MAE 222Mechanics of FluidsPrinceton UniversityAssignment # 3February 19, 1997Due on Wednesday, 2PM, February 26, 1997Chapter 4 : Fluid Kinematics. This is a chapter of tools.§4.1, The velocity field . Velocity V is a vector. When we hearsomeone say: “I know the velocity field of this fluid flow field,”we get the idea that if we point to any tiny glob of fluid in theflow field and ask him/her ”What is the velocity of this glob?”he/she will draw an arrow on the blackboard, and explain to youthe amplitude and direction of this arrow. If you then say: “Iwant numbers!” and he/she will inquire what coordinates youhave in mind, and then give you numbers appropriate only tothat coordinate system.On page 111, the formula for Vgiven is for people who preferCartesian coordinate:bi,bj andbk are unit vectors in the x, y, z di-rections, respectively. What is a unit vector? It is a vector withunit amplitude, and its direction has been explained to you. Howdo you explain direction? Usually by a picture; sometimes by anangle in reference to some other known direction, sometimes byusing your index finger.As we have already explained in class, there are the Eulerian andthe Lagrangian descriptions of a flow field. The description onpage 111 is Eulerian. When you think about it, Eulerian in theonly sensible thing to do. If that is the case, why do we evenmention the Lagrangian description? Because! Because Law ofPhysics are normally stated in the Lagrangian sense. When New-ton says: F = ma for his apple, he is applying his second law tothe apple as it falls to the ground. When he says the z-velocitycomponent of the apple is w=-gt, we all know that it refers to1the apple always, and not to some imaginary point fixed in spacewhere the apple happens to pass through.Know what a streamline is: it is a line whose tangent at everypoint is parallel to the fluid velocity vector there. The streaklineand the pathline are also defined in this section. The main thingto learn is that they are, in unsteady flow, not the same as thestreamline. If you mark the fluid passing a fixed point in space(by a dye or a source of smoke) and then take a snapshot, whatyou have is a streakline. If you mark one tiny glob of fluid, andlet it trace its trajectory for you over time, what you have is apathline. If the flow is steady, these three kind of lines are thesame. Don’t spend too much time on Example 4.2 and 4.3. Theyare merely exercises of solving (micky mouse) ordinary differentialequations—once you have understood the concepts.§4.2, The Acceleration Field. Here comes the substantial deriva-tive (or material derivative, or Lagrangian derivative). We haveseen eq.(4.4) before. But this time, you are introduced to the ele-gant representation of eq.(4.5) using vectors: eq.(4.6). You mustagree that eq.(4.6) is prettier than eq.(4.5). We are introduced tothe concept of a gradient. Given some scalar field (.), the gradientof that scalar is denoted by ∇(.). For example, given the scalarpressure field p(x, t), the gradient of pressure is ∇p(x, t). If weuse Cartesian coordinates, then we have∇p ≡bi∂p(x, y, z, t)∂x+bj∂p(x, y, z, t)∂y+bk∂p(x, y, z, t)∂z. (1)What happens if we would like to use polar coordinates? We willneed to find out how to compute ∇p in polar coordinates (look itup in your old math books). What happens if we would like touse some crazy coordinates such as the streamline coordinates in§4.2.4? In good time, we will get the answer to the last questionin this course.§4.3, Control Volume and System Representations. What is asystem? The apple is the system Newton studied as it fell on hishead. In general, a system is a collection of matter of fixed identity.In a flow field, a system is a glob of fluid I have arbitrarily identified2and have decided to put under surveillance. It moves. What is acontrol volume? A control volume is an arbitrary volume fixed inspace that I have decided to put under surveillance. It does notmove. (In more sophisticated quarters, some people do talk aboutmoving control volumes. We will stay out of that).§4.4, Reynolds Transport Theorem This is important stuff! Wehave done a lot of beating around the bush, just to get readyto deal with this important concept!!! In §4.4, YMO actually de-rive the Reynolds Transport Theorem twice: a streamtube version(Fig. 4.8), and an arbitrary glob version (Fig.4.9). Personally, Ithink the arbitrary glob version is much, much easier to read andunderstand. The punch line of the streamtube version is eq.(4.12);the punch line of the arbitrary glob version is eq.(4.17). There isno question about it: eq.(4.17) is much prettier! What isbn? Itis our friend, the unit outward normal to the element of surfacedA. You must understand the physical meaning of each term ineq.(4.17). In fact, you are expected to know this formula by heart.Problems at end of Chapter 4:• Problem 4.2. totally straightforward.• Problem 4.8. to make sure you know how to work out the sub-stantial derivative.• Problem 4.13. Another substantial derivative problem.• Problem 4.17. More substantial derivative problem.• Problem 4.23 Basically, you are asked to find the mass flow ratethrough this square duct in a steady flow of constant density fluid(water). You are asked to compute it first using the surface A−B,next using the surface C −D, and last using the wriggly surfaceE − F . Clearly you can do the first two cases, long hand. (use acoordinate on the surface, pick any infinitesimal surface elementdA, find the normal component of the fluid velocity, and inte-grate!) You should not be surprised that these two answers agree.Now, how do you do the last case? YMO did not even give youthe equation for the wriggly line. Control Volume comes to therescue!!! Watch for this in the