MECHANICAL FLUID MECHANICS - · PDF filemechanical fluid mechanics

MECHANICAL FLUID MECHANICS

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FLUID MECHANICS

&

HYDRAULIC MACHINES

CONTENTS

1 INTRODUCTION TO FLUIDS

Solid, Liquid and Gases 6

Concept of continuum 9

Mass density, specific weight, and specific

gravity

9

Compressibility and bulk modulus 12

Viscosity 14

Surface tension and capillarity 19

Capillary or meniscus effect 24

Vapour pressure 26

Newtonian and Non-Newtonian fluids 28

2 PRESSURE AND ITS MEASUREMENT

Pressure and its relationship with height 40

Pascal’s law 40

Pressure density height relationship: hydrostatic

law

42

Manometers 46

Simple manometers 47

Differential manometers 53

3 HYDROSTATIC FORCES ON

Force on a horizontal submerged plane surface 55

Force on a vertical plane submerged surface 56

Force on an inclines submerged plane surface 58

Force on curved submerged surfaces 60

Pressure Diagram 62



SUBMERGED BODIES

4 BUOYANCY AND FLOATATION

Archimedes’ principle 74

Principle of floatation 75

Equilibrium of a floating body 76

Metacentric height 80

Oscillation of a floating body 81

5 LIQUIDS IN RELATIVE EQUILIBRIUM

Liquid in a container subjected to uniform acceleration in the horizontal direction

88

Liquid in a container subjected to uniform acceleration in the vertical direction

89

Liquid in a container subjected to uniform acceleration along inclined plane

91

Liquid in a container subjected to constant rotation 92

6 FLUID KINEMATICS

Fluid flow 97

Classification of fluid flow 98

Streamlines, Path lines, Streak lines 107

Flow rate and Continuity Equation 109

Differential equation of continuity 110

Rotational Flow 111

Stream function 114

Potential function 117

Circulation 121

7 FLUID DYNAMICS

Euler’s equation along a straight line 126

Euler’s equation in Cartesian coordinate 129

Bernoulli’s theorem 130

Kinetic energy correction factor 134

Venturi meter 135

Orifice meter 137



Pitot tube 137

8 LAMINAR AND VISCOUS FLOW

Navier Stokes equation 142

Relationship between pressure gradient and shear stress 143

Laminar flow between parallel stationary plates 144

Laminar flow between parallel plates having relative motion

147

Laminar flow in circular pipes 150

9 LAMINAR AND VISCOUS FLOW

Introduction 159

Flow losses in pipe 160

Darcy Equation 161

Minor Head Loss 164

Pipes in series and parallel 169

Concept of equivalent pipe 170

Hydraulic gradient lines and total energy lines 171

10 DIMENSIONAL ANALYSIS

Reyleigh Method 178

Buckingham pi-theorem 178

Model Analysis 180

Dimensionless No. 182

11 Boundary Layer theory 186

Boundary layer theory over a smooth plate 186

Drag force on a flat plate 189



BOUNDARY LAYER FLOW

Thermal boundary layer 191

Separation Boundary layer 192

12 NOTCHES AND WEIRS

Introduction 193

Geometry of flow motion 194

Discharge over a rectangular weir 197

Discharge over a submerged rectangular weir 199

Discharge over a broad weir 200

Discharge over a triangular weir or V-notch 201

Discharge over a trapezoidal weir 202

13 FLOW THROUGH ORIFICE AND MOUTHPIECES

Introduction 204

Hydraulic coefficient 205

Discharge through a sharp edged large orifice 207

Discharge through a submerged orifice 208

Discharge through a partially submerged orifice 209

Flow through external cylindrical mouthpiece 210

Flow through re-entrant 211

14 OPEN CHANNEL FLOW

Related terms 216

Classification 218

The Chezy Equation 218

Economic section for maximum discharge 220

Most economical rectangular section 220

Most economical trapezoidal section 221

Most economical circular section 224

15 IMPACT OF FREE JETS

Impulse momentum theorem 229

Jet striking a stationary flat plate 230

Jet striking a stationary flat plate inclined at an angle 231

Jet striking a moving straight plate 232

Jet striking a moving straight plate inclined at an angle 234

Jet striking at centre of stationary vane 235

Jet striking at centre of moving vane 236

Jet striking stationary vane tangentially at one tip 238

Jet striking moving vane tangentially at one tip 239



16 HYDRAULIC TURBINE

Impulse and reaction turbine 243

Pelton turbine 244

Head , power and efficiency 248

Design aspects of Pelton Wheel 250

Radial flow impulse turbine 252

Francis turbine 255

Propeller and Kaplan turbine 259

Model Relationships 265

17 HYDRAULIC PUMP

Classification 273

Centrifugal pump 273

Velocity diagram 276

Pump losses and efficiency 280

Pressure rise in impeller 282

Priming 285

Axial flow pump 286

Reciprocating pumps 287

18 HYDRAULIC SYSTEMS

Hydraulic accumulator 296

Torque convertor 297

Hydraulic Ram 297



1. BUOYANCY AND FLOATATION

ARCHIMEDE’S PRINCIPLE

Fluid exerts pressure on the surfaces of a body which is either partially or completely

submerged into it.

We consider a potato shaped irregular body ABCD (fig.) completely submerged in

a fluid at rest. The body may be of any size, shape and weight distribution.

The body consider is acted upon by two system of forces:

I. A downward gravitational force due to weight of the body; this body force

acts through the centre of mass or centre of gravity of the body.

II. An external pressure force acting all around the surface of the body.

Horizontal force



A horizontal component of force equal to the force on a horizontal projection into a

vertical plane. We consider any vertical plane drawn through the body.

The projected areas of the two sections ABD and BCD of the body onto a vertical plane

are equal.

Consequently the horizontal forces on these two surfaces are equal and opposite, and so the net horizontal force is zero.

Vertical force

A vertical component of force equal to the weight of the fluid vertically above the curved

surface.

For the submerged body under consideration, a vertically downward force F1 acting on

the upper surface equals the weight of volume of fluid ABCEF.

Likewise, a vertically upward force F2 acting on the lower surface equals the weight of volume of fluid ADCEF.

Buoyant force

A body submerged in a fluid experience an upward thrust due to fluid pressure. This force is called buoyant force;

Buoyancy

The tendency of the body to be lifted upward in a fluid due to buoyant force is called buoyancy.

Centre of Buoyancy

The line of action of the buoyant force is vertical and passes through the centre of gravity

of the displaced fluid, i.e. the centroid of the displaced volume. This centroid of the displaced fluid volume is called the centre of buoyancy.



Depending on the ratio of the weight W of a body and the buoyant force Fb, three cases

are possible:

W>Fb; the body tends to move downwards and eventually sinks

W=Fb; the body floats and is only partially submerged W<Fb; the body is lifted upward and rises to the surface.

These aspects are known as Archimedes’ principle.

“A body immersed in a fluid is buoyed or lifted up by a force equal to the weight of

the fluid displaced by the body. The body apparently loses as much of its weight as

the weight of the fluid displaced by it. A floating body displaces volume of fluid just

sufficient to balance its weight”.

PRINCIPLE OF FLOATION

It states that the weight of a body floating in a fluid is equal to the buoyant force which in turn equal to the weight of fluid displaced by the body.

EQUILIBRIUM OF A FLOATING OR SUBMERGED BODY

For a body to be in equilibrium, gravitational force and the buoyant force must be

collinear, equal in magnitude and act in opposite direction.

Stability characterises the response of the system to small disturbing influences such as gust of air, waves on oceans etc.

Possible conditions of equilibrium

A floating or submerged body can have three possible conditions of equilibrium: stable,

unstable and neutral.

STABLE EQUILIBRIUM

A small displacement from the equilibrium position produces a righting moment tending

to restore the body to the original equilibrium position.

UNSTABLE EQUILIBRIUM

A small displacement produces an overturning moment tending to displace the body

further to a condition different from the initial equilibrium position.



NEUTRAL EQUILIBRIUM

The body remains at rest in any position to which it may be displaced. No net force tends

to return the body to its original state or to drive it further away from the original

condition.

SUBMERGED BODY

Stability of a submerged body is determined by the location of centre of gravity G with respect to centroid of the displaced volume, i.e., the centre of buoyancy B.

Stable: G is located below B

Neutral: G is located at B

Unstable: G is located above B

Stable configuration of an aerostatic balloon

The aerostatic balloon considered above will be in unstable when placed in the title

position. When displaced from the position, it will return to original position.

Likewise a system comprising a ship with a small hull and a tall mast which is

very heavily weighed at the top would constitute an unstable system.



Unstable configuration of an aerostatic balloon

When the balloon is displaced from the position, it will never return to original

position

Floating Bodies

ROTATIONAL STABILITY OF A FLOATING BODY

The facets of rotational stability of a floating body are different from those of a wholly

immersed body. For instance, here the mass centre G may lie above

buoyancy centre B and still the floating body may constitute a stable, neutral and unstable

system depending on its geometrical configuration and on the position in which it floats in a fluid.



Neutral equilibrium of a ball

Fig. shows the cross section of a body (a boat or a ship) floating in a static mass of fluid at rest.

The weight of the ship W acts vertically downwards through centre of gravity G, and the

buoyant force Fb acts upwards through centre of buoyancy B.

The centre of buoyancy is the centre of gravity of the ship.

Let the ship tilt through a small angle of heel θ in the clockwise direction due to wind and

the wave action etc.

The ship has now submerged portion A’BCD’ as against ABCD before tilting. A

triangular wedge AOA’ on the left has emerged from the liquid and the wedge DOD’ on

the right has moved into the fluid.

Due to this redistribution, the centre of buoyancy shifts from B to B’; B’ is the centroid

of the immersed portion A’BCD’of the ship. However, with a fixed cargo the relative

position of the centre of gravity G remains unchanged.

The point of intersection of the line of action of the buoyant force before and after the

heel is called the Metacentre (M) and the distance GM is called the Metacentric height.

The stability of a floating body is governed by the position of metacentre M relative to

centre of gravity G.



Metacentre and Metacentric height for floating bodies

If M lies above G, then the metacentric height GM is regarded as positive

and the system is in stable equilibrium. The couple W x acting on the ship

in its title position is the restoring couple, i.e., the couple tries to bring the

ship back to its original position. For proper design of ships, care is taken to

ensure that the metacentre is above the centre of gravity for all angles of

heel which may be encountered.

If M lies below G, then the metacentric height GM is regarded as negative

and the system us in unstable equilibrium. The couple acting in the body

whilst in titled position is the overturning couple, i.e., the couple tries to tilt

the body still further.

Position of metacentre for stability of floating bodies



Submerged body Floating body

Stable Centre of gravity below

Centre of buoyancy

Centre of gravity below meta-

center

Unstable Centre of gravity above

Centre of buoyancy

Centre of gravity above meta-

center

Neutral Centre of gravity coincides

with centre of buoyancy

Centre of gravity lies at meta-

center

METACENTRIC HEIGHT

Analytical method:

Consider a floating object (fig.) which has been given a small tilt angle θ from its

initial upright state.

Because of an increase in the volume of displacement on the right hand side, there is

a lateral displacement of the center of buoyancy from B to B1.

This shift results in a couple of moment WBM tanθ which tends to restore the

object to its original upright position.

Further recognizing that the volume V of the liquid displaced by the object remains

the same area of the triangular wedge OAA’ that emerges out of liquid is equal to

that of wedge ODD’ that has submerged into the liquid.



If l and b are the length (measured normal to the plane of the paper) and the breadth of

the object respectively, then the weight of each wedge shaped portion is: -

Thus there acts a buoyant force δFb upwards on the wedge ODD’ and δFb downwards on

the wedge OAA, each at a distance 2/3 (b/2) = b/2 from the centre.

These two equal and opposite forces constitute a couple of magnitude

Where I is the moment of inertia of the floating object about the longitudinal axis.

This moment must be equal and opposite to the moment caused by the movement of

upward thrust from B to B1. Thus:

Then the metacentric height GM is given by



The –ve sign is used when G lies above B and + ve sign when G lies below B.

OSCILLATION OF A FLOATING BODY

When a floating body is given a tilt, it is set in a state of oscillation as if suspended at the

metacentre in a manner similar to that a simple pendulum.

Let

The time period of oscillation in seconds is given by

The time period of oscillation decreases with an increase in metacentric height h; number of oscillations increase with increase in h.

It is not desirable to keep the metacentric height very large to avoid large number of

oscillations.

Also, large value of metacentric height corresponds to improved stable equilibrium; we

find that the two requirements are contrary to each other. In actual practice, an optimum

value of metacentric height is selected and specified for the ship.

Dialogue Box

ROLLING AND PITCHING OF THE SHIP

…………………………………………………………………………………..

Rolling



The oscillatory motion of a ship or boat about its longitudinalaxis is known as

rolling.

Pitching

The oscillatory motion of a ship or boat about its transverse axis is known as

pitching.

Dialogue Box

DESIGN CRITERIA OF THE SHIP

………………………………………………………………………………….

Expressions for meta-centric height and the time period of oscillation has

been derived for rolling motion, same may be adopted for pitching motion

also.

Since the moment of inertia of the cross-sectional area of ship or boat at

liquid surface about its transverse axis is much more than the same about

longitudinal axis, hence metacentric height is invariably larger than that for

rolling motion.Hence a ship has a safe metacentric height for rolling motion

then it will be safe in pitching motion also.

Practical Applications

Increase of metacentric height gives

- Greater stability to floating body

- Reduces the time period of rolling of the body.

A smaller value of time period of rolling of a passenger ship is quite

uncomfortable for the passengers and may damage the structure of ship.

In case of warships and racing yachts, the stability is more important than the

comfort, hence have larger metacentric height.

In case of cargo ships, metacentric height varies with loading and hence some

control on the value of metacentric height as well as the time period of rolling

is possible by adjusting the position of cargo.



Type of ship or vessel Metacentric height

Ocean-going vessel 0.3 m to 1.2 m

Warships 1.0 m to 1.5 m

River crafts Up to3.6 m

SOLVED EXAMPLE

EXAMPLE 4.1

A wooden block floats in the water with 5 cm height projecting above the water

surface. The block is next launched in glycerine (relative density -1.35) and projects

7.5 cm above the surface of glycerine. Make calculations for the height of block and

its relative density.

SOLUTION

Let A and h denotes the cross sectional area and height of the wooden block.

In accordance with the principle of floatation,the weight of each displaced liquid equals

the total weight of the block.

Then

From last two,

Solving



From first two

Relative density of wood is 0.657

EXAMPLE 4.2

A wooden cylinder ( sp gr = 0.6) of circular cross-section having length l and

diameter d floats in water. Find the maximum permissible l/d ratio so that the

cylinder may float in stable equilibrium with its axis vertical.

SOLUTION

Where s is the specific gravity of cylinder material.

This also represents the weight of water displaced.

Volume of water displaced



Volume of cylinder immersed in water

Height of centre of buoyancy B above the base point O

If M is the meta-centre height, then

Distance of centre of gravity G from the base point O

For stable equilibrium, M should be greater than G. That is,



EXAMPLE 4.3

A log of wood, 1m in diameter and 3 m in length floats in fresh water (sp gr = 9810

N/m3). If the log has a specific gravity of 0.5, work out the angle subtended at the

centre and the depth of immersion.

SOLUTION

This also represents the weight of water displaced.

If A represents the sectional area of log of wood under water, then



Let 2 be the angle submerged at the centre by the immersed portion of log of wood.

A = area of section OAB – area of triangles OAO’ and OBO’.

By symmetry

Area of triangle OAO’ = Area of triangle OBO’

The sector OAB (included angle 2 ) represents part of a circle.

Solving

Evidently the floating log subtends an angle 1800

at centre. This also implies that the log is half immersed or depth of immersion is 0.5 m.

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