ME 305 Fluid Mechanics I
Fall 2023 - Study Set 1
These problems are for study purposes, only; you will not submit solutions for grading. The answers to the
problems are provided however the solutions will not be posted. Feel free to ask your instructor and/or
your TA’s for help with these problems.
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:
U
y  U

− 0 ln 1 −  , 0 0 )
ln 2  2h  2h ln 2
U0
h
y
x
2) A layer of water flows down an inclined fixed surface with the velocity profile as shown in the figure.
Determine the magnitude and direction of the shearing stress that the water exerts on the fixed surface
for U=2 m/s and h=0.1 m. Take dynamic viscosity of water as 1.12 x 10-3 N.s/m2 . (Ans: 4.48x10-2 Pa,
in the flow direction)
3) 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. Find the ratio V1/V as a function of c for 0 < c < 1.
𝑽
𝟐𝒄
(Ans: 𝑽𝟏 = 𝒄+𝟏)
V
2
b
V1

cb
1
4) 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 − 10y 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 (+x), 14x10-3 Pa (+x))
y
Interface
x
u2 Liquid 2
2 and 2
1m
Liquid 1
1 and 1
1m
u1
5) 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. Please note that oil film thickness
is defined as the difference between the pipe diameter and the piston diameter. (Ans: 6.98x10-4 m/s)
6) 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. (Ans: (a) 24.750)
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.
Surface tension
force
Blade

2