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Chapter 2 Properties of Fluids 261 Solution A thin Mt plate 1s pulled horizontally rough an oll layer sandsiched besween two plates, one stationary fn the other moving a a cunstin velocity, Ihe location in ol whee the velocity is zero and the force that ace to be applic un the plate ae io be detrined, Assumptions 1 The thickness ofthe plate Is negligible. 2 The velocity profile in each ofl ayer liner. Properties "Ie solute viscuscy uf ois given w be r= 0.027 Pus = QU2T sini Analysis _ (2) The velcity profile in each al layer elative to he xed wall sas shown i the Fguve elas. The point tof zero velocity is dicaed by pol A. and is disnee font Use lower plate 3s deterained from geome considerations {We snulariyof the ne tangs in the loser ol aye) to be > yy 0.23636 mm Fixe alt yt mm N mis F rs a nid Vind ‘Mowing wal {0 he magnitudes of shear fores acting onthe upper and lsser surfaces of he plate are Paonge Faccpee al eA 420 ers sin? }03-08m°1 ren Futuna =F yaa 214 f=, {0.027 sim8i(9.3»0amey 4 COS “i 26210? m [Noting tha bh shear forces art inthe opposite deeton of alon of the plate the force Fk determined from a force hance nn he plate tobe F Foseagae Hbosdnne 7291208 10.4 ‘Diseusslon Note that wall shea is freon force bewveen a solid and a iqud, and it acts in the oppostedicelon of [Chapter 2 Properties of Fis Solution A duch sytem i wsod eras txgue rough aol i besween nwo Mental disks, Per spect Pott spec tw tensed ongue ob determine, Avoumplons 1 Tae ticknes of the ln wom, 2 Tacroatlnal spe of Ue sks rela constant Froperes The atest viscosity of ils gen tobe ¢-O28Naint, — alll Derg Deen WLU... sks sha aia WIA, sxe (77777) ee ssescwon ness the sks anc ming in the sm tion a dilficet gular prods of 9 am of a "Thefare we som ‘source he inks lo be stony sm Ie ier berating la al perf ye. ike ely rie ‘eyes in te oof Elm thickness is Vf whune fy ep} the genta velosiy. Phen Use wall shear sess ‘eyes onthe sureee uf the Fase dis at a dace r rum he a fran eat be expressed as ett te La thes force stg eri wen aor te rte we a tong erat soe! cam be expressed <———— oF dd OOM ern or a rie BOOT gy FMEA OD psy Twegacing ov i 825 he ea oe aga pes yo Bef A) (er radies(4490 1208) revmn]| AMY) 5.45 ts Sacto, heorq uated i deteoed oe 1x SO28N SOCAN yee carson Not a ds tre toes sje 4 the fa es sk es, on sey [Wlopoticnal to he thickness fhe ry, Coapters eropertes of Paes Ferns wena block eae psn este a eguas ete: hea ao fm ppd oe ‘tiene been Avmins the inked ce py ioe 2 Ube ion cae! ale i a Propane: the asslue vacuo lis gvea obs ULL Res QUIEN vapsis (a) the vcr of te block cont, an Use “Eton tht ce im tae on Fe bey pm vine Ye-w [Bruin 204 8220 (Q27 CAT Nha 20 4007 sat oyna ee ton ce cpa by he sa face 1055.8 Rector we | SEO wa ook earYaseo2a° sss mes nye en Rant Ro 8 Maas Aso ayy atior wos & Eu abe tae (AN) sis20"=:s0 Ni eusz0 100591 Snag i Ey Aue egies aces da sb m tea mtecsa 1 gure Paveeage ela nga ae ‘oe a te force ecu o pach te block 28 Le scl! sce Leaurs sgl by ong he 2.88 Solution ‘The velocity profile for laminar ane dimensiaaal lowe through a clrular pipe 1 given, A celation For ‘ieion drag force extsed on the pipe and iss mumerical value for water are io be deveined, Properties “The viscosity of wave at 20°C is given tv be 1.0010 kins. UIA = el FE) ithe pipe, and unqy is the maximum flow velocity, which occurs at the T ° du af ry Pils it moe 2) man § ‘Nove tha Uke quantity dvdr is negative in pipe floss, andthe negative sign & added to che 5 relation Eur pipes to make scar suess in the positive (ow) direction a posive quantity. (Or, dwiir= duly since y~ Rs). Then the fiton diag force exerted hy the Fluid en the ian surface of the pipe becomes nA, B Fila (27 RL) Sp ge (0 Sostning eves we ge fy Heide A (B0O10Kgin-)0m0 8m 1 gn! Discussion Tn due warance region and during turbulent flow, the velociey gradient is greater near die wall, and thus ube ‘ag fore in such cases wil be greater (Chapter 2 Properties of Hunts 290 Solution the velucty profile for laminar one-dimensional flow through « cirlar pipe is given, A reladon fur ition dag force exerted on the pipe and is uerical value for water are to be dewrined Assuors Ie Uc ares o-ial2 The ts Nevin, Fropentes he ehcnty of er °C en O00 a~ngQiee nays (0) pes en #) Yo IR whee i the ads oF he pipe the radia distance fom the center of te pipe, an oy isthe miu Nos velocity, which wceurs at center, ~The shear sizes atthe pipe surface can be expressed 2s [Nove that the quant cv Is negative a pipe low, andthe aegis sig fs ae ta the ¢, telaton fr pipes tn ike Sar suess inthe postave (How dinecuon a postive quantity. (0, dvelr= ay since y= R —}. Then the Bedon dap arc exerted bythe ful onthe er surface of the pe bxeumnes 2, an s F, t 1) = Sage, (aSintining news Anila,, seooni0K-sicowrwe| 2S | 264M Diseusslon Tn he entance region nd ving tuueat Haw, the velocity gradient Is greater aear the wal, and this the tag fore in such eases wil Tages, (Chapter 2 Properties of Flas 2.00 Solution A frustun shaped body is rotating ata constant angular speed in an oil eomainer. Ihe power requied 12 Imainain ths motion andthe reduction in the required power Inga shen the ol wnperatre rises ae c be dewrained Assumptions. The hiekness f the al laser wemains constant Properties The ssolte vscasy of >is giver 0 best 11 Pas—0,1 Nes at 20°C and 1.0078 Pass BO"C. Anaisis "The velocity gradieat anrwhere in de oll of fin thickness Asis Wh where Vo ~ ur i the tangential ‘oleity, ten the wall shear sess anywhere on de surface of Uh astm ta distance rire the as of ration ie eV ar Fe be ‘he shear fore acting un differential area un te surface, the torque generate, andthe sat power atociated with t nroexpressed as hateem SAE 10W oil of lm dikes ae ryt wath oth : 1 fra tig or 2 vi ot Sfp apse Fos theop sic, d= 2d. Sato on pra, ax? Oe 2ape? pt, 2sum? |"? un? nt Ware Pema Per oe [ema mah ah Ballou suc’ A rltion forthe boom surface is obtained by replacing Dby a. Hi, ‘oreion of rds with axa stones expressed a8 + Pg op feat Ditfrentiting gives ee! 2 ba ae ‘le, Sustain and negating, ba wit a) WHI se? 9 HO i? 97 og Aa abe” pa ae AEP La cagt 2a teat hg gg = TOMI se ee oy 2 20-2 =} 1w ooo | Swit ond btm Noting hac pase is propetional to viscosity the par emule at °C UTA sin? ‘O1N-sion™ i, “Thocefore the recto in the requires peer input a ROC Is Redution = Hae Ha Hy, lem) =21.1W 2mn21.1=240 W which i hot 92 isewssion —_Notethat the power rogue o overcome sat forces in viscous fla realy depends un tempera Assumptions 1 veloey pln oe Arabs ang soy pn ol ag: eal a2 tonal nboou =i ent ppg wee ceramide were mgt ome he eg Ds. La le ia eh a ee aa is Peevey win vo ow we ies ele Vee Beton wat z “hic comsnor seo steed mas yay eRe vce we ave peers CoA Uta in cs se ae as pa ae ark Chapter 2 Properties of Pius 292 Solution A layge plate is pulled at 8 constant speed aver fixed plate. The space between the plates Is fled wid fenxine oi, Ite shear stress developed on the upper plate adits dieetlon are to be detnmined for parabolic and ina lock ple eases Assumptions 1 The thickness ofthe plate negligible Properties bo viseusiy of lls j= U8874 Pos (Vable A) Anatysis nN y Engine ol Consiering 4 parablis patie we would have VE om Ay, shee fe is a costa, Sinoe Ym Um m/s when ym he Seam 840" we wri (42) wie (5210-8 m) +m S200 m/e ‘Then the vetocty profile becomes 31200) + V = 58566 /F Assunsiny, Newtonian behavior, the shear sess on tae upper wall ae a omar AG 8 egy B86) = sz.71a4y | es w= S7.712 x (0.8574Pa-s) x (0.0185) aaa Nau? Since dynatnie viscosity of oll is QBB7E Para (See able A 7). IC wesassume a hinear profile we will have ay U__ amp dy hSxi0?m ‘hen the shear srs inthis case would be B00 st (0.8574 Pass) x (G00) * 670 N/m? eae wey here we conclue tat the finer assumption is nat ealste since it gles aver prediction. Chloe? Propet af ois Solution st spill! with» eon ley trang erg Ue spare Beta die sal an Bong Filed he Rie ec wie a ain hdl sie shal diet evumplione 1 tid Newtown, Propose ‘The viscosity ofthe tude gven be 0.1 Pas Anais esring ocous oly. shot the varying rane cam be ox a Kuso oan sume x ie Bg. Aamir thse ws e feta tha aT Assunng tlacrvloety ds oton lth gap, he vsous Free acting the dirt ap ements at atte panes a po Qa dG pe loon nn 2 oy Chapter 2 Property of Mis one Solttion staf rates th aeons aga spe a bring ‘These heen he Satan earn is Fille ‘ith 2 ld The erg i to main the mao so he etemine. Assumptions 1 Tye Olds Newontan Propertes ‘The viscosity ofthe Ads given to be .1 Pas Analysis The varyag clearance ft canbe oxpessed 3a functon of xl coordinate ce awe bear) Ancol tnt serch ean hwy ha AIT Assuing Hoes velocity sition nthe gap, the viscous foe acting on th differen ste peer 6 era tte Barve Hy where Us 21/60 inthis case. Thon ihe vss trque developed onthe shat at 2 By nz) xy weeded arearxde A — a = hha ad ha) Thy in z Imeraing pemaot pa pimp? tas — ha = =n 2 pte 20 haw Vung Ls “220 Ah hy rahe giver deta, we bea 2 (ot Peery (602 L0-Fuum)*(400 x 10-tmm) 4.2 eT aaa 
