Anna University, Chennai
CE 6303 MECHANICS OF FLUIDS
UNIT I FLUID PROPERTIES AND FLUID STATICS PART – B
1. If the velocity profile of a liquid over a plate is a parabolic with the vertex 202 cm from the plate, where the velocity is 120 cm/sec. calculate the velocity gradients and shear stress at a distance of 0, 10 and 20 cm from the plate, if the viscosity of the fluid is 8.5 poise.
2. A 15 cm diameter vertical cylinder rotates concentrically inside another cylinder of diameter 15.10 cm. both cylinders are 25 cm high. The space between the cylinders is filled with a liquid whose viscosity is unknown. If a torque of 12.0 Nm is required to rotate the inner cylinder at 100 rpm determine the viscosity of the fluid.
3. The dynamic viscosity of oil, used for lubrication between a shaft and sleeve is 6 poise.
The shaft is of diameter 0.4 m and rotates at 190 rpm. Calculate the power lost in the bearing for a sleeve length of 90mm. the thickness of the oil film is 1.5 mm.
4. If the velocity distribution over a plate is given by u=2/3 y – y
in which U is the velocity
in m/s at a distance y meter above the plate, determine the shear stress at y = 0 and y =
5. Derive Pascal’s law.
6. Derive expression for capillary rise and fall.
7. Two large plane surfaces are 2.4 cm apart. The space between the gap is filled with glycerin. What force is required to drag a thin plate of size 0.5 m between two large plane surfaces at a speed of 0.6 m/sec. if the thin plate is (i) in the middle gap (ii) thin plate is
0.8 cm from one of the plane surfaces? Take dynamic viscosity of fluid is 8.1 poise.
8. Calculate the capillary rise in a glass tube of 2.5 mm diameter when immersed vertically in (a) water (b) mercury. Take surface tension = 0.0725 N/m for water and = 0.52 N/m for mercury in contact with air. The specific gravity for mercury is given as 13.6 and
angle of contact of mercury with glass = 130o.
9. The diameters of a small piston and a large piston of a large piston of a hydraulic jack at
3 cm and 10 cm respectively. A force of 80 N is applied on the small piston. Find the load lifted by the large piston when:
a. The pistons are at the same level
b. Small piston in 40 cm above the large piston.
The density of the liquid in the jack is given as 1000 kg/m3.
10. A U - Tube manometer is used to measure the pressure of water in a pipe line, which is in excess of atmospheric pressure. The right limb of the manometer contains water and mercury is in the left limb. Determine the pressure of water in the main line, if the difference in level of mercury in the limbs. U tube is 10 cm and the free surface of mercury is in level with over the centre of the pipe. If the pressure of water in pipe line is
reduced to 9810 N/m2, Calculate the new difference in the level of mercury. Sketch the
arrangement in both cases.
11. A vertical sluice gate is used to cover an opening in a dam. The opening is 2 m wide and 1.2 m high. On the upstream of the gate, the liquid of sp. Gr 1.45, lies up to a height of 1.5 m above the top of the gate, whereas on the downstream side the water is available up to a height touching the top of the gate. Find the resultant force acting on the gate and position of centre of pressure. Find also the force acting horizontally at the top of the gate which is capable of opening it. Assume the gate is hinged at the bottom.
UNIT II FLUID KINEMATICS AND DYNAMICS .
1.Water flows through a pipe AB 1.2 m diameter at 3 m/s and then passes through a pipe BC 1.5 m diameter. At C, the pipe branches. Branch CD is 0.8 m in diameter and carries one – third of the flow in AB. The flow velocity in branch CE is 2.5 m/s. find the volume
rate of flow in AB, the velocity in CD, the velocity in BC and the diameter of CE.
2.A fluid flow field is given by V = x2yi+ 2zj – (2xyz+yz
2)k prove that it is a case of
possible steady incompressible flow. Calculate the velocity and acceleration at the point
(2, 1, 3).
3.Derive the continuity equation for a three dimensional incompressible flow.
Derive the Euler’s equation of motion and deduce that to Bernouillie’s equation.
The water is flowing through a taper pipe of length 100 m having diameters 600 mm at the upper and 300 mm at the lower end, at the rate of 50 litres /s. the pipe has a slope of 1 in 30. Find the pressure at the lower end if the pressure at the higher level is 19.62 N/cm2.
4.An oil of sp .Gr. 0.8 is flowing through a venturimeter having inlet diameter 20 cm and throat diameter 10 cm. the oil mercury differenyial manometer shows a reading of 25 cm. Calculate the discharge of oil through the horizontal venturimeter, Take CD = 0.98.
250 litres/s of water is flowing in a pipe having a diameter of 300 mm. of the pipe is bent by 135o, find the magnitude and direction of the resultant force on the bend. The pressure of water flowing is 39.24 N/cm2.
5.A vertical wall is of 8 m height. A jet of water is coming out from a nozzle with a velocity of 20 m/s. The nozzle is situated at a distance of 20 m from the vertical wall. Find the angle of projection of the nozzle to the horizontal so that the jet of water just
clears the top of the wall.
UNIT III FLOW.THROUGH PIPES
1. Find the head lost due to friction in a pipe of diameter 300 mm and length 50 m, through which water is
flowing at a velocity of 3 m/s using (i) Darcy formula, (ii) Chezy’s formula for .which C = 60.
2. An oil of sp.Gr 0.9 and viscosity 0.06 poise is flowing through a pipe of diameter 200 mm at
the rate of 60 litres/sec./ find the head lost due to friction for a 500 m length of pipe. Find the power required to maintain this flow.
3. The rate of flow of water through a
horizontal pipe is 0.25 m3/s. The
diameter of the pipe which is 200 mm is suddenly enlarged to 400 mm. The
pressure intensity in the smaller is
11.772 N/cm2. Determine: (i) loss of
head due to sudden enlargement, (ii) pressure intensity in the large pipe, (iii) power lost due to enlargement.
4. A horizontal pipe line 40 m long is connected to a water tank at one end discharges freely into the atmosphere at the other end. For the first 25 m of its length from the tank, the
pipe is 150 mm diameter and its diameter is suddenly enlarged to 300 mm. The height of water level in the tank is 8 m above the centre of the pipe. Considering all losses of head which occur, determine the rate of flow. Take f = 0.01 for both sections of the pipe.
UNIT IV BOUNDARY LAYER
1. Briefly explain the boundary layer definitions.
2. Find the displacement thickness, the momentum thickness and energy thickness for the velocity distribution in the boundary layer given by u/U =y/δ, where u is the velocity at a distance y from the plate and u = U at y = δ, where δ = boundary layer thickness. Also calculate the value of δ*/θ.
3. Find the displacement thickness, the momentum thickness and energy thickness for the velocity
distribution in the boundary layer given by u/U = 2 (y/ δ) – (y/ δ) .
4. For the velocity profile u/U = 2 (y/ δ) – (y/ δ)2, find tghe thickness of boundary layer at the end of the plate and the drag force on one side of a plate 1 m long and 0.8 m .
when placed in water flowing with a velocity of 150 mm/sec. Calculate the value of co - efficient of drag also. Take μ for water = 0.01 poise.
5. For the velocity profile for laminar boundary layer u/U = 2 (y/ δ) – (y/ δ 3
+(y/ δ 4
an expression for boundary layer thickness, shear stress, drag force on one side of the plate and co – efficient of drag in term of Reynold number.
6. For the velocity profile for laminar boundary flow u/U = sin (π y/2 δ ). Obtain an expression for boundary layer thickness, shear stress, drag force on one side of the plate
and co – efficient of drag in terms of Reynold number. .
7. For the velocity profile for laminar boundary layer u/U = 3/2 (y/ δ) –1/2 (y/ δ)
thickness of the boundary layer and the shear stress 1.5 m from the leadingcomedge of a plate. The plate is 2 m long and 1.4 m wide and is placed in water which is moving with a
velocity of 200 mm per second. Find the total drag force on the plate if μ for water = 0.01 poise.
8. For the velocity profile for turbulent boundary layer u/U = (y/ δ 1/7
, obtain an expression
for boundary layer thickness, shear stress, drag force on one side of the plate and co –
efficient of drag in terms of Reynolds Number. Given the stress (ζo) for turbulent boundary layer as
ζo = 0.0225 ρ 2
(μ/ ρ U g 1/4.
9. Determine the thickness of the boundary layer at the trailing edge of smooth plate of length 4 m and of the width 1.5 m, when the plate is moving with a velocity of 4 m/s in
stationary air. Take kinematic viscosity of air as 1.5 x 10-5 m2/s.
10. For the following velocity profiles, determine whether the flow has or on the verge of separation or will attach with the surface:
(i) u/U = 3/2 (y/ δ) –1/2 (y/ δ 3
(ii) u/U = 2 (y/ δ 2
– (y/ δ 3
(iii) u/U = - 2 (y/ δ) + (y/ δ)
UNIT V DIMENSIONAL ANALYSIS AND MODEL STUDIES
PART - B
1 .Explain Buckingham’s theorem.
2. The resisting force (R) of a supersonic flight can be considered as dependent upon length of aircraft (l), velocity (V), air viscosity ‘μ’, air density ‘ρ’, and bulk modulus of air ‘ k’. Express the functional relationship between these variables and the resisting force.
3. A ship is 300 m long moves in sea water, whose density is 1030 kg/m3. A 1:100 model of this to be tested in a wind tunnel. The velocity of air in the wind tunnel around the model
is 30 m/s and the resistance of the model is 60 N. Determine the velocity of ship in sea water and also the resistance of the ship in sea water. The density of air is given as 1.24 kg/m3. Take the Kinematic viscosity of sea water and air as 0.012 stokes and 0.018 stokes
4. A 7.2 m height and 15 m long spillway discharge 94 m3/s, under a head of 2.0m. If a 1:9 scale model of this spillway is to be constructed, determine model dimensions, head over
spillway model and the model discharge. If model experience a force of 7500 N (764.53
Kgf), determine force on the prototype.
5. A quarter scale turbine model is tested under ahead of 12 m. The full scale turbine is to
work under a head of 30 m and to run at 428 rpm. Find N for model. If model develops
100 kW and uses 1100 l/s at this speed, what power will be obtained from full scale
turbine assuming its n is 3% better than that of model.
6. Using Buckingham’s π theorem, show that the drag force FD = ρ L V φ (Re,M) which Re
= ρ LV/μ; M = V/C; ρ = fluid mass density; L = chord length: V= velocity of aircraft; μ =
fluid viscosity; C = sonic velocity = √K/ ρ where K = bulk modulus of elasticity.
7. The resistance ‘ R’ experienced by apartially, submerged body depends upon the velocity
‘V’, length of the body ‘l’, viscosity of fluid ‘μ’, density of the fluid ‘ρ’, and gravitational acceleration ‘g’; obtain expression for R.
8. Derive the relation using Buckingham’s π theorem F = ρ U
2f (μ/UD ρ), ND/U).
9. State the reasons for construction distorted model of rivers and discuss the various types of distortion in models. What are the merits and demerits of distorted models as compared to undistorted model?
10. In an aeroplane model of size 1/10 of its prototype the pressure drop is 7.5 kN/m3. The
model is tested in water. Find the corresponding pressure drop in the prototype. Take density of air is 1.4 kg/ m3, density of water is 1000 kg/ m3, viscosity of air is 0.00018