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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