FALL 2020

MIDDLE EAST TECHNICAL UNIVERSITY

DEPARTMENT OF MECHANICAL ENGINEERING
ME 305 FLUID MECHANICS I
HOMEWORK 1
1. The space between two very long parallel plates separated by a distance of h is filled
with a fluid. The fluid is heated in such a way that its viscosity decreases linearly
from 0 at the lower surface to 0/2 at the upper surface. The upper plate moves
steadily at a velocity U0 relative to the lower one and the pressure everywhere is
constant. Determine the velocity profile and the shear stress distribution. (Ans.
U0  y  U
− ln 1 −  , 0 0 )
ln 2  2h  2h ln 2

U0

h
y

Problem 1

2. Two flat plates are oriented parallel above a fixed lower plate as shown in the figure.
The top plate, located a distance b above the fixed plate, is pulled along with speed
V. The other thin plate is located a distance cb, where 0 < c < 1, above the fixed plate.
This plate moves with speed V1, which is determined by the viscous shear forces
imposed on it by the fluids on its top and bottom. The fluid on the top is twice as
viscous as that on the bottom. Plot the ratio V1/V as a function of c for 0 < c < 1.
 V

2
b
V1

cb 
b

Problem 2

3. Two Newtonian fluids, which do not mix, flow between two stationary flat plates
under the influence of a pressure gradient. The lower half is filled with liquid 1, while
the upper half is filled with liquid 2. The gap between the plates is 2 m. The velocity
profiles in the lower and upper halves are given as
u1 = 4 + Ay − 2.5 y 2 −1  y  0

u2 = B + Cy − 10 y 2 0  y 1
where A, B and C are constants.
a) What are the two interface conditions? Determine the values of the
constants A, B and C.
b) Find the viscosity, 2, of liquid 2, if the viscosity of liquid 1 is
1 = 4x10-3 Pa s
c) Determine the location where the shear stress is zero.
d) Determine the magnitude and direction of shear stresses applied by the
liquids on the lower and upper plates.
(Ans. a) 1.5, 4, 6, b) 1x10-3 Pa s, c) 0.3 m, d) 26x10-3 Pa, 14x10-3 Pa)

y
u2 Liquid 2 1m
2 and 2
Interface x

u1 Liquid 1
1m
1 and 1

Problem 3

2
4. A piston having a diameter of 13.92 cm and a length of 24.13 cm slides downward
with a velocity V through a vertical pipe. The downward motion is restricted by an
oil film between the piston and the pipe wall. The film thickness is 0.005 cm and the
cylinder weights 0.23 kg. Estimate V if the oil viscosity is 0.766 N.s/m2. Assume the
velocity distribution in the gap is linear. (Ans. 6.98x10-4 m/s)

5. A new computer drive is proposed to have a disc, as shown in the figure. The disc is
to rotate at 10,000 rpm, and the reader head is to be positioned 0.012 mm above the
surface of the disc. Estimate the shearing force on the reader head as result of the air
between the disc and head. The viscosity of air is 1.8x10-5 Pa.s. (Ans. 1.542x10-3 N)

Stationary reader head 5 mm dia.

A A

0.012 mm

Rotating disk

Section AA
50 mm
Problem 5

6. A 30 cm diameter circular plate is placed over a fixed bottom plate with a 0.25 cm
gap between the two plates filled with glycerin having a viscosity of 0.407 Pa.s, as
shown in the figure. Determine the torque required to rotate the circular plate slowly
at 2 rpm. Assume that the velocity distribution in the gap is linear and that the shear
stress on the edge of the rotating plate is negligible. (Ans. 0.02711 N.m)

Rotating Torque
plate

0.25 cm
Problem 6 gap

3
7. A hydraulic coupling, which is shown in the figure, is used to transmit a torque of T
for constant angular speeds 1 and 2. Derive an expression for the torque in terms
of the oil viscosity, , the disc diameter, D, the distance between the discs, h, and the
slip, (1 - 2). Neglect the effect of gravity and assume that the pressure is constant.
 (2 − 1 ) D 4
Oil is of Newtonian type. (Ans. )
32h

Driver Driven

D
2 1
Oil of
viscosity 
h

Problem 7

8. A soap bubble 0.05 m in diameter contains a pressure of 20.07 Pa in excess of the

atmosphere. Determine the surface tension of the soap film. (Ans. 0.2509 N/m)

9. Surface tension forces can be strong enough to allow a double-edge steel razor blade
to “float” on water, but a single-edge blade will sink. Assume that the surface tension
forces act at an angle of  relative to water surface and the temperature of water is
200C, as shown in the figure.
a) The mass of the double-edge blade is 0.64x10-3 kg and the total length of
its sides is 206 mm. Determine the value of  required to maintain equilibrium
between the blade weight and the resultant tension force.
b) The mass of the single-edge blade is 2.61x10-3 kg and the total length of
its sides is 154 mm. Explain why this blade sinks. Support your answer with the
necessary calculations.
(Ans. a) 24.750)

4
 Surface tension
Blade force

Problem 9

10. The vertical cylinder of diameter 0.1 m is pulled up from an inviscid liquid. Assume
that the angle of contact is 300 and the surface tension for the liquid in presence of
air is 0.073 N/m. Find the force, F, required to pull the cylinder at a constant velocity
at the instant that it leaves the liquid, if the mass of the cylinder is 0.1 kg. Neglect
buoyant and viscous effects. (Ans. 1.001 N)

Air

0.1 m
Liquid
300

Problem 10

QUIZ 1 will be a conceptual quiz and will be given at week 3

QUIZ 2 based on problems of this homework will be given at week 4