FLUID MECHANICS (2141906) - Darshan Institute of ... · PDF fileFLUID MECHANICS (2141906) ......

FLUID MECHANICS (2141906)

B.E. Semester IV Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot

CHAPTER – 1 PROPERTIES OF FLUID

Theory

1. Explain following terms in brief:

a) Density or Mass density f) Kinematic viscosity

b) Specific weight or Weight density g) Viscosity or Dynamic viscosity

c) Specific volume g) Bulk modulus and compressibility

d) Specific gravity or Relative density h) Cavitation

2. Gives classification of fluid.

3. Define surface tension. Derive an expression for surface tension for the following

cases:

i. Water droplet

ii. Hollow bubble

iii. Liquid jet

4. Define capillarity effect. Derive an expression for Capillarity rise and Capillarity fall

(Capillarity depression).

Class Tutorial

1. The pressure outside the droplet of water of diameter 0.04 mm is 10.32

N/cm2(atmospheric pressure). Calculate the pressure within the droplet if surface

tension is given as 0.0725 N/m of water.

2. A 50 mm diameter shaft rotates with 500 rpm in a 80 mm long journal bearing with

51 mm internal diameter. The annular space between the shaft and bearing is filled

with lubricating oil of dynamic viscosity 1 poise. Determine the torque required and

power absorbed to overcome friction.

3. A plate 0.03 mm distant from a fixed plate, moves at 70 cm/s and requires a force of

3 N/m2 to maintain this speed. Calculate the fluid viscosity between the plates.

Tutorial

1. Calculate dynamic viscosity of an oil, which is used for lubrication between a square

plate of size 0.8m 0.8m and an inclined plane with angle of inclination 30. The

weight of square plate is 300 N and is slide down the inclined plane with a uniform

velocity of 0.3 m/s. The thickness of oil film is 1.5mm. [Ans: 11.7 poise]

2. Calculate capillarity effect in millimeters in a glass tube of 4mm diameter, when

immersed in (a) water and (b) mercury. The temperature of liquid is 20C and value

of surface tension of water and mercury at 20C in contact with air are 0.073575

N/m and 0.51 N/m respectively. The angle of contact for water is zero and that for

mercury is 130. Take the density of water at 20C as equal to 998 kg/m3.

[Ans: 7.51 mm , -2.46 mm]

Fluid Mechanics (2141906)

B.E. Semester IV Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 1

CHAPTER – 2 PRESSURE AND HEAD

Theory

1. Define the pressure. Obtain an expression for the pressure intensity at a point in a

fluid.

2. State and prove the Pascal’s law. Also mention its application.

3. Prove that “Intensity of pressure at any point in a fluid at rest is same in all

direction”.

4. What do you understand by Hydrostatic law?

5. What do you mean by gauge pressure, vacuum pressure and absolute pressure?

Explain the working principle of U-tube differential manometer with neat sketch.

6. What do you mean by single column manometers? How are they used for the

measurement of pressure?

7. Explain following with neat sketch:

a. Diaphragm pressure gauge

b. Bourdon tube pressure gauge

c. Dead weight pressure gauge

d. Bellows pressure gauge

Class Tutorial

1. A hydraulic press has a ram of 30 cm diameter and plunger of 4.5 cm diameter. Find

the weight lifted by the hydraulic press when the force applied at the plunger is

500N. [Ans: 22.22kN]

2. An open tank contains water up to a depth of 2 m and above it an oil of sp. gr. 0.9 for

a depth of 1 m. Find the pressure intensity: (i) at the interface of the two liquids, and

(ii) at the bottom of the tank. [Ans: 8829 N/m2; 28449 N/m2]

3. The diameters of a small piston and a large piston of a hydraulic jack are 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, and (b) small piston

is 40 cm above the large piston. The density of the liquid in the jack is given as 1000

kg/m2. [Ans: 888.96N; 919.7N]

4. 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



mercury and is open to atmosphere. The contact between 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 of U-tube is 10 cm and free surface of mercury is in

level with 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.

[Ans: 12360.6 N/m2; 8.016cm]

5. A single column vertical manometer (i.e. micrometer) is connected to a pipe

containing oil of sp. gr. 0.9. The area of the reservoir is 80 times the area of the

manometer tube. The reservoir contains mercury of sp. gr. 13.6. The level of

mercury in the reservoir is at a height of 30 cm below the centre of the pipe and

difference of mercury levels in the reservoir and right limb is 50 cm. Find the

pressure in the pipe. [Ans: 6.48 N/cm2]

6. A differential manometer is connected at the two points A and B as shown in figure

2.1. At B air pressure is 7.848 N/cm2 (abs.); find the absolute pressure at A.

[Ans: 6.91 N/cm2]

7. In figure 2.2 shows an inverted differential manometer connected to two pipes A and

B containing water. The fluid in manometer is oil of sp. gr. 0.8. For the manometer

readings shown in the figure, find the difference of pressure head between A and B.

[Ans: 0.26 m of water]

Figure 2.1 Figure 2.2



Tutorial

1. Calculate the pressure due to a column of 0.3 m of (a) water, (b) an oil of sp. gr. 0.8,

and (c) mercury of sp. gr. 13.6.[Ans: 0.2943 N/cm2; 0.2354 N/cm2; 4.002 N/cm2]

2. The pressure intensity at a point in a fluid is given 3.249 N/cm2. Find the

corresponding height of fluid when it is: (a) water, and (b) an oil of sp. gr. 0.9.

[Ans: 4 m of water; 4.44 m of oil]

3. A hydraulic press has a ram of 20 cm diameter and plunger of 3 cm diameter. It is

used for lifting a weight of 30 kN. Find the force required at the plunger.

[Ans: 675.2N]

4. A simple manometer is used to measure the pressure of oil (sp. gr. = 0.8) flowing in a

pipe line. Its right limb is open to the atmosphere and left limb is connected to the

pipe. The centre of the pipe is 9 cm below the level of mercury in the right limb. If

the difference of mercury level in the two limbs is 15 cm, determine the absolute

pressure of the oil in the pipe in N/cm2. [Ans: 12.058 N/cm2]

5. A simple manometer (U-tube) containing mercury is connected to a pipe in which an

oil of sp. gr. 0.8 is flowing. The other end of the manometer is open to atmosphere.

Find the vacuum pressure in pipe, if the difference of mercury level in the two limbs

is 20 cm and height of oil in the left limb from the centre of the pipe is 15 cm below.

[Ans: -27.86 N/cm2]

6. An inverted differential manometer containing an oil of sp. gr. 0.9 is connected to

find the difference of pressures at two points of a pipe containing water. If the

manometer reading is 40 cm, find the difference of pressures. [Ans:392.4 N/m2]

7. An inverted differential manometer is

connected with two pipes A and B in which

water is flowing as shown in Fig. 2.3. The

manometric fluid is oil of specific gravity 0.8.

Refer the figure and find the pressure

difference between A and B.

[GTU Dec-2014]

Fig. 2.3


B.E. Semester – IV Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 1

CHAPTER – 3 STATIC FORCES ON SURFACE AND BUOYANCY

Theory

1. Derive the expression for total pressure and centre of pressure for a vertical plate

submerged in the liquid with usual notations.

2. Define metacenter and metacentric height. Explain analytical method for

determination of metacentric height.

3. Define buoyant force, center of buoyancy, metacenter and metacentric height. Also

describe conditions of equilibrium for floating and submerged bodies.

4. Define the following terms:

(I) Total pressure (II) Centre of pressure

(III) Force of buoyancy (IV) Metacentre

Class Tutorial

1. Determine the total pressure and depth of centre of pressure on a plane rectangular

surface of 1 m wide and 3 m deep when its upper edge is horizontal and (a) coincides

with water surface (b) 2 m below the free water surface.

2. A solid cylinder of diameter 4 m has a height of 4 m. Find the metacentric height of

the cylinder if the specific gravity of the material of cylinder is 0.7 and it is floating in

water with its axis vertical. State whether the equilibrium is stable or unstable.

3. A rectangular pontoon 8.0 m long, 7 m board and 3.0 m deep weighs 588.6 kN. It

carries on its upper deck an empty boiler of 4.0 m diameter weighing 392.4 kN. The

centre of gravity of the boiler and the pontoon are at their respective centers along a

vertical line. Find the meta-centric height. Weight density of sea-water is 10104

N/m3.

4. A wooden block of specific gravity of 0.7 and dimensions 18 cm wide, 30 cm deep

and 100 cm long floats horizontally on 18 cm wide surface in water. Calculate the

metacentric height and comment on the stability of the block. If the block is given a

tilt of 6o in the clockwise direction. Calculate what should be the mass should be

kept at a distance from the centre 5 cm on the opposite side of offset the tilt.

Tutorial

1. A square plate of diagonal 2m is immersed in a liquid with its diagonal vertical and upper corner is 0.5m below the free surface of the liquid. The specific gravity of the liquid is 1.4. Find, (i) The force exerted by liquid on the plate. (ii) The position of its centre of pressure

2. A pontoon of 15696 KN displacement is floating in water a weight of 245.25 KN is moved through a distance of 8 m across the deck of pontoon which tilts the pontoon through an angle of 4° find the 1etacentric height of the pontoon.



CHAPTER – 4 & 6 KINEMATICS OF FLOW

Theory

1. Explain various types of fluid flow.

2. Derive an expression for continuity equation for three dimensional flow and

reduce it for steady, incompressible 2-D flow in Cartesian co-ordinate system.

3. Explain the following in brief:

a. Total acceleration, Convective acceleration & Local acceleration

b. Velocity potential function & Stream function

c. Vorticity & Circulation

d. Steam line, Streak line & Path line

4. Prove that equipotential line and stream line are perpendicular to each other.

5. Define circulation. Prove that circulation Γ = ∫ ξdA

6. Define flow net. Also describe the use and limitations of flow net.

7. Define vortex flow. Also derive expressions of potential function and stream

function for vortex flow.

8. Differentiate between free and forced vortex flow.

Examples

1. The velocity vector in a fluid flow is given by,

Find the velocity and acceleration of a fluid particle at (2,1,3) at time t = 1.

2. Velocity component of a fluid flow are given as

Where, x, y, z are given in meter and time t in sec.

Determine velocity vector at point P (4, 1, 2) at time t = 4 sec. Also determine the

magnitude of velocity and acceleration of the flow for given location and time.

3. The velocity components in a 2-D flow are,

Show that these components represent a possible case of an irrotational flow.



4. Determine whether the following flows are rotational or irrotational.

a. u = -2y v = 3x

b. u = 0 v = 3xy

c. u = 2x v = -2y

5. An open circular cylinder of 15 cm diameter and 100 cm long contains water up to a

height of 80 cm. Find the maximum speed at which the cylinder is to be rotated

about its vertical axis so that no water spills.



CHAPTER – 5 THE ENERGY EQUATION AND THEIR

APPLICATION

Theory

1. Derive Euler’s equation of motion along a stream line and hence obtain Bernoulli’s

equation. Also state Bernoulli’s theorem with its assumptions. 2. Explain Venturimeter in brief and Derive an expression for discharge through

Venturimeter. 3. What is pitot tube. Derive an expression for the measurement of velocity of flow at

any point in a pipe by pitot tube. 4. Derive an expression for discharge over triangular notch. 5. Derive an expression for discharge over rectangular orifice. 6. Define following terms

I. Kinetic energy correction factor

II. Momentum energy correction factor

Class Tutorial

1. The water is flowing through a pipe having diameters 20 cm and 15 cm at sections 1

and 2 respectively. The rate of flow through pipe is 40 litres/s. The section 1 is 6 m

above datum and section 2 is 3 m above datum. If the pressure at section 1 is 29.43

N/cm2, find the intensity of pressure at section 2. 2. In a duct of 400 mm diameter, a pitot static tube is placed in the centre. The mean

velocity in the duct is 0.85 of the central velocity. Determine the discharge through

the duct if the difference between the static and total pressure is 80 mm of water.

The co-efficient of pitot tube as Cv = 0.98

3. An oil of sp. gr. 0.9 is flow through a venturimeter having inlet diameter 20 cm and

throat diameter 10 cm. The oil-mercury differential manometer shows a reading of

20 cm. Calculate the discharge of oil through the horizontal venturimeter. Take

Cd=0.98.

4. An orifice-meter with orifice diameter 15 cm is inserted in a pipe of 30 cm diameter.

The pressure gauges fitted upstream and downstream of the orifice meter give

readings of 14.715 N/cm2 and 9.81N/cm2 respectively. Find the rate of flow of water

through the pipe in litres/s. Take Cd = 0.6.

Tutorial

1 The water is flowing through a taper pipe of length 100 m having diameter 600 mm

at the upper end and 300 mm at the lower end, at the rate of 50 litre/sec. 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 * 104 N /m2 & lower end is 10 m above datum. [Ans p =22.857 N/cm2]



2 In a laboratory experiment with a right angled V- notch the discharge is calculated to

be 0.06 m3/min. Taking discharge co-efficient as 0.62. However, the actual discharge

is known to be 0.065 m3/min. Assuming that this discrepancy is due to an error in

measuring head above the sill, determine the error in mm. [Ans dH = 2.146 m]



CHAPTER – 7 DIMENSIONAL ANALYSIS & SIMILARITIES

Theory

1. State Buckingham’s π-theorem. What do you mean by repeating variables? How the

repeating variables are selected in dimensional analysis? 2. State the various dimensionless numbers with their significance in fluid flow

situations. Explain Froude, Euler and Weber model law with applications. 3. Discuss different types of similarities that must exist between a prototype and its

model. Class Tutorial

Rayleigh’s Method

Three Variables 1. Find an expression for the power P developed by pump. When P depends upon the

head H, discharge Q and specific weight w of the fluid.

[Ans : R.K Bansal Pg No 584]

Four Variables 2. Find an expression for the drag force F on smooth sphere of diameter D, moving with

a uniform velocity V in a fluid of density ρ and dynamic viscosity µ.

[Ans : (

) R.K Bansal Pg No 562]

Buckingham’s π-Theorem

3. The efficiency η of a fan depends on density ρ, dynamic viscosity μ of the fluid,

angular velocity ω, diameter D of the rotor and the discharge Q. Express η in terms

of dimensionless parameters.

[Ans : *

+ R.K Bansal Pg No 568]

4. The pressure difference ΔP in a pipe of diameter D and length l due to turbulent flow

depends on the velocity v, viscosity μ, density ρ and roughness k. Using

Buckingham’s π-theorem, obtain an expression for ΔP. Also show that

[Ans : *


Model laws

5. A pipe of diameter 1.5m is required to transport an oil of sp. gravity 0.90 and

viscosity 3*10-2 poise at the rate of 3000 lit/sec. Tests were conducted on a 15cm

diameter pipe using water at 20˚C. Find the velocity and rate of flow in the model.

Viscosity of water at 20˚C = 0.01 poise. [Ans : Qm= 89.9 litre/s R.K Bansal Pg No 584]



6. A ship model of scale 1/50 is towed through sea water at a speed of 1 m/sec. A force

of 2N is required to tow the model. Determine the speed of ship and the propulsive

force on the ship, if prototype is subjected to wave resistance only.

[Ans : Fp=250000 N R.K Bansal Pg No 590]

Tutorial

1. The time period of a pendulum depends upon the length of pendulum and

acceleration due to gravity. Derive an expression for the time period.

[Ans : √

R.K Bansal Pg No 562]

2. Derive on the basis of dimensional analysis suitable parameters to present the thrust

developed by propeller. Assume that thrust P depends upon the angular velocity ω,

speed of advance V, diameter D, dynamic viscosity µ, mass density ρ, elasticity of the

fluid medium which can be denoted by speed of the sound in the medium C.

[Ans *


3. The pressure drop in an airplane model of size 1/40 of its prototype is 80 N/cm2. The

model is tested in water. Find the corresponding pressure drop in the prototype.

Take density of air = 1.24 kg/m3, viscosity of water is 0.01 poise while the viscosity of

air as 0.00018 poise. [Ans : Pp= 0.01306 N/cm2 R.K Bansal Pg No 597]



CHAPTER – 8 VISCOUS FLOW

Theory

1. For the viscous flow through circular pipe show that the velocity distribution across the section is parabolic also prove that the mean velocity is equal to one half the maximum velocity.

2. Derive the expression for Hagen-Poiseuille’s formula.

3. Obtain Expression for the power required to overcome the viscous resistance of Journal bearing and Foot-step bearing.

4. Explain falling sphere resistance method to determine the co-efficient of viscosity.

5. Explain (i) Say Bolt viscometer (ii) Redwood viscometer.

Class Tutorial

1. A laminar flow is taking place in a pipe of diameter of 200 mm. The maximum velocity is 1.5 m/sec. Find the mean velocity and the radius at which this occurs. Also calculate the velocity at 4 cm from the wall of the pipe.

[Ans: 0.75m/s, 70.7mm, 0.96m/s]

2. What power is required per km of a line to overcome the viscous resistance to the flow of glycerine through a horizontal pipe of diameter 100 mm at the rate of 10 lit/sec?

Take µ = 8 poise and kinematic viscosity ν = 6 stokes. [Ans: 32.588 KW]

3. Two parallel plates kept 80 mm apart have laminar flow of oil between them maximum velocity of flow is 1.5 m/sec, Calculate:

a. Discharge per meter width

b. Shear stress at the plate

c. The difference in pressure between two points 20 m apart

d. Velocity gradient of plates

e. Velocity at 20 mm from the plate

Assume viscosity of oil 24.5 poise.

[Ans: 0.08 m3/sec, 183.75 N/m2, 91.875 kpa, 75 s-1, 1.125 m/sec]

4. The radial clearance between a hydraulic plunger and the cylinder walls is 0.1 mm. The length of the plunger is 300 mm and diameter 100 mm. Find the velocity of leakage and rate of leakage past the plunger at an instant when the difference of the pressure between the two ends of the plunger is 9 m of water.

Take µ = 0.0127 poise. [Ans: 19.3 cm/sec, 6.06E-3 lit/sec]

5. A container full of oil has a horizontal parallel crack in its end wall which is 500 mm wide and 50 mm thick in the direction of flow. The pressure difference between two faces of the crack is 10 kPa and the crack forms a gap of 0.4 mm between the parallel surfaces. Calculate:

a. Volume of oil leakage per hour through the crack;



b. Maximum leakage velocity;

c. Shear stress and velocity gradient at the boundary.

Take specific gravity and viscosity of oil equal to 0.85 and 1.8 poise respectively.

[Ans: 2.96E-06, 0.0222 m/sec, 40 N/m2, 222 per sec]

6. A shaft having a diameter of 50 mm rotates centrally in a journal bearing having a diameter of 50.15 mm and length 100 mm. The angular space between the shaft and the bearing is filled with oil having viscosity of 0.9 poise. Determine the power absorbed in the bearing when the speed of rotation is 60 rpm.

[Ans: 46.41 W]

Tutorial

1. An oil of viscosity 0.1 N-s/m2 and relative density 0.9 is flowing through a circular pipe of diameter 50 mm and of length 300 m. The rate of flow of fluid through the pipe is 3.5 lit/sec. Find the pressure drop in a length of 300 m and also the shear stress at the pipe wall. [Ans: 684288 N/m2, 28.512 N/m2]

2. A viscous flow is taking place in a pipe of diameter 100mm. The maximum velocity is 2 m/s. Find the mean velocity and the radius at which this occurs. Also calculate the velocity at 30mm from the wall of the pipe.

[Ans: 1m/s, r=35.35mm, u=1.68m/s]

3. Oil of specific gravity 0.82 is pumped through a horizontal pipe line 15 cm in diameter and 3 km long at the rate of 900 liters per minute. This pump has an efficiency of 68% and requires 7.35 kW to pump the oil. Determine the dynamic viscosity of oil and verify whether the flow is laminar.

[Ans: 0.092 Ns/m2, Re = 1135]

4. There is a horizontal crack 50 mm wide and 3 mm deep in a wall of thickness 150 mm. Water leaks through the crack. Find the rate of leakage of water through the crack if the difference of pressure between the two ends of the crack is 245.25 N/m2. Take the viscosity of water as 0.01 poise. [Ans: 183.9 cm3/sec]

5. A thrust bearing having external and internal diameters 20 cm and 10 cm respectively is used to take the thrust of a shaft. An oil film of thickness 0.03 cm is maintained between the collar surface and the bearing. Find the power lost in overcoming the viscous resistance when the shaft rotates at 250 rpm. Take µ = 0.9 poise. [Ans: 30.165 W]

6. A capillary tube of diameter 4 mm and length 150 mm is used for measuring viscosity of a liquid. The difference of pressure head between the two ends of the tube is 0.7848 N/cm2 and the viscosity of the liquid is 0.2 poise. Find the rate of flow of liquid through the tube. [Ans: 16.43 cm3/sec]



CHAPTER – 9 TURBULENT FLOW

Theory

1. Derive an expression for loss of head due to friction in pipe flow.

OR

Derive Darcy-Weisbach equation with usual notations.

2. What do you understand by the terms major energy loss and minor energy losses in

pipe?

3. Write a short note on moody diagram for calculating the head loss due to friction.



CHAPTER – 10 FLOW THROUGH PIPE

Theory

1. Derive the expression for Darcy- Weisbach formula for friction loss in the pipe.

2. Explain total energy line (T.E.L) and Hydraulic gradient line (H.G.L).

3. Derive the expression for maximum efficiency corresponding to the maximum power

transmitted for flow through the pipes.

4. Explain equivalent pipe and Syphone.

Class Tutorial

1. Two reservoirs are connected by a pipeline which is 15 cm in diameter for the first 5

m and 25 cm diameter for the remaining 15 m. Entry to and exit from the pipe is

sharp and the water surface in the upper reservoir is 7.5 m above that in the lower

reservoir. Represent layout and calculate the head losses and flow rate by assuming

the friction co-efficient is 0.01 for both the pipes. Also draw hydraulic gradient line

(H.G.L) and Total energy line (T.E.L).

Tutorial

1. A pipe AB branches into two pipes BC and BD. The pipe has diameter of 30 cm at A,

20 cm at B, 15 cm at C and 10 cm at D. Determine the discharge at A if flow velocity

at A is 2.5 m/s. Also find the velocity at B and D, if the velocity at C is 4.2 m/s.


B.E. Semester – IV

Department of Mechanical Engineering

Darshan Institute of Engineering and Technology, Rajkot

CHAPTER – 11 COMPRESSIBLE FLOW

Theory

1. Prove that velocity of sound wave in a compressible fluid is given by C= √(k/ρ)

Where k= Bulk modulus of fluid and ρ = Density of fluid.

2. Explain Zone of action, Zone of silence, Mach angle, Mach number and Mach-cone

with the help of diagram.

3. Derive an expression for the velocity of sound wave in a compressible fluid in terms

of change of pressure and change of density.

Class Tutorial

1. Certain mass of air is passing through a horizontal pipe with a velocity of 350 m/s, at

a section with corresponding pressure of 80 KN/m2 absolute and temperature 45oC.

There is a change in diameter of the pipe at a section and pressure at this section is

128 kN/m2 absolute. Find the velocity of air stream if the flow is adiabatic.

2. At what speed the shock wave propels in the flow in the air at 1750 kN/m2, absolute,

is moving at 150 m/s in the high pressure wind tunnel at 40o C. Take R=287 J/Kg K.

State whether the flow is supersonic or not.

3. Air has velocity of 1000 Km/hr at pressure of 9.81 kN/m2 vacuum and temperature

of 470C. Compute its stagnation properties and the local Mach number. Take

atmospheric pressure = 98.1 KN/m2, R= 287 J/Kg K and γ = 1.4.

Tutorial

1. A projectile is travelling in air having pressure and temperature as 0.1 N/mm2 and

00C. The mach angle is 380. Calculate the velocity of the projectile. Assume R=0.287

kJ/kg k.

2. A supersonic aircraft flies at an altitude of 1.8 km where temperature is 4o C.

Determine the speed of the aircraft if its sound is heard 4 seconds after its passage

over the head of an observer. Take R = 287 J/kg K and = 1.4.

FLUID MECHANICS (2141906) - Darshan Institute of ... · PDF fileFLUID MECHANICS (2141906) ......

Documents

Transcript of FLUID MECHANICS (2141906) - Darshan Institute of ... · PDF fileFLUID MECHANICS (2141906) ......