ME 309 – Fluid Mechanics LAST NAME (print): ___________________________________ Spring 2015 FIRST NAME (print): __________________________________ 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 1

ME 309 – Fluid Mechanics LAST NAME (print): ___________________________________ Spring 2015 FIRST NAME (print): __________________________________

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):

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ME 309 – Fluid Mechanics LAST NAME (print): ___________________________________ Spring 2015 FIRST NAME (print): __________________________________

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

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ME 309 – Fluid Mechanics LAST NAME (print): ___________________________________ Spring 2015 FIRST NAME (print): __________________________________

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 b. 6” of water c. -2” of water d. -6” of water e. 0” of water f. 8” of water

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 FH and the vertical force is FV

a. (3 pts) Write the expression

for the horizontal force per unit length FH/L

b. (5 pts) Write the expression for the vertical force per unit length FV/L

1. Two tanks filled with air and connected by water-filled manometers are shown below. The water levels are as shown. Circle the letter of the correct answer for the gage pressure (the pressure relative to atmospheric) for Tank A.

A. PA equals + 3" water B. PA equals + 1" water C. PA equals + 0 " water D. PA equals – 1 " water E. PA equals – 3 " water

A

2”1”0”- 1”

BAtmosphere

4"in""2"in"0"in"'2""

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ME 309 – Fluid Mechanics LAST NAME (print): ___________________________________ Spring 2015 FIRST NAME (print): __________________________________

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 two-dimensional (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

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ME 309 – Fluid Mechanics LAST NAME (print): ___________________________________ Spring 2015 FIRST NAME (print): __________________________________

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.

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ME 309 – Fluid Mechanics LAST NAME (print): ___________________________________ Spring 2015 FIRST NAME (print): __________________________________

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:

________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

!

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ME 309 – Fluid Mechanics LAST NAME (print): ___________________________________ Spring 2015 FIRST NAME (print): __________________________________ Part%2%% SCORE:

A water tank with cross-sectional area, Atank, stands on a frictionless cart. A jet with area, Ajet, 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 Atank >> Ajet. 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.

Page 8

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

2 2 2

31 4

2 5

x

u u u u P u u uu v w g

t x y z x x y zρ ρ µ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + = − + + +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

��������� ���������

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, � � = ��̂ + ��̂ + ���.

(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.

��

��

ℎ

Note: ��� > ���

�

�

(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?