A textbook of fluid mechanics and

a Acceleration A Area As Area of suction pipe, surge tank Ad Area of delivery pipe B Width of wheel turbine b Width, bed width of rectangular or trapezoidal channel cp Specific heat at c

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A TEXTBOOK OF FLUID MECHANICS

AND HYDRAULIC MACHINES

in

SI UNITS

www.EasyEngineering.net

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A TEXTBOOK OF FLUID MECHANICS

M.E (Hons.), Gold Medallist; Grad (Mech.Engg & Elect Engg.); M.I.E (India);

M.S.E.S.I.; M.I.S.T.E.; C.E (India)

Recipient of:

‘‘Best Teacher (Academic) Award’’

‘‘Distinguished Author Award’’

“Jawahar Lal Nehru Memorial Gold Medal’’

for an outstanding research paper (Institution of Engineers–India)

Principal (Formerly):

PATIALA

S CHAND & COMPANY LTD.

(AN ISO 9001 : 2008 COMPANY) RAM NAGAR, NEW DELHI-110 055

MULTICOLOUR EDITION

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copying or storing it in any medium in form of graphics, electronic or mechanical means and whether or not transient

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Jurisdiction : All desputes with respect to this publication shall be subject to the jurisdiction of the Courts, tribunals

and forums of New Delhi, India only.

First Edition 1998

Subsequent Editions and Reprints 2002, 2005, 2006, 2007, 2008 (Twice), 2009 (Twice),

2010 (Twice), 2011

Fully Revised Multicolour Edition 2013

printed in india

By Rajendra Ravindra Printers Pvt Ltd., 7361, Ram Nagar, New Delhi -110 055

and published by S Chand & Company Ltd., 7361, Ram Nagar, New Delhi -110 055

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To my wife

Ramesh Rajput

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PREFACE TO THE FIFTH EDITION

I am pleased to present the Fifth Edition of this book The warm reception, which the

previous editions and reprints of this book have enjoyed all over India and abroad has been

a matter of satisfaction to me

Besides revising the whole book two new chapters numbered 17 in “Fluid Mechanics”

(Part – I) and 8 in “Hydraulic Machines” (Part – II), the title of both being “Universities’

Questions (Latest) with Solutions”, have been added separately to update the book

comprehensively

I’m thankful to the Management Team and the Editorial Department of S Chand &

Company Ltd for all help and support in the publication of this book

Any suggestions for the improvement of this book will be thankfully acknowledged

and incorporated in the next edition

Er R.K Rajput

(Author)

Disclaimer : While the author of this book has made every effort to avoid any mistake or omission and has used his skill, expertise

and knowledge to the best of his capacity to provide accurate and updated information The author and S Chand do not give

any representation or warranty with respect to the accuracy or completeness of the contents of this publication and are selling

this publication on the condition and understanding that they shall not be made liable in any manner whatsoever S.Chand and

the author expressly disclaim all and any liability/responsibility to any person, whether a purchaser or reader of this publication

or not, in respect of anything and everything forming part of the contents of this publication S Chand shall not be responsible

for any errors, omissions or damages arising out of the use of the information contained in this publication.

Further, the appearance of the personal name, location, place and incidence, if any; in the illustrations used herein is purely

coincidental and work of imagination Thus the same should in no manner be termed as defamatory to any individual.

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PREFACE TO THE FIRST EDITION

The main object of writing this book on the subject of Fluid Mechanics and Hydraulic

Machines is to present to the student community, a book which should contain

compre-hensive treatment of the subject matter in simple, lucid and direct language and envelope

a large number of solved problems properly graded, including typical examples, from

examination point of view

The book comprises 22 chapters and is divided into two parts: Part I deals with ‘Fluid

Mechanics’ while Part II deals with ‘Hydraulic Machines’ (Fluid Power Engineering) All

chapters of the book are saturated with much needed text supported by simple and

self-explanatory figures and large number of Worked Examples including Typical Examples (for

competitive examinations) At the end of each chapter Highlights, Objective Type Questions,

Theoretical Questions and Unsolved Examples have been added to make the book a

compre-hensive and a complete unit in all respects

The book will prove to be a boon to the students preparing for engineering

under-graduate, AMIE Section B (India) and competitive examinations

The author’s thanks are due to his wife Ramesh Rajput for extending all cooperation

during preparation of the manuscript

In the end the author wishes to express his gratitude to Shri Ravindra Kumar Gupta,

Director, S Chand & Company Ltd., New Delhi, for taking a lot of pains in bringing out

the book, with extremely good presentation, in a short span of time

Although every care has been taken to make the book free of errors both in the text as

well as solved examples, yet the author shall feel obliged if errors present are brought to

his notice Constructive criticism of the book will be warmly received

Er R.K Rajput (Author)

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a Acceleration

A Area

As Area of suction pipe, surge tank

Ad Area of delivery pipe

B Width of wheel (turbine)

b Width, bed width of rectangular or trapezoidal channel

cp Specific heat at constant pressure

CP Centipoise

Cv Specific heat at constant volume

C Chezy’s discharge coefficient

C Celerity of a pressure wave

d Diameter of orifice plate, pipe, particle

D Diameter of pipe, wheel

dd Diameter of delivery pipe

ds Diameter of suction pipe

e Linear strain

E Young’s modulus of elasticity of material

f Darcy Weisbach friction coefficient, frequency

F Force

FB Force exerted by boundary on the fluid

FD Drag force on the body

H Total energy head, net head

had Acceleration head for delivery pipe

has Acceleration head for suction pipe

I Moment of inertia (of area), moment of inertia (of mass)

ld Length of delivery pipe

ls Length of suction pipe

ld´ Length of delivery pipe between cylinder to air vessel

ls´ Length of suction pipe between cylinder and air vessel

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k Roughness height

K Conveyance

K Head loss coefficient, bulk modulus of elasticity, blade friction coefficient

Kt Vane thickness factor

p, ps Pressure, stagnation pressure

P Power, shaft power (turbine), Poise, force

q Discharge per unit width, discharge per jet

Q Discharge, heat

r Distance from the centre

R Radius of pipe, hydraulic radius, radius of pipe bend

Ro Universal gas constant

Re Reynolds number

S Specific gravity, bed slope of channel

t Thickness, time

T Absolute temperature in Kelvins

T Torque, water surface width

u Instantaneous velocity at a point in X-direction

uf Shear friction velocity

U Free stream velocity

Vd Velocity of flow in delivery pipe

V Velocity of flow in the cylinder

Vs Velocity of flow in suction pipe

v Instantaneous velocity at a point in Y-direction

Vf Velocity of flow (in turbines and pumps)

Vw Velocity of swirl (in turbines and pumps)

V Volume

w Weight density, Instantaneous velocity at a point in Z-direction

W Weight of fluid, workdone

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Greek Notations

α Energy correction factor, Mach angle, angle

β Momentum correction factor, angle

γ Ratio of specific heats

δ Boundary layer thickness

δ´ Laminar sub-layer thickness

δ Displacement thickness of boundary layer

*∆s Change in entropy

η Efficiency, dimensionless distance (y/δ)

θ Angle, momentum thickness of boundary layer

µ Coefficient of dynamic viscosity

ν Kinematic viscosity

ρ Mass density of fluid

σ Coefficient of surface tension, cavitation number (Thoma number)

τ0 Bottom shear stress

φ Angle, velocity potential

Subscript 0 refer to any quantity at reference section

Subscripts 1, 2 refer to any quantity at section 1 or 2

Subscripts x, y, z refer to any quantity in x, y, z direction

Subscripts m, p refer to any quantity in model and prototype

Subscript r refer to the ratio of any quantity in model to that in prototype

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1.8.1.1 Pressure inside a water droplet, soap bubble

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2.6 Pressure at a Point in Compressible Fluid 83

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6.6 Practical Applications of Bernoulli’s Equation 291

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6.7 Free Liquid Jet 313

6.12.1 Liquid in a container subjected to uniform acceleration in the

7.8.1 Reynolds number (Re) 418

7.8.2 Froude’s number (Fr ) 419

7.8.3 Euler’s number (Eu ) 419

7.8.4 Weber number (We ) 419

7.8.5 Mach number (M ) 420

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7.11 Froude Model Law 434

8.4 Experimental Determination of Hydraulic Co-efficients 460

8.5.1 Determination of co-efficient of velocity (Cv) 460

8.5.2 Determination of co-efficient of discharge (Cd) 461

8.5.3 Determination of co-efficient of contraction (Cc) 462

8.9 Time Required for Emptying a Tank Through an Orifice at its Bottom 474

8.10 Time Required for Emptying a Hemispherical Tank 483

8.11 Time Required for Emptying a Circular Horizontal Tank 487

8.14 Discharge Through a Convergent-divergent Mouthpiece 493

8.15 Discharge Through an Internal Mouthpiece (or Re-entrant or Borda’s

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9.7 Effect on Discharge Over a Notch or Weir due to Error in the

9.9 Empirical Formulae for Discharge Over Rectangular Weir 518

9.15 Time Required to empty a Reservoir or a Tank with Rectangular and Triangular

10.4 Relationship between Shear Stress and Pressure Gradient 540

10.5 Flow of Viscous Fluid in Circular Pipes—Hagen Poiseuille Law 541

10.7 Flow of Viscous Fluid Between Two Parallel Plates 570

10.7.1 One plate moving and other at rest—couette flow 570

10.7.3 Both plates moving in opposite directions 572

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11.2 Loss of Head due to Friction in Pipe Flow–Darcy Equation 606

11·6·1 Velocity distribution for turbulent flow in smooth pipes 613

11·6·2 Velocity distribution for turbulent flow in rough pipes 615

11.7 Common Equation for Velocity Distribution for both Smooth

11.8 Velocity Distribution for Turbulent Flow in Smooth Pipes by Power Law 620

11.9 Resistance to Flow of Fluid in Smooth and Rough Pipes 621

12·3·2 Chezy’s formula for loss of head due to friction 639

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12·4·1 Loss of head due to sudden enlargement 645

12·4·3 Loss of head due to obstruction in pipe 656

12·11·2 Condition for transmission of maximum power through nozzle 707

12·11·3 Diameter of the nozzle for transmitting maximum power 708

12·12·2 Instantaneous closure of valve in rigid pipes 712

12·12·3 Instantaneous closure of valve in elastic pipes 713

12·12·4 Time required by pressure wave to travel from the valve to the tank

13.2 Boundary Layer Definitions and Characteristics 726

13.2.1 Boundary layer thickness (δ) 727

13.2.3 Momentum thickness (θ) 728

13.3 Momentum Equation for Boundary Layer by Von Karman 739

13.6 Total Drag due to Laminar and Turbulent Layers 769

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14 FLOW AROUND SUBMERGED BODIES—DRAG AND LIFT 785—824

14.8.3 Pressure at any point on the cylinder surface 807

14.8.4 Expression for lift on cylinder (kutta- joukowski theorem) 807

14.8.5 Expression for lift coefficient for rotating cylinder 809

15.5 Propagation of Disturbances in Fluid and Velocity of Sound 837

15.5.1 Derivation of sonic velocity (velocity of sound) 837

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15.7 Propagation of Disturbance in Compressible Fluid 841

15.8.1 Expression for stagnation pressure (ps ) in compressible flow 844

15.8.3 Expression for stagnation temperature (Ts ) 847

15.9 Area-velocity Relationship and Effect of Variation of Area for Subsonic,

15.10 Flow of Compressible Fluid Through a Convergent Nozzle 852

15.12 Flow Through Laval Nozzle (Convergent-Divergent Nozzle) 860

15.15 Flow of Compressible Fluid Through Venturimeter 870

16.1.2 Comparison between open channel and pipe flow 880

16.2.2 Uniform and non-uniform (or varied) flow 882

16.2.4 Subcritical flow, critical flow and supercritical flow 882

16.5.1 Most economical rectangular channel section 890

16.5.2 Most economical trapezoidal channel section 892

16.5.3 Most economical triangular channel section 908

16.5.4 Most economical circular channel section 910

16.6 Open Channel Section for Constant Velocity at all Depths of Flow 914

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B NON-UNIFORM FLOW

17 UNIVERSITIES’ QUESTIONS (LATEST) WITH “SOLUTIONS” 959—994

OBJECTIVE TYPE TEST QUESTIONS 995—1046

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PART – II HYDRAULIC MACHINES

1.2 Force Exerted on a Stationary Flat Plate Held Normal to the Jet 1

1.3 Force Exerted on a Stationary Flat Plate Held Inclined to the Jet 2

1.5 Force Exerted on a Moving Flat Plate Held Normal to Jet 11

1.6 Force Exerted on a Moving Plate Inclined to the Direction of Jet 12

1.7 Force Exerted on a Curved Vane when the Plate Vane is Moving

1.8 Jet Striking a Moving Curved Vane Tangentially at One Tip and

2.3.1 Construction and working of Pelton wheel/ turbine 552.3.2 Work done and efficiency of a Pelton wheel 57

2.4.1.1 Work done and efficiencies of a Francis turbine 842.4.1.2 Working proportions of a Francis turbine 852.4.1.3 Design of a Francis turbine runner 862.4.1.4 Advantages and disadvantages of Francis turbine over a

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2.6 Tubular or Bulb Turbines 130

2.13 Performance Characteristics of Hydraulic Turbines 154

2.13.1 Main or constant head characteristic curves 1542.13.2 Operating or constant speed characteristic curves 1562.13.3 Constant efficiency or iso-efficiency or Muschel curves 157

3.6 Work done by the Impeller (or Centrifugal Pump) on Liquid 182

3.8 Losses and Efficiencies of a Centrifugal Pump 186

3.8.3 Effect of outlet vane angle on manometric efficiency 1873.9 Minimum Speed for Starting a Centrifugal Pump 217

3.10 Effect of Variation of Discharge on the Efficiency 220

3.11 Effect of Number of Vanes of Impeller on Head and efficiency 222

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3.12 Working Proportions of Centrifugal Pumps 222

3.15 Model Testing and Geometrically Similar Pumps 229

4.3 Main Components and Working of a Reciprocating Pump 249

4.4 Discharge, Work Done and Power Required to Drive Reciprocating

4.5 Co-efficient of Discharge and Slip of Reciprocating Pump 252

4.6 Effect of Acceleration of Piston on Velocity and Pressure in the Suction

4.7.2 Effect of acceleration in suction and delivery pipes on indicator

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6.2.6 Safety measures in hydro-electric power plants 337

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6.2.10 Hydro-power development in India 339

6.2.12 Comparison of hydro-power station with thermal power station 341

7.2 Advantages, Disadvantages and Applications of Fluidic Devices/Fluidics 360

8 UNIVERSITIES’ QUESTIONS (LATEST) WITH “SOLUTIONS” 371—401

LABORATORY PRACTICALS (Experiments: 1–26) 1—64

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PART – I FLUID MECHANICSwww.EasyEngineering.net

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1.1 INTRODUCTION Hydraulics:

Hydraulics (this word has been derived from a Greek work ‘Hudour’ which means water) may be defined as follows :

“It is that branch of Engineering-science, which deals with water (at rest or in motion).”

or

“It is that branch of Engineering-science which

is based on experimental observation of water flow.”

Fluid Mechanics:

Fluid mechanics may be defined as that branch of

Engineering-science which deals with the behaviour of fluid under the conditions of rest and motion.

The fluid mechanics may be divided into three

parts: Statics, kinematics and dynamics.

1

Statics. The study of incompressible fluids under static conditions is called hydrostatics and

that dealing with the compressible static gases is termed as aerostatics.

Kinematics It deals with the velocities, accelerations and the patterns of flow only Forces or

energy causing velocity and acceleration are not dealt under this heading.

Dynamics. It deals with the relations between velocities, accelerations of fluid with the forces

or energy causing them.

Properties of Fluids–General Aspects:

The matter can be classified on the basis of the spacing between the molecules of the matter as

1.8 Surface tension and capillarity

1.9 Compressibility and bulk

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2 Fluid Mechanics

In solids, the molecules are very closely spaced whereas in liquids the spacing between the

different molecules is relatively large and in gases the spacing between the molecules is still large It

means that inter-molecular cohesive forces are large in solids, smaller in liquids and extremely small

in gases, and on account of this fact, solids possess compact and rigid form, liquid molecules can

move freely within the liquid mass and the molecules of gases have greater freedom of movement

so that the gases fill the container completely in which they are placed

A solid can resist tensile, compressive and shear stresses upto a certain limit whereas a fluid has

no tensile strength or very little of it and it can resist the compressive forces only when it is kept in a

container When a fluid is subjected to a shearing force it deforms continuously as long as the force

is applied The amount of shear stress in a fluid depends on the magnitude of the rate of deformation

of the fluid element

Liquids and gases exhibit different characteristics The liquids under ordinary conditions are

quite difficult to compress (and therefore they may for most purposes be regarded as incompressible)

whereas gases can be compressed much readily under the action of external pressure (and when the

external pressure is removed the gases tend to expand indefinitely)

1.2 FLUID

A fluid may be defined as follows:

“A fluid is a substance which is capable of flowing.”

or

“A fluid is a substance which deforms continuously when subjected to external shearing

force.”

A fluid has the following characteristics:

1. It has no definite shape of its own, but conforms to the shape of the containing vessel

2. Even a small amount of shear force exerted on a liquid/fluid will cause it to undergo a

de-formation which continues as long as the force continues to be applied

A fluid may be classified as follows:

A. (i) Liquid, (ii) Gas, (iii) Vapour.

B. (i) Ideal fluids (ii) Real fluids.

Liquid

 A liquid is a fluid which possesses a definite volume (which varies only slightly with

tem-perature and pressure)

 Liquids have bulk elastic modulus when under compression and will store up energy in the

same manner as a solid As the contraction of volume of a liquid under compression is extremely

small, it is usually ignored and the liquid is assumed to be incompressible A liquid will withstand

a slight amount of tension due to molecular attraction between the particles which will cause an

apparent shear resistance, between two adjacent layers This phenomenon is known as viscosity.

 All known liquids vaporise at narrow pressures above zero, depending on the temperature

Gas It possesses no definite volume and is compressible.

Vapour. It is a gas whose temperature and pressure are such that it is very near the liquid state

(e.g., steam).

Ideal fluids. An ideal fluid is one which has no viscosity and surface tension and is incompressible

In true sense no such fluid exists in nature However fluids which have low viscosities such as water

and air can be treated as ideal fluids under certain conditions The assumption of ideal fluids helps

in simplifying the mathematical analysis

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Real fluids. A real practical fluid is one which has viscosity, surface tension and compressibility

in addition to the density The real fluids are actually available in nature.

Continuum. A continuous and homogeneous medium is called continuum From the continuum

view point, the overall properties and behaviour of fluids can be studied without regard for its

atomic and molecular structure.

 Liquid can be easily distinguished from a solid or a gas

 Solid has a definite shape

 A liquid takes the shape of vessel into which it is poured

 A gas completely fills the vessel which contains it

The properties of water are of much importance because the subject of hydraulics is mainly

concerned with it Some important properties of water which will be considered are:

(i) Density, (ii) Specific gravity, (iii) Viscosity,

(iv) Vapour pressure, (v) Cohesion, (vi) Adhesion,

(vii) Surface tension, (viii) Capillarity, and (ix) Compressibility.

1.4 DENSITY

1.4.1 Mass Density

The density (also known as mass density or specific mass) of a liquid may be defined as the

mass per unit volume m

V

 

 at a standard temperature and pressure It is usually denoted by ρ (rho)

1.4.2 Weight Density

The weight density (also known as specific weight) is defined as the weight per unit volume at

the standard temperature and pressure It is usually denoted by w.

For the purposes of all calculations, relating to Hydraulics and hydraulic machines, the specific

weight of water is taken as follows:

In S.I Units: w = 9.81 kN/m3 (or 9.81× 10–6 N/mm3)

Specific gravity is the ratio of the specific weight of the liquid to the specific weight of a standard

fluid It is dimensionless and has no units It is represented by S.

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4 Fluid Mechanics

For liquids, the standard fluid is pure water at 4°C

∴ Specific gravity = Specificweight of purewaterSpecificweight of liquid liquid

water

w w

=

Example 1.1 Calculate the specific weight, specific mass, specific volume and specific gravity

of a liquid having a volume of 6 m3 and weight of 44 kN.

Solution: Volume of the liquid = 6 m3

Weight of the liquid = 44 kN

Viscosity may be defined as the property of a fluid which determines its resistance to shearing

stresses It is a measure of the internal fluid friction which causes resistance to flow It is primarily

due to cohesion and molecular momentum exchange between fluid layers, and as flow occurs, these

effects appear as shearing stresses between the moving layers of fluid

An ideal fluid has no viscosity There is no fluid which can be classified as a perfectly ideal fluid

However, the fluids with very little viscosity are

sometimes considered as ideal fluids

Viscosity of fluids is due to cohesion and

interaction between particles.

Refer to Fig 1.1 When two layers of fluid,

at a distance ‘dy’ apart, move one over the other

at different velocities, say u and u + du, the

viscosity together with relative velocity causes a

shear stress acting between the fluid layers The

top layer causes a shear stress on the adjacent

lower layer while the lower layer causes a shear

stress on the adjacent top layer This shear stress

is proportional to the rate of change of velocity

with respect to y It is denoted by τ (called Tau).

Upper layer Lower layer

Solid boundary

y u

Fig 1.1 Velocity variation near a solid boundary.

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where, µ = Constant of proportionality and is known as co-efficient of dynamic viscosity or only

viscosity.

du

dy = Rate of shear stress or rate of shear deformation or velocity gradient.

From Fig 1.1, we have µ = du

Note. The viscosity of water at 20°C is 1

100 poise or one centipoise.

Kinematic Viscosity :

Kinematic viscosity is defined as the ratio between the dynamic viscosity and density of fluid

It is denoted by ν (called nu).

Mathematically, v = ViscosityDensity = µ

1.6.1 Newton’s Law of Viscosity

This law states that the shear stress (τ) on a fluid element layer is directly proportional to the

rate of shear strain The constant of proportionality is called the co-efficient of viscosity.

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6 Fluid Mechanics

1 Newtonian fluids. These

fluids follow Newton’s viscosity

equation (i.e eqn 1.7) For such

fluids µ does not change with rate

of deformation.

Examples. Water, kerosene, air

etc

2 Non-Newtonian fluids

Fluids which do not follow the

linear relationship between shear

stress and rate of deformation (given

by eqn 1.7) are termed as

Non-Newtonian fluids Such fluids are

relatively uncommon

suspensions (slurries), mud flows,

polymer solutions, blood etc

These fluids are generally complex

mixtures and are studied under

rheology, a science of deformation

and flow

3 Plastic fluids. In the case of

a plastic substance which is non-Newtonian fluid an initial yield stress is to be exceeded to cause a

continuous deformation These substances are represented by straight line intersecting the vertical

axis at the “yield stress” (Refer to Fig 1.2).

An ideal plastic (or Binigham plastic) has a definite yield stress and a constant linear relation

between shear stress and the rate of angular deformation Examples: Sewage sludge, drilling muds

etc

A thyxotropic substance, which is non-Newtonian fluid, has a non-linear relationship between

the shear stress and the rate of angular deformation, beyond an initial yield stress The printer’s ink

is an example of thyxotropic substance

4 Ideal fluid An ideal fluid is one which is incompressible and has zero viscosity (or in

other words shear stress is always zero regardless of the motion of the fluid) Thus an ideal fluid is

represented by the horizontal axis (τ = 0)

A true elastic solid may be represented by the vertical axis of the diagram.

Summary of relations between shear stress (τ) and rate of angular deformation for various types

of fluids:

(i) Ideal fluids: τ = 0, (ii) Newtonian fluids:τ = µ.du dy,

(iii) Ideal plastics: τ = const + µ du dy , (iv) Thyxotropic fluids: τ =const.+ µ  .du dyn , and

(v) Non-Newtonian fluids: τ =  du dyn

In case of non-Newtonian fluids, if n is less than unity then are called pseudo-plastics

(e.g., paper pulp, rubber suspension paints) while fluids in which n is greater than unity are known

as dilatents (e.g., Butter, printing ink).

Velocity gradient ,

DilatentfluidNewtonian

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Ostwald-de-Waele Equation It is an empirical solution to express steaty-state shear stress as

a function of velocity gradient, and is given as

τ yx = α du dy n−1du dy

If n = 1, this reduces to Newton’s law of viscosity, with α = µ

Example 1.2. (a) What are the characteristics of an ideal fluid ?

(b) The general relation between shear stress and velocity gradient of a fluid can be written as

τ = Adu dy +n B

where A, B and n are constants that depend upon the type of fluid and conditions imposed on

the flow Comment on the value of these constants so that the fluid may behave as:

(i) an ideal fluid,

(ii) a Newtonian fluid and

(iii) A non-Newtonian fluid.

(c) Indicate whether the fluid with the following characteristics is a Newtonian or

 Incompressible (i.e., ρ = constant)

An ideal fluid can slip near a solid boundary and cannot sustain any shear force however small

it may be

(b) τ = Adu dy +n B

(i) An ideal fluid:

Since an ideal fluid has zero viscosity (i.e., shear stress is always zero regardless of the

motion of the fluid), therefore

(ii) A Newtonian fluid:

Since a Newtonian fluid follows Newton’s law of viscosity;

 n = 1 and B = 0

 The constant A takes the value of dynamic viscosity µ for the fluid.

Air, water, kerosene etc behave as Newtonian fluids under normal working conditions

(iii) A non-Newtonian fluid:

Depending on the value of power index n, the non-Newtonian fluids are classified as:

 If n > 1 and B = 0 Dilatant fluids

Examples: Sugar solution, aqueous suspension and printing ink.

 If n < 1 and B = 0 Pseudo plastic fluids.

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8 Fluid Mechanics

Examples : Blood, milk, liquid cement and clay

 If n = 1 and B = τ0 Bingham fluid or ideal plastic.

An ideal plastic fluid has a definite yield stress and a constant-linear relation between shear

stress developed and rate of deformation:

Since this has the same form as the given shear stress, therefore the fluid characteristics

correspond to that of an ideal fluid.

(ii) τ = Ay n(n – 1) and u = Cy n

Now, du dy = dy d (Cy n ) = Cn(y) n – 1

For a Newtonian fluid τ = du Cn y( )n 1

This expression does not conform to the value of shear stress and as such the fluid is

non-Newtonian in character.

1.6.3 Effect of Temperature on Viscosity

Viscosity is effected by temperature The viscosity of liquids decreases but that of gases

increases with increase in temperature This is due to the reason that in liquids the shear stress

is due to the inter-molecular cohesion which decreases with increase of temperature In gases the

inter-molecular cohesion is negligible and the shear stress is due to exchange of momentum of

the molecules, normal to the direction of motion The molecular activity increases with rise in

temperature and so does the viscosity of gas

where, µT = Dynamic viscosity at absolute temperature T,

a, b = Constants (for a given gas).

1.6.4 Effect of Pressure on Viscosity

The viscosity under ordinary conditions is not appreciably affected by the changes in pressure

However, the viscosity of some oils has been found to increase with increase in pressure

Example 1.3. A plate 0.05 mm distant from a fixed plate moves at 1.2 m/s and requires a force

of 2.2 N/m2 to maintain this speed Find the viscosity of the fluid between the plates.

Tiêu đề	A Textbook Of Fluid Mechanics And Hydraulic Machines
Tác giả	Er. R.K. Rajput
Trường học	Thapar Polytechnic College
Chuyên ngành	Mechanical Engineering
Thể loại	textbook
Thành phố	New Delhi

Định dạng
Số trang	1.570
Dung lượng	19,34 MB