problem set 5 solutions

problem set 5 solutions - BME C101/C201 INTRODUCTION TO...

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Unformatted text preview: BME C101/C201. INTRODUCTION TO BIOMEDICAL ENGINEERING FALL 2010 ‘ PROBLEM SET 5 Due November 3, 2010 (Wednesday) Please either (1) put your homework in my mailbox in 5121 Engineering V by noon or (ii) give them to the teaching assistants in their discussion sections or ofiice hours. 1. Let’s consider the van der Waals interactions between two cylinders separated by a closest distance of approach equal to d. If d is kept the same, but the cylinders are increased dramatically in size, would the overall van der Waals interaction energy between the two cylinders increase or decrease? Why? Plot shear stress vs. strain rate for both a pseudo-plastic fluid and a dilatant fluid. Also explain why the slope changes as strain rate is increased. Consider laminar flow of a Newtonian fluid down and through a thin rectangular plate (see figure below). The distance between the two parallel walls is 21. The height and width of the plate are h and w, respectively. Note that the flow is due to both gravity and a pump. Accordingly, the pressure at any 2 position is notjust 1 atm + pgh. However, you are told that dP/dz = or, where 0t is a constant. Assuming fully developed flow, derive an expression for the velocity profile, keeping in mind that x=0 is located at the center of the plate. % <——> 2‘ i1. fluid out i i l i 4. An incompressible Newtonian fluid has the following velocity profile \7 = 4xy>‘< - 2y29 in units of m/s. Let the density and viscosity be p kg/m3 and u kg/(m*s), respectively. Derive an expression for the pressure gradient in the x-direction _a_P_ 6x and evaluate this gradient at (x,y,z)=(2,2,1), where each coordinate is in meters. 5. An incompressible Newtonian fluid has the following velocity profile \7 = 2x2)? — 229 - 4xz:3'~ in units of m/s. Let the density and viscosity be p kg/m3 and u kg/(m*s), respectively. Derive expressions for all 9 stresses (i.e. 3 normal and 6 shear stresses). firm—72m mm d“, Waco/5 aH‘r-ach‘w rWVJLQf‘VJnJ WI'I/ (hangar; (/26. Me Inflraa’v'm energy aw” WW more nejav‘I'I/‘e ) I/C 7%8 57/4114er ”401111536! r74 513,36 Thu“ occurf éecwoue Van filer Wad ”474376194077! are Fal'fufl'fe 40105762» any] you are [mamas/rag 7942. 2515 025’ flare V Inwraof'm: by [McKay/161 flu; #‘ d/ nap/Eta“ I'M 646/2 C7/fléztif A PSCMJLO'Phnsfia [In-‘0‘ rs Shear 'Hm'nninj_ agmpau‘nd flu. S’of; In [egian Z [/010 .fhér) +5 HUA— l‘" 243:5" 7’1: 013% tharL H4110“. Jamar,“ a: #0“ 54‘" mare, 50%; #4.. sic/ea. ,umms’ Hz vixaosify damage“ a: yams/‘44? mart, . A dl'lafi’h-f- Fla-d I'S I/uar Mathening, leaany {/u, Slo’ee, I?» leejo'cm I Nomi/tear) 7‘6 Mom" :31 £0741,” IE. {Adl‘ SW)/ flu f/o‘ge inwaxe: a: you .r/war mom. 5fn¢¢ #u. .r/afe in creases, {41¢ W359“? Moreen: a: flush” mag-C, 9.’ ”TWIMIQWM .L—"CWL 6mm?! (MM I: :/W af‘ gremr x VW ‘ v’f ”WWW” I‘M/6’1”“ Ma“ MMWW . P! (”Wu”) Causing vflm 6'an SfreJJ' fay; ) gm” ‘ ii 44% “Iva-how £- #1!”th 73,: am"? Mud 41: 01400” A49"? ca. . . m Fla-b , since. MN. 19 ' I .7 WV a? N, {14° «F (Maul— X’VW‘K 6’6” 5’” 'i" Va 9;: “5' 537+an (5,4,3 aMLM- "mum . . ' w ' x-wlw, barf). '77“: '1‘- Fully may '50 5313’?» and \/E 22am; amulet m. We ' ' 5/4. Na Flow M11: 4;“ MM wall V” a V” =0. ’ . Pjgutg 'f/‘fznf, I10 / am.“ 21 44 w, Va *3” 47‘)- [M figmfinggrgy My) Nah, 44‘” IX?) 0 "’1 a, . {F 1;"- ‘Hfl-L amr #mefly" V’Véffi) half 0F PMS] 3 9V '7‘ 0 + (0,) 9'3? *(v)(a)+ 14; (9;) =0 *3? 1" ‘0 '4. F‘ ' x: r: (M) f ' 0 ,a aflmwg «firm!!! (439019) 7‘74, MORE) + 13:55ng («139603) _ Pwe (Way) A um... .n'~.~.am owl-"w : | Applqm 44w. \oounoKn-rg Walk-5m: Ill-#24:?“ “F ”(7“? #4. 3:: at; =.[&MK'/::L W 5'" “19‘ W )5 7!.” “725.: ”Misflfl: 5?“:n: we; 0 “1%! >0 “My, 0%! fan‘- 0/19”“! at 7: l' .3 =1 p a V _ 21....7’6 Ma ’ w!‘ 2‘.— mi. ”U; L7 WK we. a? up d—«Enmo m tVMER-pfi azure amr WINSJ‘ Ma}! W haw «1,3: -9 a ”fig “fit“awrfl' Kw Is 79qu «may 743m MW/a E i i i i =‘(fw-.—-- ”gh-‘wav-ma "’1' K’Jza'vz; wa ' , D CL9+W)(€B+,6LV?’€3 = “'(zfiifid/fia We can app/5‘, {+5 Navier—Sv‘v/‘es %u¢+'ans, sine; we Adv-Cay; lbwm/nreSSI'Lle/ Newhm'an f/w/. Sung m are 071/091, thylcrgJ/eg/ ”'4 gig, we orI/y mega/7‘0 765M 0” 79”! cham/ommf of 5%; [Mr mamem‘um. 924$ 9U): 921' a ' A 4, y _.._ x 97/ .9 2/” 2 ' p at (9 X 3): +10%?” +/OU?3§ z,“ . x +9 ”9: +91% 'aF Cmbinina 53:. [/)~(/29-' ”(0) +,a(‘/xy) My) +/f2y’jl4x) +,a lo) 10) Luz-0 + 0 +5] - 3’3 + filo} 9X WW7?“— W‘Ia = "$5 \211. we " "By/”X72 at; (Ly/:9) : (2121') U56 'H’Lo— egwavhaws 740m [adum 74" 2399 EV, t%%/ Ix”, t9”; 1"ng t%xl 792" “M fay .743)" an inwmpness/éé. MEW'I‘vnIh-n 747qu 9 TX): 5 2A 5}?(2X2)= 2M(L/X) == SMX (K s: q H N 3: 621 T m \‘S l! N k E? V ‘ 0 Tata- = 2/4 3‘ (“L/Xe): 2M (”L/)0 = ”3"”! TM! = 21,, ”1 [%/ZX‘)4 5c (-2913,; (0+0) =0 » : ~4a4+0):~4 a. TK%:ZZX=M %¢(“/X?)+§E (2741)] /“( M 2. _, _. 4"):a2M3' 1712:7527 =M[§g[—%}+§3v(”‘/W>'J*M(Za 0 E i . , ...
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