chapter 1 fluids and their properties · 2019-11-19 · gtu paper analysis fluid mechanics...

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Fluid Mechanics (2141906) Department of Mechanical Engineering Darshan Institute of Engineering & Technology

Chapter 1 – Fluids and their Properties

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Theory

1. Explain the following terms 1. Relative Density 2. Kinematic Viscosity 3. Cavitation 4. Vapour Pressure 5. Continuum 6. Compressibility 7. Capillary Effect 8. Newton’s law of Viscosity 9. Viscosity 10. Specific Gravity 11. Surface Tension 12. Density

7 2 3 3 3

2. State Newton’s law of viscosity and give an example. 4 3. Define gauge pressure, absolute pressure and atmospheric pressure. 3 1 4. Explain surface tension and derive its equation for liquid droplet. 4

Examples

1. Calculate the shear stress developed in oil of viscosity 1.2 poise, used for lubricating the clearance between a shaft of diameter 12 cm and its journal bearing. The shaft rotates at 180 rpm and clearance is 1.4 mm.

7

3. Two large plane surfaces are 2.4 cm apart. The space between the surfaces is filled with glycerin. What force is required to drag a very thin plate of surface area 0.5 m2 between two large plane surfaces at a speed of 0.6 m/s, if (i) The thin plate is in the middle of the two plane surfaces and, (ii) The thin plate is at a distance of 0.8 cm from one of the plane surface. Take the dynamic viscosity of glycerin= 8.10 x 10-1 Ns/m2.

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2. A 150 mm diameter vertical cylinder rotates concentrically inside cylinder of diameter 151 mm. both the cylinders are 250 mm high. The space between the cylinders is filled with a liquid whose viscosity is unknown. If a torque of 12 Nm is required to rotate the inner cylinder at 100 rpm, determine the viscosity of the liquid.

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4. A flat plate 40 cm x 30 cm slides on oil (μ = 0.7 N-s/m2) over a large plane surface. What is the force required to drag the plate at 3 m/s if separating oil film is 0.5 mm thick ?

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Chapter 2 – Pressures and Head

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Theory

1. State and derive Pascal’s law with usual notations. States its applications. OR

State and prove Pascal’s law with usual notations.

4 4 7 4 7

2. Enlist the various Mechanical gauges for pressure measurement and describe their working with suitable diagram.

OR Explain with neat diagram construction and working of bellow and diaphragm pressure gauge.

OR Explain with neat diagram construction and working of bourdon tube pressure gauge.

9 7 + 7

4

3. State advantages and limitation of manometer OR

What are the advantages and disadvantages of manometers as pressure measuring instrument?

3 3

4. Define and derive hydrostatic law. 4 5. Differentiate between (i) simple and differential manometer (ii) piezometer and

pressure gauge 2

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Examples

1. Find out the differential reading “h” of an inverted U-tube manometer containing oil of sp.gr. 0.7 as the manometric fluid, when connected across pipes A and B as shown in figure.1, conveying liquid of sp.gr.1.2 and 1.0 and immiscible with manometer fluid. Pipes A and B are located at the same level and assume the pressure at A and B to be equal.

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2. A U-tube manometer is used to measure the pressure of water in a pipe line, This 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 15 cm and the free surface of mercury is in level with the center of the pipe.

4

3. Find the depth of point below sea water surface where the pressure intensity is 404.8 kN/m2. The specific gravity of sea water is 1.03.

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Chapter 3 – Static Forces on Surface and Buoyancy

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Theory

1. Derive expressions for total pressure and centre of pressure for vertically immersed surface.

OR Define total pressure and centre of pressure. Derive expression of total pressure force and center of pressure for vertical plane surface remains submerged in liquid.

7 7 4 7 7

2. Explain the conditions of stability for a submerged and floating body with neat diagrams.

OR Discuss stability of submerged and floating bodies.

OR Describe stability condition of floating bodies.

7 4 3 3

3. Derive equation of total pressure and center of pressure for inclined submerged body.

OR Explain pressure diagram for inclined and submerge surface.

7 7

4. Show that the distance between the meta-centre and centre of buoyancy is given

byI

BM

.

OR Define metacenter and metacentric height. Derive expression to find metacentric height using analytical method for floating body.

7 7

5. Derive an expression for calculating time period of oscillation of floating body. 7

6. Write the practical significance of metacentric height. 4

7. Determine the metacentric height of a floating vessel if the angle of tilt ϴ caused by moving load P placed over the center of the floating body.

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8. Explain Archimedes Principle. 3

Examples

1. A circular plate 1.5 m diameter is submerged in water, with its greatest and least depths below the surface being 2 m and 0.75 m respectively. Determine: (i) Total pressure on one of the face of the plate (ii) The position of centre of pressure.

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2. A rectangular plane surface 2 m wide and 3 m high immerged in water, it plan is making an angle 45° with the free surface of water. The upper edge of rectangular plate is 1.5 m below the free surface. Calculate the position of center of pressure.

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3. A rectangular gate that is 2 m wide is located in the vertical wall of a tank containing water as shown in figure.2. It is desired to have the gate open automatically when the depth of water above the top of the gate reaches 10 m. (a) At what distance “d” should the frictionless horizontal shaft be located? (b) What is the magnitude of the force on the gate when it opens?

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4. A rectangular pontoon 8 m long, 6 m wide and 2 m deep, floats in sea water (sp. weight = 10000 N/m3). It carries an empty boiler on its upper dock of 4 m diameter. The weight of pontoon and boiler are 600 kN and 200 kN respectively. The center of gravity of each unit coincides with geometric centre of the arrangement and lie on same vertical line. Find the metacentric height of arrangement and check the stability.

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5. A block of wood has a horizontal cross section 500 mm X 500 mm and height h. It floats vertically in water. If the specific gravity of wood is 0.6, find the maximum height of block so that it can remain in stable equilibrium.

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6. A barge in the shape of a rectangular block 8 m wide, 12.8 m long and 3 m deep floats in water with a draft of 1.8 m. the centre of gravity of the barge is 0.3 m above the water surface. State whether the barge is in stable equilibrium. Calculate the righting moment when the barge heels by 10°.

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7. A solid cylinder 2 m in diameter and 2 m high is floating in water with its axis vertical. If the specific gravity of the material of cylinder is 0.65 find its metacentric height. State whether the equilibrium is stable or unstable.

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8. A solid cylinder of diameter 4 meters has a height 3 meters. Find the meta centric height of the cylinder when it is floating in water with its axis vertical. The specific gravity of the cylinder is 0.6.

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9. A rectangular pontoon 9 m long, 8 m wide and 3 m deep, floats in sea water (sp. weight = 10000 N/m3). It carries an empty boiler on its upper dock of 5 m diameter. The weight of pontoon and boiler are 600 kN and 300 kN respectively. The center of gravity of each unit coincides with geometric center of the arrangement and lie on same vertical line. Find the metacentric height of arrangement and check the stability.

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10. A body of dimensions 2m x 2m x 1m, weights 1970 N in water. Find its weight in air. Find its mass.

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Chapter 4 & 6 – Fluid Kinematics

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Theory

1. Explain clearly: Stream line; Path line, Streak line, Uniform flow, Steady flow. 7 3 3 3 3 3 2. Explain different types of fluid flows.

OR Classify fluid flows.

OR

Distinguish clearly between:

1. Rotational and Irrotational flow

2. Laminar and Turbulent flow

7 7 4 3

3. Derive an equation for continuity equation for 3D flow and reduce it for steady, incompressible 2D flow.

OR Derive an equation of continuity for three dimensional Cartesian coordinate system.

7 7 7 7 7

4. Distinguish between free vortex flow and forced vortex flow. 4 4 5. Define vortex flow. Derive an expression of stream function and velocity

potential function for vortex flow. 7

6. Explain terms Rotation and Vorticity. 3 3 7. Explain Eulerian frame of reference. 4 8. What do you understand by stream function, velocity potential function and

flownet. Give the relation between stream function and velocity potential function.

OR Define velocity potential function and stream function. Derive relation between them.

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Examples

1. The stream function of a two dimensional flow is given by ψ = 2xy + 25. Calculate the velocity at the point (1, 2). Also find the velocity potential function Ø.

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2. Velocity for a two dimensional flow field is given by V=(3+2xy+4t2)i+(xy2+3t)j

Find the velocity and acceleration at appoint (1,2) after 2 sec.

5

3. The velocity components in a two- dimensional flow are 3 2( / 3) 2u y x x y

and 2 32 ( / 3)v x y y x . Show that these functions represent a possible case of

an irrotational flow.

4

4. What is the irrotational velocity field associated with the potential ɸ = 3x2 -3x +3y2 +16t2 +12zt. Does the flow field satisfy the incompressible continuity equation?

4

5. The velocity potential function is given by Ø = 4(x2-y2). Calculate the velocity components at the point (2, 3).

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6. A stream function is given by ψ = 5x – 6y. Calculate magnitude and direction of resultant velocity.

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7. A fluid flow is given by V = 18x3i – 20x2yj. State the flow is rotational or irrotational.

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Chapter 5 – Fluid Dynamics

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Theory

1. Derive the expressions for discharge over (i) Rectangular notch and (ii) Triangular notch.

7 4 3 4 7

2. Derive Euler’s equation of motion for flow along a stream line. Obtain Bernoulli’s from it. State assumptions clearly. OR Derive the Euler’s equation for motion along a stream line.

7 3 7 4

3. Derive an expression for the discharge through a venturimeter and compare it with orifice meter for measurement of flow through pipes.

3 7 7

4. Derive the expression for time required to emptying a tank through an orifice at its bottom.

7

5. Derive the relation among different energies using Bernoulli’s theorem. 7

6. State the momentum correction factor and list the momentum correction factor for different flow in pipes.

3

7. Compare rectangular and triangular notches. 3

Examples

1. A 300 mm X 150 mm Venturimeter is placed vertically with throat 250 mm above the inlet section conveys kerosene of density 820 kg/m3. The flow rate is 140 liter/sec. Calculate the pressure difference between inlet and throat section. Take Cd = 0.97.

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2. A horizontal Venturimeter 20 cm x 10 cm is used to measure the flow rate of oil having specific gravity 0.9. The discharge through the venturi meter is 3600 liters / minute. Find the reading of oil – mercury differential manometer. Take Cd = 0.98.

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3. A rectangular air duct of 1.5 m2 cross sectional area at section 1 which is gradually reduced to 0.75 m2 area at section 2. The velocity of flow at section 1 is 12m/s and pressure is 30 kN/m2. If the duct is bent by 45°, find the magnitude and direction of the force required to hold the duct in position. Take density of air is 1.15 kg/m3.

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4. 200 liters/s of water is flowing in a pipe having a diameter of 250 mm through which the water is flowing having pressure 38 N/cm2. If the pipe is bent by 125° (that is change from initial to final direction 125°). Find the magnitude and direction of the resultant force on the bend.

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5. A horizontal venturimeter with inlet diameter 20 cm and throat diameter 10 cm is used to measure the flow of water. The pressure at inlet is 17.658 N/cm2 and the vacuum pressure at throat is 30 cm of mercury. Find the discharge of water through venturimeter. Take Cd = 0.98.

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6. A horizontal Venturimeter with inlet diameter 20cm and throat diameter 10cm is used to measure the flow of oil of sp.gr 0.8. The discharge of oil through Venturimeter is 60 liters/Second. Find the reading of the oil mercury differential manometer take Cd = 0.98.

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7. A horizontal Venturimeter with inlet diameter 150 mm and throat diameter 75 mm is used to measure discharge. The differential manometer gives reading of 150 mm of mercury. Determine the rate of flow if Cd is 0.98.

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Chapter 7 – Dimensional Analysis & Similarities

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Theory

1. What are repeating variables? How are they selected for dimensional analysis? 5 3 4

2. State the dimensional homogeneity. Prove that the following equations are

homogeneous equation.

i. Q AV ii. 2 /T L g iii. 2V gH

7

3. Explain different types of hydraulic models. 4

4. Define dimensional analysis with an example. 3

5. Define following non dimensional numbers:

Reynold’s Number, Weber’s Number, Froude’s Number

3

Examples

1. The frictional torque T of a disc of diameter D rotating at a speed N in a fluid of

viscosity μ and density ρ in a turbulent flow is given by,

𝑇 = 𝐷5𝑁2𝜌 𝜑 [𝜇

𝜌𝑁𝐷2]

Prove this by Buckingham’s π method.

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2. Using Buckingham’s π- theorem show that the velocity through a circular orifice

is given by,

2 ,D

V gHH VH

Where H is head causing flow, D is the diameter of the orifice, μ is the coefficient

of viscosity, ρ is mass density and g is the acceleration due to gravity.

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

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

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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.

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

flow depends upon the velocity, density and roughness. Using Buckingham’s π

theorem obtain expression of ΔP.

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5. Using Buckingham’s π- theorem, show the efficiency ɳ 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.

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Chapter 8 – Viscous Flow

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Theory

1. Derive the Hagen – Poiseuille equation for laminar flow in the circular pipe.

OR

Derive an expression for Hagen-Poiseuille’s formula for viscous flow.

OR

For viscous flow through pipe, derive equation of velocity distribution and prove

that ratio of maximum velocity to average velocity for this flow is 2.

OR

Prove that the average velocity is half of the maximum velocity in circular pipe

with steady laminar flow.

OR

Derive an expression for the velocity distribution for viscous flow through a

circular pipe. Also sketch the velocity distribution and shear stress distribution

across a section of the pipe.

7 7 7 7 7 7 7

2. List different methods of viscosity measurement. Explain any one of them.

OR

Explain capillary tube viscometer.

4 3

Examples

1. A shaft of 100 mm diameter rotates at 60 rpm in a 200 mm long bearing. Taking

the two surfaces are uniformly separated by a distance of 0.5 m and taking linear

velocity distribution in the lubricating oil having dynamic viscosity of 4

centipoises, find the power absorbed in the bearing.

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2. A crude oil of viscosity 0.1Ns/m2 and relative density 0.9 is flowing through a

horizontal circular pipe of diameter 60 mm and length 200 m. calculate the

difference of pressure at the two ends of the pipe if flow rate through the pipe is

4 liters/s.

4

3. In a pipe of 200 mm diameter water is flowing, there is a shear stress of 0.12

kN/m2 at a point distant 30 mm from the pipe axis. If the coefficient of friction

between the pipe and fluid is 0.04, calculate the shear stress at the pipe wall.

3

4. Calculate (i) the pressure gradient along flow (ii) the average velocity and (iii)

the discharge for an oil of viscosity 0.02 Ns/m2 flowing between two stationary

parallel plates 1 m wide maintained 10 mm apart. The velocity midway between

the plates is 2 m/s.

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Chapter 9 – Turbulent Flow

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Theory

1. Derive an expression for the loss of head due to friction in pipes.

OR

Obtain Darcy-Weisbach formula for head loss due to friction.

7 7 4 7 7 5 7

2. State the Karman-Prandtl equation for the velocity distribution near hydro

dynamically smooth boundaries.

7

Examples

1. An oil of specific gravity 0.9 and viscosity 0.06 poise is flowing through a pipe of

diameter 200 mm at the rate of 60 liters/s. Find the head lost due to friction for

a 500 mm length of pipe. Also find the power required to maintain the flow. Take

f = 0.079/(Re)1/4

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2. Calculate the loss of head and power required to maintain the flow in a

horizontal circular pipe of 40 mm diameter and 750 m long when water flow at

a rate of 30 liters/minute. Take Darcy’s friction factor is 0.032.

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Chapter 10 – Flow Through Pipes

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Theory

1. Prove the friction head losses is equal to one third of total head at inlet for maximum power transmission through pipe.

7 7

2. List out major and minor losses for flow through pipe and explain any one in detail.

2 4

3. Derive an equation for loss of head due to sudden enlargement 4 4. Define or explain following terms

1. Water Hammer 2. Summit 3. Hydraulic Radius

3

5. What do you understand by frictional resistance offered by pipe? 3 6. Explain the phenomenon of water hammer. 3

Examples

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.

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2. A pipe of diameter 225 m is attached to a 150 mm diameter pipe in a straight line by means of a flange. Water flows at the rate of 0.05 m3/s. the pressure loss at transition as indicated by differential gauge length on a water-mercury manometer connected between two pipes equals 35 mm. calculate the loss of head due to contraction.

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3. A syphon of diameter 200 mm connects two reservoirs having a difference in elevation of 20 m. The length of the syphon is 500 m and the summit is 3.0 m above the water level in the upper reservoir. The length of the pipe from upper reservoir to the summit is 110 m. Determine the discharge through the syphon and also pressure at the summit. Neglect minor losses. The co-efficient of friction f= 0.005.

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4. The rate of flow of water through a horizontal pipe is 0.25 m/s. The diameter of the pipe which is 200 mm is suddenly enlarged to 400 mm. The pressure intensity in the smaller pipe is 11.772 N/cm2. Determine (i) loss of head due to sudden enlargement (ii) pressure intensity in large pipe.

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5. Find the loss of head when a pipe of diameter 250 mm is suddenly enlarged to a diameter of 500 mm. the rate of flow through the pipe is 300 liters/ s.

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6. Determine the head lost due to friction in a pipe using Chezy’s formula. Diameter and length of pipe = 300 mm and 50 m Velocity of water flowing in pipe = 2.5 m/s Chezy’s constant = 60

3

7. Determine the head lost due to friction in a pipe using Chezy’s formula. Diameter and length of pipe = 250 mm and 60 m Velocity of water flowing in pipe = 2.5 m/s Chezy’s constant = 60

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Chapter 11 – Compressible Flow

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Theory

1. Prove that the velocity of sound wave in compressible fluid is given by

KC

.

OR

Prove the velocity of a sound wave in a compressible fluid is given byC RT .

OR

Prove the velocity of a sound wave in a compressible fluid is given bydP

Cd

.

OR Find the expression of velocity of sound in terms of Bulk Modulus.

7 3 4 3 7

2. Explain propagation of sound waves for Sub sonic and Sonic flow. 4 3. Derive Bernoulli’s equation for adiabatic process in compressible fluid flow. 4 4. Prove the compressibility is reciprocal of bulk modulus. Also derive relation

between bulk modulus and pressure for a gas for isothermal and adiabatic process.

4

5. Define Mach number. Classify compressible flow based on Mach number. 3 6. Define compressible and incompressible. 3

Examples

1. A supersonic aircraft flies at an altitude of 1.8 km where temperature is 4°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 γ = 1.4.

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2. A projectile is travelling in air having pressure 0.1 N/mm2 and temperature -1℃. If the Mach angle is 39°, find the velocity of projectile and Mach number. Take k = 1.4 and R = 287 J/kgK

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3. An aeroplane is flying at height of 15 km where the temperature is – 50°C. Speed of the plane is corresponding to M = 2. Assuming k = 1.4 and R = 287 J/kg K, find the speed of the plane.

4

4. A projectile is travelling in air having pressure 8.83 N/cm2 and temperature -2℃. If the mach angle is 40°, find the velocity of projectile. Take k = 1.4 and R = 287 J/kgK.

3

chapter 1 fluids and their properties · 2019-11-19 · gtu paper analysis fluid mechanics...

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