Exam9_2008_Solutions

Exam9_2008_Solutions - NAME 5”“ng g S 5% £33...

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Unformatted text preview: NAME 5”“ng g S 5% £33 Officially Enrolled in ME or BME? (circle one) Fluid Mechanics Exam‘lq, Spring 2008 _ Please circle your section! There are 5 problems on this exam: Problem 1: 28 points Problem 2: 24 points Problem 3: Q points ’25 Problem 4: 21 points Problem 5: 25 points By signing below I agree to abide by the following honor code: As a member of the Northwestern community, I will not take unfair advantage of any other member of the Northwestern community. I specifically agree not to reveal to any other student the contents of this exam. I will also remember to provide UNITS on my answers Signature: Date: At the end of the exam, please use the big magic marker to write the first four letters of your last name in the four boxes below as you sign out. Thanks. Exam 9 2008 page 2 of 8 Problem 1. Parts A through G are each worth 2 points: (A) WhatFare the dimensions of pressure in terms of kilograms, meters, and econds? ?; .— — a i V W i: I - WK ' V 9 . /r* F we — twat A») 7; ' traffic M) : . . a (B) F1u1ds for which the shearing stress is linearly related to the rate of shearlng strain (also referred to as rate of angular deformation) are designated as Newtonian fluids. What are the dimensions of shearing stress in terms of kilograms, meters, and seconds? 45 M AM “~63, m3 1‘. “Mia; f... (C) What are the dimensions of rate of shearin s ck train in terms of kilograms, meters, and seconds? m f 7; ya. ’ A»: . egg, ; w». r. lav (D) If u 18 velocxty eters/sec) and x IS posrtion’Cmeters), what are the dimensions of du/dx ? 4 (E) If u is velocity (meters/sec) and t is time seconds , what are the dimensions of du/dt ? 7.. (F) If u is velocity (meters/sec), t is time Seconds), and x, y, and z are position (meters), what are the dimensions of Du/Dt? (AL 152’ (F) If F is force (Newtons) and x is position (meters), What are the kilograms, meters, and seconds? (L8 M ( F§ _ 13L§( M r A?“ (G) Fill in the blanks: If you look at a particular region iof space as a fluid flows past that region, your description of the flow will be from a s s.“ cu») point of View, whereas if you follow the motion of a single particle within a fluid flow, your description of the flow will be from a 351 ‘8 Q! 53 gm : ,_. point of view. Circle true or false (1 point each) ions of dF/dx in terms of (H) In uniform flow, the velocity at every point in the fluid is necessarily the same ----- «(1‘3 F (1) Uniform flow is never accelerating, either in space or in time. ------------------------- -- T (J) In steady flow, the velocity at every point in the fluid is necessarily the same. ——————— —— T E) (K) If a fluid is incompressible, it is necessarily Newtonian --------------------------------- -— T (L) The pressure at any point in a moving, Viscous fluid isthe same in all directions, as long as that fluid is contained in a single continuous volume. -------------- -- T E) @L) (M) Steady flow is never accelerating, either in space or in time. --------------------------- -- T (N) (2 pts) Write a brief English phrase or a mathematical‘expression to define the word "incompressible" (O) (4 pts) List the four assumptions necessary to apply Bernoulli’s equation, either in a mathematical expression, or in an English phrase: txam 9 2008 page 3 of 8 Problem 2. (24 points) _. Consider the flow field V : 003(wt)i + (A) (3 pts) Is the flow steady? Show why or why not; av “WSLAQQ’EQ At] 127‘) (ms‘k‘cfifig w; #6 5’3— 125;: (B) (3 pts) Is the flow uniform? Show why or why not: ’3? r L rr- * W ' w 7: - "a “ v 1:) ’lOtK" aw’vv new“ \ r—r Q j”) 3.x U36) “Ki/Q i 9 All“ lg d—x 7t (C) (3 pts) Is the flow incompressible? Show why or why not: N) ’Jr ‘ - “x 9 7‘ 5 ~ \ . ,_\ ‘ _ K V c \f ’ t/ + ‘43 B LL19 Let L- «if ': C)+G _: Q. (D) (5 pts) Draw a standard Cartesian coordinate system (x-axis horizontal, y—axis vertical) and sketch the flow field at the time t = 7t/2Q). his t: 1; “Ur ngW£>’7 A, sm<w>§< : 03+ 5%“)? . ‘1 ‘J V w W muted? - _n_ _ we ' " W i H 1 ‘ l U “- Sm W» I , 1/}: “’ W yo. 9 L 7:32 i?’ L ‘ lT Snag”!!! axis (E) (5 pts) What is the acceleration of the flow field? /{\ —Z J ._I r ,l f l_) V V l J I 4“ w M :‘f '4’ (v t ‘7) \l 5.9A deew [P ‘ W Dr 9*: / I! I / ‘ ‘ I V : t x 0k - 9: 1' pk Il’ \i’ % 'K’ W 3 L/kfiwx O ‘| K 3‘5 K31" ’9‘? 253/ 1 Ft .M 541 1; / 1 \ @9557 W 1’ 93/ {- VQX: l l “H: 3% i” V é? : CCSQWJC) wwfiwa 'q x l MiSIMj QI‘H" chm“: 13/; : "‘d LC“ :5 JV’M (.53 va‘K\ J (F) (5 pts) Show how to derive the material derivative lusing the chain rule: D/V‘ , .a y éfi: J— .t. a 42 9t 9t: 9% CH: 92% $3 9’ 339 ..—j [ —‘“’ W! N g-V 3‘“ gig— ” ‘~ i' Ut r“ Ar ‘v A 't' tju q 4'2 at X *‘é‘r 9% MB or BME Class? SOLUTEONS 5:1? WW ,3 ,_, MW . “3” u “x 2:? 1 43m( 32L + g (pvfil 113 iw .553 TWCJW \{Mi-km A; vs av *5” JP 4‘“ R “1’13 5, w ) a 1 \Jo LAX Cl (ch—3“ {‘MK RB l_ “M” (I n “a? 1 f; 13 f - ‘ __ —~«- R : D L“? SYN-V35 iqjj 5 ‘ V»): \ 2 rs\ ». ' ' U ,4 R3 j M (W AA “ “Um “‘ 6M «k: ,DMMN wile»; “5 v w— v ( um \ $3”! ‘3‘” “M x tux R at Na at '31s ‘1». 1L 1,! . m M A W . W‘” P‘ngé'wfi“ AV: \J‘QWV: ’— Wiéufit.‘ : f— 5 $99 +£3kjf3xim:% p ‘5? Quimx MM” 5M3 Va; V, x}, me x m \VawsrLH: {V3 AHA (M M V. "7 {V mgk‘gv? Fug; (—fk‘flH—h/mwe +ij)”" (fflbv.\(m\ F? : ZfAB‘VD AIUEmI’B-klék 3 RR, 8 f 0 C59 S '3 R g 1:: \M’ ’l ADV‘, Am m: n rg—KU. 3 ’ or?) W 9" /fl:-mm~ 2:? 4% M W t ’3‘” W 9;» EM (E, W F a a «— 2fAuv.m3 NAME SQEXFTEONS MB or BME Class? _Sf}l iU'i‘léfNS_ NAME SOLE: IONS 6') B);Water (density p, assumed inviscid) streams steadily from a pipe into the atmosphere. It bounces off t e ground as shown in the figure below. The fluid emerges from the pipe with uniform velocity V0 and area A0. The water hits the ground at an angle 6 with respect to the horizontal axis. The pipe is located a distance h above the ground. Assume that the water does not lose any speed during its collision with the ground. Find the force (magnitude and direction) that the water exerts on the ground in terms of V0, A0, h, g, p, and 8. Warning: There will be little or no partial credit given on this problem 0 One way to check your answer is to be certain your units work out correctly 0 Double check to be certain that you have correctly determined sin(6) and cos(6). Write your final, correct answer in this box: 1 V0. A0 h M x\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\‘ g p‘t A} i J #2— l ‘éwfri—K—fi‘“ vlur: a} $33 13L SiAe VI} 3 aft}: {Zak we /> _, “t Z? : j N (mm; + Sigma,” if A: ‘71 w) ‘ a}, W m»; ‘ widen BARK!” {"5 i) \A’ A -»- _ \ " wt? 4» new Ed" V16 ~ “£3 A ' r (g “Ex, " 2:: iwxqa 1 Vega V, J ‘\ $4} \Aéor a» C L/ \ ) M 2 L s ‘ Page 8 of 10 Exam 9 2008 page 6 of 8 Problem 4. For the problems below, do NOT assume steady flow. Do NOT assume constant density unless otherwise stated. Each part is three points each, NO PARTIAL CREDIT 3 This was cweéA 4m WSVJV ' (A) Write the Reynolds Transport Theorem: DES 9 ' l 1 fl E8 M) :99” + S filo (wag M: DJ: , C i (B) Write conservation of massi’n integral form: is W“ F i .4 A (9%" ;L 0' gag WW}? lMVwHA Guam (C ) (i) Write conservation of mass in differential form using vector notation: Q A . 1 0;;Abel/ (ii) You used vector notation to write your answer to part (C(i)) Does the equation you wrote result in a scalar or a vector? 4’ S (D) Write conservation of mass in differential form writing out each component 3 ’3 12% “p 3'99? + 94L “3 + 9L L03 . . . . . . 9 . I 8 . (E) Write conservation of mass 1n differential form usmg vfitor notatlon agimlng constant density 7 "J (F) Write conservation of mass in differential form, writing out each component and assuming constant density 3 o / —— —— t _/ El K 9 If 93/ (G) Write conservation of mass in differential form, writing out each component and assuming A constant density and uniform but unsteady flow. 1: CL MA ‘ —9 flag)? ‘9 A _ was fa W \ fl/éfi +7/D/95‘l'1p “g . . . . . I N-Q (H) What is another name for the equation that expresses conservation of mass in dlfferential form? M , ~_ ' MK; 5 3 c (I) Write conservation of momentum in integral form a '2 r :—> A A mwtk’me, 9» iéjgfgldeqt+ X/DVQQWS AJAX gawk C c (J) What is another name for conservation of momentu in differential form? Exam 9 2008 page 7 of 8 Problem 5. (25 points) The following equation expresses conservation of momentum in differential form: R “xx. Mort 1:3: v vaattv DV / ,4 2—1 [OB—Z—VP‘l-pg-I'IIJV V t i (A) (4 points) Draw horizontal lines over every variable in the equation that is a vector (B) (12 points) Working in Cartesian coordinates, anii assuming that gravity has a component only in the z-direction, write out all components of the conservation of momentum equation in differential form. Be very clear to indicate which terms contain partial derivatives and which terms contain full derivatives. Also be sure to clearly indicate which component is which. We cannot give you credit unless your answers are clear! 1 (C ) (9 points) Being sure that your answer to (B) is still legible, go through and cross out all terms that fall out if we assume that the flow is uniform , B ’M f“ M1, beam 77\ i AT? U 63 i W16 EMU/3: L5 all .. i I i l ...
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Exam9_2008_Solutions - NAME 5”“ng g S 5% £33...

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