Page 1 Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Circle your lecture section (-1 point if not circled, or circled incorrectly): Prof. Vlachos Prof. Ardekani Mr Berdanier 10:30 11:20 A.M. 1:30 2:20 P.M. 3:30 4:20 P.M. Please note the following: 1. The exam is closed notes and closed book. You may use only the formula sheet provided with the exam, a pen/pencil/eraser, and a calculator fitting the policy stated in the course syllabus. 2. Show all of your work in order to receive credit. An answer without supporting work will not receive a full score. Also, write neatly and organized and clearly box your answers. 3. Clearly state your assumptions, draw control volumes and coordinates systems, and include other significant information in order to receive full credit. 4. Only turn in those pages you wish to have graded. Do not turn in your formula sheets. 5. The honor code is in effect. 6. Write only on one side of the paper. Work on the backside of a page will not be graded. 7. Only the first solution approach encountered when grading will be scored. (-2 points if the following instruction is not followed) Write your name on all pages that are to be considered for grading. If you do not write your name, that page will NOT be graded SCORE: TOTAL (100 out of 100 points available ):
Page 2 Part%1%% SCORE: 1. (4 pts) Fill in the blanks (complete answer, no partial credit) a. When the flow is inviscid it means that: viscous effects are fluid viscosity is b. The streamlines are everywhere to the local vector and coincide with pathlines and streaklines when the flow is c. Bernoulli s equation corresponds to the conservation of. In order to use Bernoulli s equation the following assumptions must be satisfied (all must be listed): i. ii. iii. iv. d. For steady incompressible flow the continuity equation takes the form (write the equation):
Page 3 2. (1 pts) For the vessels shown below, which one experiences higher pressure at the bottom: a. b. c. d. e. f. they are all the same 3. (2 pts) In the figure below fill in the correct type of fluid based on the behavior shear stress as a function of rate of shearing strain
Page 4 4. (1 pts) Reservoirs A and B contain pressurized air. Using the water manometer shown below how much is the gauge pressure (measured in inches of water height) in the reservoir A (circle the correct answer): a. 2 of water Atmosphere b. 6 of water A B c. -2 of water d. -6 of water 4"in"" 2 e. 0 of water 2"in" 1 f. 8 of water 0"in" 0 '2"" - 1 5. (8 pts total) A cylindrical log of diameter D rests against the top of a dam as shown in the figure. The horizontal force is F H and the vertical force is F V a. (3 pts) Write the expression for the horizontal force per unit length F H /L b. (5 pts) Write the expression for the vertical force per unit length F V /L
Page 5 6. (5 pts) A thin layer of viscous fluid flows upward with velocity U along an inclined plate as shown in the picture. As it moves upwards the fluid height increases. Assume twodimensional (x-y plane only) laminar, steady and incompressible flow and the coordinate system as shown in the figure. a. Cancel all the terms of the Navier-Stokes equations as appropriate: 7. (1 pts) In the fan-engine shown below air in sucked-in at the inlet and accelerated through the fan to exit the engine through an annular jet outlet. For analyzing this engine, which of the following equations can be applied along a streamline from the inlet to the outlet (circle all that apply-no partial credit). a. Conservation of mass b. Conservation of momentum c. Bernoulli d. Conservation of energy
Page 6 8. (3 pts) In the velocity field shown below, determine the location (center) of any vortices and draw two representative streamlines for each vortex you found. You can draw on the vector field or on the space below.
Page 7 9. (5 pts total) A red blood cell flows along an artery that is partially blocked as shown below. Assume that the red blood cell is elliptically shaped and axisymmetric and that the flow is steady and incompressible and viscous effects are negligible. a. (2 pts) On the figure, sketch the flow streamlines in the artery b. (2 pts) As the red blood cell moves across the blocked region where will it migrate towards (select/circle the correct answer): i. Upwards ii. Downwards iii. Will continue straight iv. Will reverse and flow backwards (if cannot pass the blockage) c. (1 pts) Explain your answer:!
Page 8 Part%2%% SCORE: A water tank with cross-sectional area, A tank, stands on a frictionless cart. A jet with area, A jet, at height, h, below water surface leaves the tank and is deflected by a vane at an angle, θ, from the horizontal direction. Assume that A tank >> A jet. Also assume that the velocity and area of the jet remain constant after leaving the tank. The water density is ρ and gravitation acceleration is g. a) Evaluate the jet velocity V. b) Compute the tension, T, in the supporting cable, in terms of the given parameters. c) We now replace the cable with a mechanism that moves the cart with a constant velocity U. Compute the force we need to apply to the cart to move it with a constant velocity U towards left. d) In this part, we remove the cable and mechanism so the cart freely accelerates. Compute the acceleration of the cart when the mass of the cart and water inside it is M.
Problem 3 (a) The Navier-Stokes momentum equation in the x-direction for rectangular coordinates is shown here (for an incompressible fluid): 2 2 2 u u u u P u u u ρ + u + v + w = ρ g x + µ + + 2 2 2 t x y z ( 3) x x y z ( 1) ( ) ( 4 2 ) ( 5) Describe in words what each bracketed set of terms represents (Hint: Recall how these equations were derived!). (1) (2) (3) (4) (5)
(b) A viscous Newtonian fluid exists between two infinite, parallel plates separated by a distance h. The bottom plate moves to the right with a constant speed, and the top plate moves to the left with a constant speed such that >. There is no pressure gradient =0. You are asked to solve for the velocity distribution between the plates, = + +. Note: > h (i) If you assume the flow is fully-developed, show the y-component of velocity must be zero =0. Carefully and clearly define any required assumptions and/or boundary conditions.
(ii) Assuming the flow between the plates is fully-developed, begin with the Navier-Stokes equations to determine the velocity field between the plates,. V = i + j + k
(iii) Sketch the velocity profile between the plates. (iv) Using the velocity profile you calculated above, determine the volumetric flow rate between the plates per unit width: =
(v) Calculate the friction force exerted by the fluid on the bottom plate. (vi) Is the net fluid flow to the left or to the right? (vii) How would the velocity profile change if this problem could be approximated as inviscid?