «a , we , w N "5" N “1: ‘9 )- V's-“At: z: "‘7‘ -' ’ l d x; 3"" \' (Aya‘cj‘ Eli é-ivtﬁﬁhin-‘f _ i, I M W m E, {'53:- 5 Q9; (it: C it g :7ng A.” 3 A? > ‘ ' 3 e a e f 1:.» , x. i If”) .4 v' i or; V: UL i + ’U‘ CU} “'2 v x v N N '31] "I: 1 £1, .4 k 6} a}: at "‘T £4" £5“. :5 a E I! d" J ﬁt“ a A A L f . ' 4 "X g \ at) C 2w ,, - . , M" / '1 ,1; ~; :2: -' M“? V 5% 3‘ u. r‘ 31‘; ' ‘— -..z m» amww . ‘3? i 5! 1" I i L051: mm it» . ~ mot'H/l eMY‘Ca/‘SI "'5_ why“— ctt'd not Undeﬁme, uhﬂ' m5) *- 5 CCWQAOQ. . vac-101‘s amok \$CQlOU’5J - 10 (b) By considering the stagnation streamline, explain why a 7-5 0 in this ﬂow. a EA . \. f I L; w 35%,“? Iva~f’5t,min a: ctr i if?“ zit-"ﬂ W-” ‘ ‘- «-‘r in": .2 ' ::- " -"-‘ . \ v 3’ ; "\‘l 2’_LQ;,"\V ‘5. VMQQWE a,-"~&?'r«.ﬁlil 1,1:- L... “3 :3“? it“ “a 1‘ _ r I . t , ' :54; «Lilia-“t. diet-1’} as i?" U‘ ND Orcdx-lr 9w vagumewie 4mm- ! - ' Phlébi'c.a\ mm‘ IN BLOCK LETTERS PRINT YOUR NAME ON THIS PAGE “9 1508—1 2. (65) The vector ﬁeld V : iécrf—F-vﬁ‘, 2367+ c212 {c > 0, constant) stretched in the axial direction. For this ﬂow, the ﬂuid acceleration is 2 a= (ice * 3:)H (9‘1 e if? at ’1‘ WKQ Swag Mam/1‘. LiOQ/(of) a» Dex/'- 332A represents a vortex that is being given by the expression )é + 62217:. (You are not required to Show that.) Find the partial differential equation satisﬁed by the function v(r, if). (You are not asked to solve it.) Given. In cylindrical polar coordinates 136', 2, if F : frf' + fag + fzfi is an arbitrary vector1 then » wJun 61"": 5m 3 i E I 61 Z § p. z i i Z ‘1' T V “2'1 54L {A {:7 M, V ‘9ch ‘ ~36 “1 \$1M attempkeci to Eula“? CCIUQhCV" Quad \ﬁuﬂtﬁ. ~ Em Cceﬂtlm L513 , 4: '50 1808—2 cm at smut-H04 I Pod/rt Wat-“(One 1 —- Did Vick ace vi. 912 o midi“ - ML‘S‘aM% v“ tom/13) —- 5)— ? 7‘9 3:: ' ¥ 5‘ 3 3 fr TfB fz r s “7;. (w 2;; ,h “r Iii” ‘y ’ p 1;": , 3:4 I“ {Lizzy} 1'" e V"- T—1 #7:. ‘ r?»- V‘ ii i b g L. Di. v x . Z; <— + .2 W_WWWM__W'N£ v+w Expanding V “9 '2 C3 5 1,; w» i , 1%; - " m3. :- T in: m L: ‘I‘ 3. vi (“a V; 3,. 1‘ C‘ "r 5- ' h -. g 5 Jr \6 the CLWPlE’i’t and " _. (1 ﬁt {it 70 3. ('35) The figure shows the side view of two discs each of radius R, and separated by ﬁxed distance h. Air enters the gap between the discs with speed Vb through the supply tube of radius a, ﬂows radiain and leaves as a free jet into the atmosphere Atmospheric pressure is pa. Within the gap, the Speed V varies with distance r from the axis according to the rule v={ (You are not required to show that.) The pressure within the supply tube is unknown; it is not atmospheric. _ hm Ax isymmclric ,1; HQ QM- v:UOJ?”-"? WOLdM “3/ CM” I'i V0 ifr<a 2V0 ifr>a. s. l. ssh—- rah—- kin .q Atmosphcl'c ‘a - U '— r "_ frecjel e ’9‘ W id, ": Wynn J 83% .3 L93 v Bfch C) Go 4' 5 £0?“ a. Au—nospl‘mm|a 0 Cog . vm P Wm (LUZ? (3.) Using the Bernouili equation, ﬁnd the pressure p within the gap as a function of r, p, Vb, h and a. W Sketch p as a function of 7" showing clearly any maxima, minima and zeros. (You need to consider the cases 7“ < a, r > a separately.) 4' ‘ “Pr g ' \ + 520 For WW5 ‘ "" -' - 5 ' t;qu wagon ewe/We “Tl/\L‘S welt/idea Haase. who +0 'de Pew‘vg gay-ea. = tr: —- , ‘ _-_- - f”; 11.. 5 v 1508—3 .7 a“? R a +5 ‘Er caveat a *7 \ pm; 100,. amen“ 45 \Moalmgl Jr WW5 go 3 (b) Noting that the ﬂow is axisymmetric, derive the integral expressing the force F exerted By the air "70 On both sides of the lower disc in terms the pressure difference p(r} -— pa. Pi: ’- ‘7 I, “r 3 1;” "9"5k-5w‘--I.o *2 3 ' w iii. 1» ' ﬂautRE _ 7 | 7‘ _ 1. ,7 FOR mu. ORbe ‘ W 0U“ ' W4“ 46? Wt“ YZETM-w‘ ' 77,777 m WW“ m » ~--‘--w~-'-- -— ‘---~Wr~ _i Iﬁwiw__'ﬂﬁﬂ_r_ﬁ):‘E——H_MMJ ﬁt“, A“ j“ y ‘M" r ' SE ; ‘2 1-” 5 , . WU; 3 “1 3h _‘ If , a. ‘7. . .: - . n I 091‘? “afﬁx, : ‘ I “g f, \ jg {x (c) Hence show that F is given by the expressionk m‘l‘ k.) H \O 93v can“: (9 1‘ 3.. Fan...) Via-1N mammal? . a4 R Explain physically why, despite the presence of a stagnation point at O, the force is upwards if R > a. (2 ksk K; .24, s r-—-»-—."-—’_'-— a 1L aft \ J ,Q J é 3‘ a “w”. w 5f; rev-m! 1 e in +5 _ 1508—4 emu W smwame prB "3:? \$30; , OW mask 0’? m, dxeic we, (stiﬂe-Poe wees amsewmim. FCRMM, as war, v” dean ' memes, be We W Pa 0* sac-n paws _ 4’01" - v“ 5 CA - - ...
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