MAE 222Mechanics of FluidsPrinceton UniversityAssignment # 1February 2, 1998Due on Wednesday, 2PM, February 11, 1998• Assignments will be posted on the web, usually by noon each monday.The URL is:http://lam95.princeton.edu/mae222.htmlYou can download the assignments directly each week right after class.• The assignments are posted in .pdf file formal. You will need AcrobatReader, available free from Adobe’s web page, to decipher the .pdffiles. Most university computers should already have Acrobat readeravailable. If not, click on Acrobat on the course web page and followthe instructions.• I have also set up a MAE 222 newsgroup, called “pu.mae.222”. If youare using Netscape, go to the Netscape icon on the menu bar, and select“Collabra Discussion Groups.” Go inside “nntpserver.princeton.edu”to subscribe to “pu.mae.222.” You are encouraged to ask me ques-tions and give me feedbacks using this newsgroup. I will also use thisnewsgroup to provide brief clarifications of lectures.• Additional detailed information about the course can be found on thecourse web page.In doing homeworks, do not substitute numbers in your formula until thevery last step (whenever that is possible). Formula in symbolic form is muchless prone to errors and is much easier to check for correctness.For the first week, we will cover Chapters 1 and 2 of Young, Munsonand Okiishi (YMI).1Chapter 1 :§1.1 We all instinctively know what a fluid is. Air and water imme-diately come to mind. How about milk? Yes! How about honey?Well, yes. How about tooth paste? Now I am not too sure. Forthe moment, we will let the pompous definition given by YMI onthe bottom of page 1 stand. At the end of the course, we will seewhether we can come up with a better definition of what we hadbeen studying.§1.2 We will use both SI and the British units in this course. It shouldbe obvious to all that if an equation is correct, then each term inthe equation must have the same units (you can’t add appled andoranges). However, some equations as presented to you can beused in any system of units you may choose, and some equations(as presented to you) are valid only when a specific unit is used.The former is obviously theoretically preferred, but the latter isfrequently pragmatically preferred. The latter occurs when somevariables in the former formula have been substituted by theirnumerical values in one specific unit.§1.3-1.6 These are the major properties of fluids we shall be concernedwith: density (mass per unit volume), pressure (normal force perunit area), and viscosity (the Newtonian description of fluid fric-tion). You should know by heart the “order of magnitude” ofthese properties for air and water. See Table 1.4-1.7 of the insidefront cover of YMI. Do you know the relation between the GasConstant R and the “Universal Gas Constant” and the molecularweight of the gas? If not, ask me in class.§1.7 Focus your attention only on §1.7.3 where you are told the for-mula for the speed of sound eq.(1.15). Please note that the speedof sound for a perfect gas is solely a function of the (absolute)temperature of the gas.§1.8-1.9 Just scan these sections. This course does not emphasizethese topics.• Problem 1.5,• Problem 1.7,2• Problem 1.15 (an estimate is a good estimate when you are “inthe ball park.”),• Problem 1.29, (you know the definition of viscosity eq.(1.8), andyou can compute du/dy and evaluate it at y = 0).• Problem 1.42, (if you inhale helium (don’t do it) then try to talk,you will sound funny).Chapter 2 : Fluid Statics. This is a course in fluid dynamics, but we willget started by first consider the simple case when the fluid is not movingat all. Fluid statics is also called hydrostatics.§2.1 This is to convince you that pressure is a scalar. It is representedby a number, and not a vector. This is called Pascal’s Law.§2.2-2.3 Eq.(2.2) gives us Newton’s law for fluid dynamics. What isa? It is the “acceleration” of a glob of fluid. When the glob of fluidis at rest, we have a = 0, and eq.(2.4) is the governing equation forpressure in hydrostatics (γ ≡ ρg). If density is constant, we haveeq.(2.8) where h is vertical distance from a reference “altitude,”its positive direction is downward.§2.4 Appendix C has the standard atmosphere. Note that the pressureat about 20,000 ft is about half an atmosphere. So, if you aredealing with problems which vertical extent is a few tens of feet,we can say ”the air pressure is a constant” with good justificationfor this approximation.§2.5-2.6 When the specific weight γ is a constant as is the case of amanometer, Eq.(2.8) is all you need.§2.7 Just scan this section.§2.8-2.9 If you have a plane surface under water, how much (normal)force is being exerted by the water on the surface? This is just anexcercise in multi-variable integration. §2.9 just shows you whatyou are integrating to obtain is just the volume of the “pressureprism.”§2.10 What happens if the surface in contact with water is curved?The force (vector) experienced by the curved surface must be com-puted one component at a time, that is all! You just need to be3careful with your trigonometry and geometry. (The unit normaln of a surface element, and the “dot” product of two vectors,are useful tools which you must master) In general, the calcula-tion is usually messy but straightforward (examples in homeworkuse only simple geometries to reduce the messiness). Remember,whenever you see a frictionless hinge, the moment about the hingeis zero.§2.11 Archemedes Principle comes to the rescue! We will see in classthat Archemedes Principle (and an elegant trick to be given inclass) can greatly simplified the calculation of the total verticalforce vector on some curved surfaces (It does not provide helpin problems involving frictionless hinges). Watch for this is thelecture. Scan the interesting stability section.§2.12 If the fluid is in rigid-body rotation, then a is known. The staticpressure distribution is then easily calculated.• 2.13 (you may assume that the static pressure distribution in thespace filled with air is constant. However, the static pressuredistribution in the water is not a constant),• Problem 2.30 (the force exerted by the water is resisted by P andby the hinge. Since the hinge is frictionless, the moment aboutthe hinge must be zero. This determines P),• 2.44 No hinge here. (Archemedes is at your service! So is myelegant trick),• Problem