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Exam9_Solutions2007 - MB or BME Class DULUU a N 3 NAME...

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Unformatted text preview: MB or BME Class? DULUU a N 3 NAME f’oL» on <3 M5 Fluid Mechanics _ {/i Q3551 fat/«emwawgr Exam 9, Spring 2007 Km ; There are 6 problems on this exam for a total of 107 points Problem 1: 16 points Problem 2: 14 points Problem 3: 15 points Problem 4: 32 points Problem 5: 16 points Problem 6: 14 points By signing below 1 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. Si gnature: Date: Page 1 ol‘ ll) MB or BME Class? NAME Problem 1: 16 points Water (density p, assumed inviscid) flows steadily from a right—angle elbow as shown. It emerges with an initial velocity V0 and an area A0. Its initial velocity is entirely in the positive X-direction. (A) 4 points. Find the (x, y) location where the fluid hits the ground. Hint for part (A): In addition to a value for x and a value for y, there is something you need to indicate, either in the drawing or in words, in order for us to give you credit for your answer: r \z i x“ mama) (B) 12 points. Find the force (magnitude and direction) of the ground on the water pit“; ‘r‘ 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(0) and 005(9). Write your final, correct answers in this box: (Q T”: " Page 2 of 10 MB or BME Class? NAME You can usethis page to work on problem 1 MB or BME Class? NAME Problem 2. (14 points) For the problems below, do NOT assume steady flow. Do NOT assume constant density unless otherwise stated A) Write the Reynold’s Transport Theorem F) Write conservation of mass in differential form, writing out each vector component and assuming constant density. r} x m 32‘ r r: w»! {~1an WWW. (mm was“ We _ an”; ,«3 \ G) Does the equations you wrote in (F) hold for steady flow, unsteady flow, or both? 94 t n: H) What is another name for the equation that expresses conservation of mass in differential form? I) Write conservation of momentum in integral form Page 4 ol’lO MB or BME Class? NAME Problem 3) 15 points The following equation expresses conservation of momentum in differential form: wig?“ fl: p§ + ,uV2V DZ Term 1 Term 2 Term 3 Term 4 ,0 (A) Draw horizontal lines over every variable in the equation that is a vector. (B) Which term(s) do you need to leave out of the equation in order to obtain Euler’s equations? (C) l7 ill in the blanks. (mass)* (acceleration) of the system is reflected 111 term number(s) t; reg. 22W :22» , iiimw? M“ wwwfirmw 4‘ 1.: 6,6»: energy is reflected in term number(s) The energy is reflected in term number(s) energy is reflected in term number(s) WM“ ,e mus-ovum :4 Mm W. WWW», MN, Wflmwwbmmw m o N Na (iii) Working in Cartesian coo1dinates and assuming that grav1tymhfigvswa”)gompwonentmonly 1.9.3132? write out all components ol the conservation of momentum equation in differential form. Be very ca1eful to indicate which of your terms contain partial derivatives; and which contain full derivatives. Also be sure you indicate which component is which. 'We cannot give you credit unless you make it clear! ‘~ MB or BME Class? NAME Problem 4. 32 points) A rectangula1 plate is fixed at the ground and a second parallel plate with a1ea 4 m2 moves above it at a distance of 10 mm with a speed of 2 m/s. The shear stress developed 15 4O N/m2. Assuming steady flow of a constant density Newtonian fluid and that edge— effects are negligible, please do the following: (A) Sketch a plot of velocity as a function of distance y between the plates after the flow is fully developed. Indicate the origin, label the axes, and indicate the numerical value of the slope at each region of the curve you have sketched. (B) Here are six assumptions needed to derive the sketch that you drew in part (A) from the conservation of momentum equations. In the blanks next to each assumption write down the mathemancal ex' ‘ ss (l) Steady flow (4) No pressure gradient (2) Constant density (5) Fully developed flow (3) Newtonian fluid (6) TWO-dimensional flow {hwfiés W if? 411) Page (1 of m ME or BME Class? NAME D. Write down the x—component of the conservation of momentum equation (note: you already wrote it down in the previous problem). Cross out each term of the conservation of momentum equation that is zero and indicate which assumption you used to state that it is zero: {a (E) Rewrite the equation you wrote in part (D) as a differential equation you can solve. Clearly list all assumptions you are making in this step (G) Use the information given in the first part of the problem to apply appropriate boundary conditions to get a full solution ME or BME Class? NAME Problem 5 The drawing below shows a tank that contains two incompressible liquids. The liquids have specific weights VA and VB. ya” << “(A and VA < YB- The top of the tank is open to the atmosphere. Side View A. Write an expression for the pressure P as a function of y between heights hl and 112. Work in absolute pressure, and assume that that atmospheric pressure is Palm B. Write an expression for the pressure P as a function of V between heights 0 and hi, Wor< in absolute pressure, and assume that that atmospheric pressure is Pm C. Now assume that Pam = 0. Find the magnitude of the force due to fluid pressure acting on the side of the tank . Please put a box around your final answer. Page 8 of ill NAME 01' BME Class? ‘ MP .1 Blank page. 5 You can use this page to work on problem xwwdzs MB or BME Class? NAME Problem 6: 14 points Water (density p, assumed inviscid) flows steadily from a pipe with initial velocity V0, area A0 and angle of emergence 8. It pools up on the ground without splashing and rolls smoothly away into the page. Find the force (magnitude and direction) of the water on ground 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(9) and 008(9). Write your final, correct answer in this box: LA, x If?” y“ w " {V Page ll) ol‘ 10 g _i r w fiflwflw XE: ,a w § w M3 4 4 4 2/;qu «a “m «W ...
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