MECHANICAL FLUID MECHANICS

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MECHANICAL FLUID MECHANICS

2 | P a g e THE GATE COACH All Rights Reserved 28, Jia Sarai N.Delhi-16, 26528213,-9998

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

Pascals 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

MECHANICAL FLUID MECHANICS

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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

Eulers equation along a straight line 126

Eulers equation in Cartesian coordinate 129

Bernoullis theorem 130

Kinetic energy correction factor 134

Venturi meter 135

Orifice meter 137

MECHANICAL FLUID MECHANICS

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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

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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

MECHANICAL FLUID MECHANICS

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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

MECHANICAL FLUID MECHANICS

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1. BUOYANCY AND FLOATATION

ARCHIMEDES 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

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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.

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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

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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