Mech. Eng. Ph.D. Preliminary Qualifying Examination
January 22, 2009
Fluid Mechanics
This is one of five problems. You are required to do four out of five problems. Clearly indicate
which four problems you are selecting. Show all work on the exam sheets provided and write
your student number on each sheet. Do not write your name on any sheet.
Student Number ______________________________________________________
1.
Consider the low-speed flow of air between parallel disks as shown. Assume that the
flow is incompressible and inviscid, and that the velocity is purely radial and uniform at any
section. The flow speed is V = 15 m/s at R = 75 mm. Simplify the continuity equation to a
form applicable to this flow field. Show that a general expression for the velocity field is
V  V ( R / r )eˆr for ri  r  R . Calculate the acceleration of a fluid particle at the
locations r  ri and r  R .
Fig. 1
Mech. Eng. Ph.D. Preliminary Qualifying Examination
January 22, 2009
Fluid Mechanics
This is one of five problems. You are required to do four out of five problems. Clearly indicate
which four problems you are selecting. Show all work on the exam sheets provided and write
your student number on each sheet. Do not write your name on any sheet.
Student Number ______________________________________________________
2.
Consider steady laminar flow (  ,  ) in a parallel-plate device where one wall is lined
with endothelial cells to demonstrate elongated cell alignment with substantial shear flow.
Given  water  103 kg / m3 ,  water  103 Ns / m 2 and h  250  m as well as measured
 water  10,50,100 dyn / cm2 , calculate the operational conditions, i.e., associated Reynolds
numbers and entrance length (for cell placement in the fully-developed flow region).
(Assume Poiseuille-type flow.)
Fig. 2
Mech. Eng. Ph.D. Preliminary Qualifying Examination
January 22, 2009
Fluid Mechanics
This is one of five problems. You are required to do four out of five problems. Clearly indicate
which four problems you are selecting. Show all work on the exam sheets provided and write
your student number on each sheet. Do not write your name on any sheet.
Student Number ______________________________________________________
3.
The pressure drop in a Venturi meter varies only with the fluid density, pipe approach
velocity, and diameter ratio of the meter. A model venture meter tested in water at 20 oC
shows a 5 kPa drop when the approach velocity is 4 m/s. A geometrically similar prototype
meter is used to measure gasoline (density 680 kg/m3) at a flow rate of 9 m3/min. If the
prototype pressure gauge is most accurate at 15kPa, what should the upstream pipe
diameter be?
Solution:
Given: ΔP=fn(ρ, U, d/D)
Using Buckingham Pi theorem to obtain the useful relationship of
non-dimensional pressure drop: ΔP/(ρU2)= fn(d/D)
For Water model:
5000/(998. 42 ) = 0.313
For Gasoline prototype:
15000/(680 . U2 ) = 0.313
Solve for U
 U = 8.39 m/s
By definition, Flow rate : Q =U A= U Π(D2 )/4, and is given as 9/60 m3/s;
Solve for D , obtain
 D=0.151 m
Mech. Eng. Ph.D. Preliminary Qualifying Examination
January 22, 2009
Fluid Mechanics
This is one of five problems. You are required to do four out of five problems. Clearly indicate
which four problems you are selecting. Show all work on the exam sheets provided and write
your student number on each sheet. Do not write your name on any sheet.
Student Number ______________________________________________________
4.
The drag force of a bullet-shaped device may be measured using a wind tunnel. The
tunnel is round with a diameter of 1 m, the pressure at section 1 is 1.5 kPa gage, the
pressure at section 2 is 1.0 kPa gage, and air density is 1.0 kg/m^3. At the inlet, the velocity
is uniform with a magnitude of 30 m/s. At the exit, the velocity varies linearly as shown in
the sketch. Determine the drag force on the device and support vanes. Neglect viscous
resistance at the wall, and assume pressure is uniform across sections 1 and 2.
Fig. 4
Mech. Eng. Ph.D. Preliminary Qualifying Examination
January 22, 2009
Fluid Mechanics
This is one of five problems. You are required to do four out of five problems. Clearly indicate
which four problems you are selecting. Show all work on the exam sheets provided and write
your student number on each sheet. Do not write your name on any sheet.
Student Number ______________________________________________________
5. A hydroelectric power plant in Fig. 5 takes in 30m3/s of water through its turbines and
discharges it at V2 = 2 m/s at atmospheric pressure. The head loss in the turbine and
penstock system is hf= 20 m.
a. Calculate the exit area
b. Estimate the power extracted by the turbine in megawatts.
Fig. 5