ESC2C5 problem set 1 answers

1


ES2B0 - Fluid Mechanics
Model Answers to Example Questions (Set I)

Question 1: Wind Tunnel

A simple wind‐tunnel is depicted schematically in Figure 1. The flow speed in the working
section is assumed to be constant with a value of VB = 60m/s at point B. The cross‐sectional
area of the working section (point B is in working section) is 1 m2.




A
B




Figure 1: Sketch of simple wind‐tunnel


i) Neglecting irreversible losses, such as those due to viscous effects, calculate the gauge
pressure in the working section. (Assume a value of  1.2 kg/m3 for the density of air).
ii) Calculate the mass flow rate across the cross‐section in the working section.

SOLUTION


(i) Consider a typical streamline AB. Point B is in the working section and point A is located
in surroundings where the pressure equals atmospheric pressure p A and the flow speed is zero.
According to the Bernoulli equation one then gets

1 1
pA   V A2  p B   V B2
2 2

with V A  0 and VB  60 ms-1

The gauge pressure is
2
 1.2 3   60    2160 Pa
1 kg m
pB  pA  
2 m  s

, 2

This means that the pressure in the working section is 2160 Pa below the atmospheric
pressure. (N.B.: The modulo of this pressure difference corresponds approximately to the
pressure resulting from a 21 cm high water column.)


(ii) The mass flow rate Q across the cross-section in the working section is given by:
kg m kg
Q    V B  cross  sec tion area  1.2 3  60  1 m 2  72
m s s



Question 2: Plunger

A plunger is moving through a cylinder as schematically illustrated in the Figure 2. The velocity
of the plunger is Vp = 10 ms‐1. The oil film separating the plunger from the cylinder has a
dynamic viscosity of μ=0.3 N.s.m‐2. Assume that the oil‐film thickness is uniform over the
entire peripheral surface of the plunger. Calculate the force and the power required to
maintain this motion.




Figure 2: Plunger moving through cylinder

SOLUTION

The oil film is sufficiently thin such that we can assume a linear velocity profile for the flow of
oil in the film. One can calculate the frictional resistance by computing the shear stress at the
plunger surface by means of Newton’s law of viscosity.
m
V Ns 10 N
   0.3 2 s  12000 2
r m 1
50  10  3 m  49.5  10  3 m 
2
m

The frictional force can now be calculated by multiplying the shear stress with the surface of
the plunger.