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INS3051 FLUID MECHANICS

CHAPTER 1

INTRODUCTION

INTRODUCTION

1.1 Definition of Fluid

Matter exists in three states in nature:

MATTER

Solid state

Liquid state

Gaseous state FLUID

When we look at the spacing and latitude of motion of molecules in each state, they are completely different from each other:

the spacing and freedom of motion of molecules:

GAS > LIQUID > SOLID

Thus it follows that intermolecular cohesive forces are larger for a solid, smaller in a liquid, and extremely small in a gas.

These fundamental facts explain the familiar compactness and rigidity of form possessed by the solids, the ability of liquid molecules to move freely within the liquid mass, and capacity of gases to fill the containers in which they are placed, while a liquid has a definite volume and well-defined surface.

SOLID

Hard and not easily deformed

Densely spaced molecules

Large intermolecular cohesive forces

FLUID

Soft and easily deformed

Sparsely spaced molecules

Smaller intermolecular cohesive forces

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Why are liquids and gases considered as fluids?

Answer: Because of their reaction to the applied shearing

stresses.

(A shearing stress is created when a tangential force acts

on a surface)

Definition of fluid:

A FLUID is a substance which deforms continuously

under the application of a shearing stress, no matter how

small the shearing stress is.

However, when common solids such as steel or other

metals are acted on by a shearing stress, they will initially

deform (usually a very small deformation), but they will

not continuously deform.

Definition of flow:

The process of continuous deformation is known as

FLOW of fluids.

F

b Dq

B B

Dx plate

fluid t0 t1

A

tD

D

qt

tD

D

qt

D

D

D ttt

qq

0lim The rate of angular deformation

Since the deformation of fluids is continuous, we talk about

the rate of deformation rather than the amount of deformation.

F

b Dq

B B

Dx plate

fluid t0 t1

A

3

0tF0D

D

t

qthen

This means that the fluids at rest do not contain shearing

stresses. Therefore, the forces in static fluids are transmitted

to solid boundaries normal to the boundary at every point.

F

b Dq

B B

Dx plate

fluid t0 t1

A

If

Solids can carry both tension and compression while fluids

can carry only compression; fluids have no tensile strength.

Compression = Pressure

FC FT

SOLID

FC FT

FLUID

1.2 Scope of Fluid Mechanics

Fluid mechanics is the branch of physics which deals with the properties of fluids, namely liquids and gases,

and their interaction with forces.

Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of

fluids in motion. Fluid mechanics, especially fluid

dynamics, is an active field of research with many

unsolved or partly solved problems.

Fluid mechanics can be mathematically complex. Sometimes it can best be solved by numerical methods,

typically using computers.

Fluid mechanics is a branch of continuum mechanics, a subject which models matter without using the information

that it is made out of atoms.

Continuum Mechanics

Fluid Mechanics Solid Mechanics

Elasticity Plasticity Newtonian Fluids Non-Newtonian Fluids

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Engineering applications of Fluid Mechanics

Water supply

Energy production

Transportation

Pipelines

Hydraulic structures

Fluvial hydraulics

Coastal hydraulics

Groundwater flow

Wind forces on structures

Heating, ventilation

Main engineering disciplines that deal with Fluid Mechanics:

Civil Engineering

Aerospace Engineering

Mechanical Engineering

Environmental Engineering

Geological Engineering

Mining Engineering

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1.3 Concept of Continuum

Atoms are widely spaced in the gas

phase.

However, we can disregard the atomic

nature of a substance.

View it as a continuous, homogeneous

matter with no holes, that is, a

continuum.

This allows us to treat properties as

smoothly varying quantities.

Continuum is valid as long as size of

the system is large in comparison to

distance between molecules.

Concept of Continuum Continuum and Fluid Particle

Continuum it is the continuous distribution of matter in the flow field without any

discontinuity.

A fluid particle is defined as the mass contained in the smallest fluid volume for

which the continuum assumption is not

violated.

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1.4 Dimensions and System of Dimensions

Describing Physical Entities

Qualitative Description Quantitative Description

Serves to identify the characteristics (such as

length, time, stress)

Provides a numerical measure of the characteristics.

Requires both a number and a standard by which various quantities

can be compared. DIMENSIONS

UNITS

1.5 Units and System of Units PRIMARY UNITS

DERIVED UNITS

Dimensions and Units

Dimensional Homogeneity:

All theoretically derived equations are dimensionally homogeneous, that is, the dimensions on the left side of the equation must be the same as those on the right side.

A dimensionally homogenous equation has therefore the same dimensions for each additive term on both sides of the equations.

Dimensions and Units

Example 2:

The equation for the velocity, V, of a uniformly accelerated body,

where V0 is the initial velocity, a the acceleration and t the time

interval. In terms of dimensions the equation is

Therefore, the equation is dimensionally homogeneous.

atVV 0

TLTLTLT 211

111 LTLTLT

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1.6 Physical Properties of Fluids A property is a characteristic of a substance, which is invariant when

the substance is in a particular state.

Properties

Extensive: depends on the amount of substance present (volume, energy, weight, momentum)

Intensive: independent of the amount of substance (specific values, namely, mass per unit volume, energy per unit mass)

The intensive properties are the values which apply to a particle of a

fluid. These are:

Density

Specific weight

Specific Gravity

Specific Volume

Viscosity

Surface Tension

Vapor Pressure

Compressibility

DENSITY, r

Density, r, of a substance is a measure of concentration of matter, and expressed as mass per unit volume.

Dimension of density

Unit of density in SI system

kg/m3

kg/m3

m

r

3MLr

),( TPrr where P=pressure and T=temperature

10004@2 COHr

SPECIFIC WEIGHT, g

Specific Weight, g, is the force due to gravity on the mass contained in a unit volume of a substance, i.e. weight per

unit volume.

Dimension of specific weight

Unit of specific weight in SI system

N/m3

N/m3

mgWg

3 FLg

grg

98104@2 COHg

SPECIFIC GRAVITY, SG

The specific gravity is a term used to compare the density of a substance with that of water if the fluid is

liquid, and with that of air or hydrogen if the fluid is gas.

Since the density r depends on the temperature and pressure, for precise values of specific gravity,

temperature and pressure values must be specified.

Since specific gravity is the ratio of densities, the value of SG does not depend on the system of units used.

w

liq

w

liq

liqSGg

g

r

r

air

gas

gasSGr

r

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VISCOSITY

Viscosity is the resistance of a fluid to shear stresses (and hence to flow).

In study of fluid flow, viscosity requires the greatest consideration among the other properties of fluid.

In order to understand the effect of viscosity, let us consider the motion of a fluid along a stationary solid boundary:

VISCOSITY

Observations show that, while the fluid clearly has a finite velocity, u, at a finite distance from the boundary, the velocity is zero at the solid

boundary.

Therefore, velocity increases with the increasing distance from the stationary boundary. That is, if we plot the velocities at different

distances from the boundary, we might obtain a picture as follows:

VISCOSITY

Let us consider two such layers, the lower layer is moving with velocity u, and the upper layer with (u+Du).

Two particles (1) and (2),

starting on the same vertical

line, will move to different

distances during the time

interval Dt:

S2=(u+Du).Dt

S1=u.Dt

S2

S1

Thus, the fluid is distorted or sheared, as the line connecting

(1) and (2) acquires an increasing slope and length as time t

increases.

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VISCOSITY

In other words, the faster moving fluid tries to speed up the slower moving layer, while the slower moving layer tries to reduce the speed of faster one.

Therefore, it is evident that a frictional or shearing force exist between the fluid layers.

This shearing force may be expressed as a shear stress t (shearing force per unit contact area).

The shear stress is directly proportional to the velocity difference and inversely proportional to the distance between two layers, that is:

y

u

D

Dt

VISCOSITY

All real fluids posses viscosity and therefore exhibit certain frictional resistance when motion occurs.

The term, u/ y, is called the velocity gradient, and it shows the angular velocity of the line ab, or the rate of increase of angle q, and hence equal to the rate of deformation.

tuyS DDDDD q

Therefore

ty

u

D

D

D

D q

Recall that

y

u

t D

D

D

D

qt

VISCOSITY

The result indicates that the shearing stress and the velocity gradient can be related with a relationship of the following form:

NEWTONS LAW OF VISCOSITY

The constant of proportionality m is called the dynamic viscosity and it is a property of fluid.

It shows the effect of different fluids. For example honey shows more resistance to flow than water. Therefore honey must have a higher m.

dy

du

dt

d

tt

D

D

D

qq

0lim

dy

dumt

VISCOSITY

Newtons Law of viscosity states that the shear stress at any point is proportional to the velocity gradient at that point:

2

2

yydy

du

mt

1

1

yydy

du

mt

21 tt

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VISCOSITY

The magnitude of the shear stress on the solid boundary is called the wall-shear stress. That is:

If we integrate the wall-shear stress over the area on which it is acting, we obtain the total frictional resistance (force) on the fluid due to solid boundary.

The force exerted by the flowing fluid on a solid boundary in the direction of flow is called the DRAG FORCE.

0

y

wdy

dumt

VISCOSITY

A typical variation of shear stress:

Solve Example 1.1

VISCOSITY

When a flow takes place between solid boundaries or over a solid boundary, the fluid layer next to the solid boundary sticks to the boundary and attains its velocity.

If the boundary is stationary, the velocity of fluid is zero on the boundary.

If the boundary is moving with a velocity of U0, the fluid next to the boundary will move with the velocity U0.

In other words, there is a NO SLIP CONDITION between the fluid and the solid for all fluids that can be treated as continuum.

Note that if a fluid is at rest, no layer moves relative to an adjacent layer, therefore, there will be no shearing stress.

Since 0dy

du0t

VISCOSITY

On the other hand, if the viscosity of the fluid is zero, regardless of the motion, no shear stress can develop. Such a fluid is called

INVISCID FLUID.

All the fluids in nature have a viscosity, and we call them as real fluids.

Inviscid fluid is only an idealization. It simplifies the solution of many real fluid flow problems.

Not all the real fluids obey the Newtons Law of viscosity. If we plot

shear stress t, versus the velocity gradient ( or the rate of angular deformation) for various fluids, the plot looks like:

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VISCOSITY

The fluids for which the shearing stress is linearly related to the velocity gradient are designated as Newtonian Fluids.

On the other hand, the fluids for which the shearing stress is not linearly related to the velocity gradient are designated as non-

Newtonian Fluids.

n

dy

duK

t

map =apparent viscosity

VISCOSITY

Shear thickening fluids:

The apparent viscosity increases with increasing shear rate. The

harder the fluid is sheared, the more viscous it becomes. For

example, water-sand mixture (quicksand).

t

(map)B

dy

du1

1

(map)A

(map)B > (map)A

VISCOSITY

Shear thinning fluids:

The apparent viscosity decreases with increasing shear rate. The

harder the fluid is sheared, the less viscous it becomes. For

example, colloidal substances like clay, milk and cement.

t

(map)B

dy

du

1

1

(map)A

(map)A > (map)B

VISCOSITY

Bingham plastic:

Neither fluid nor solid. Such material can withstand a finite shear

stress without motion, but once the yield stress is exceeded it

flows like a fluid. For example, toothpaste, mayonnaise.

t

dy

du

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VISCOSITY

Dynamic and Kinematic Viscosity:

The proportionality constant m is known as dynamic viscosity of

the fluid.

Viscosity can be made independent of fluid density; kinematic

viscosity is defined as follows:

N.s/m2

Pa.s

kg/(m.s)

m2/s

VISCOSITY

Viscosities of air and water:

VISCOSITY

Effect of Temperature on Viscosity:

Gases: As temperature increases the viscosity increases.

Because, the viscosity arises due to momentum exchange

between the randomly moving molecules in gases. As

temperature increases, the molecular activity will increase in

gases, hence more momentum exchange will take place, and as a

result viscosity will increase.

Liquids: As temperature increases the viscosity decreases.

Because, viscosity arises due to intermolecular cohesive forces. In

liquids, as temperature increases, these forces will decrease,

hence the viscosity will decrease.

VISCOSITY

Effect of Temperature on Viscosity of Water:

Temperature Viscosity

[C] [Pas]

10 1.31E-03

20 1.00E-03

30 7.98E-04

40 6.53E-04

50 5.47E-04

60 4.67E-04

70 4.04E-04

80 3.55E-04

90 3.15E-04

100 2.82E-04

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Viscosity and Drag Force

Viscosity is a property that represents the internal

resistance of a fluid to

motion.

The force a flowing fluid exerted on a body in the

flow direction is called the

drag force, and the

magnitude of this force

depends, in part, on

viscosity.

NO SLIP CONDITION

An important effect of viscosity is to cause the fluid to stick to the solid surface, this is known as the no slip condition.

No slip condition states that fluid molecules will have zero velocity with respect to the solid boundary that they stick on.

Bulk Modulus of Elasticity, Ev, and

Compressibility, K

Compressibility is a measure of change of volume and density when a substance is subject to normal pressure or tension. It is defined as:

Compressibility (K) = % change in volume (or density) for a given

pressure change

(-) sign indicates a decrease in volume and therefore an increase in

density caused by an increase in pressure.


Compressibility, K The reciprocal of compressibility is known as the bulk modulus of

elasticity.

All fluids may be compressed by the application of pressure. Elastic

energy is stored in the process. Assuming perfect energy conversions,

such compressed volumes of fluids will expand to their original

volumes when the applied pressure is released.

Thus, fluids are elastic media, and it is customary in engineering to

summarize this property by defining a modulus of elasticity as is done

for solid elastic materials, such as steel.

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Compressibility, K Since fluids do not posses a rigid form, the modulus of elasticity must be

defined on the basis of volume, and it is termed as BULK MODULUS.

Consider a cylinder full of a fluid which has a volume of V0 . Application of a force F to the piston will increase the pressure (P=F/A) in the fluid and cause the volume to decrease .

If we plot P vs V/V0, the slope of curve at any point will give the bulk modulus of elasticity at that point.


Compressibility, K The steeping of the curve with increasing pressure shows that as fluids

are compressed, they become increasingly difficult to compress further,

which is logical consequence of reducing the space between the

molecules. The bulk modulus of elasticity of a fluid is not constant, but

increases with pressure.

(Ev)air = 1.42 x 105 Pa

(Ev)water = 2.15 x 109 Pa

(Ev)steel = 2.06 x 1011 Pa

Therefore:

H2O is 100 times more compressible than steel.

Air is 20 000 times more compressible than water.

For all practical purposes, water and all liquids may be assumed as incompressible unless very large pressure ranges are involved.

Surface Tension , s

When liquid surfaces are in contact of other immiscible fluids and solids, at the boundaries, molecular attraction introduces forces which cause the interface of liquid to behave like membrane under tension. This apparent tension effects depend fundamentally upon relative sizes of intermolecular cohesive and adhesive forces.

Surface Tension , s

Although such forces are negligible in practice, they become important in capillary rise of liquids in narrow tubes, formation of liquid drops, etc.

Consider a free liquid surface in contact with the atmosphere:

On the free surface in contact with atmosphere, there is little force

attracting molecules away from the liquid because there are relatively

few molecules in the gas above the surface. Within the liquid bulk, the

intermolecular forces of attraction and repulsion are balanced in all

directions.

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Surface Tension , s

However, for liquid molecules at the free surface, the cohesive forces of the next layer below are not balanced by an identical layer above.

This situation tends to pull the surface molecules tightly to the lower layer and to each other and cause the surface to behave as though it were a membrane: hence, the name SURFACE TENSION.

Surface tension is a property of liquid and depends on temperature as

well as the other fluid it is in contact with the surface.

s = s(T), as T s

Surface Tension , s

Another characteristic of liquids in contact with solid surfaces is the adhesion of liquid molecules to the solid surface.

If Adhesive forces > Cohesive forces

The liquid wets the solid surface.

Surface Tension , s

If Cohesive forces > Adhesive forces

Then the solid surface is not wetted.

Capillary Effects

Capillary effect is the rise or fall of a liquid in a small-diameter tube.

The curved free surface in the tube is called the meniscus.

Water meniscus curves up because water is a wetting fluid.

Mercury meniscus curves down because mercury is a non-wetting fluid.

Force balance can describe magnitude of capillary rise.

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Wetting and Non-wetting Fluids

The level h, that the water or mercury can rise or drop can be computed

from the relation derived below.

q

q < 90 q > 90


By equating the lifting force created by the surface tension to the gravity force:

FST = W

gqs hrr 2cos2 r

rh

g

qs cos2

The same equation can be applied to the other case.

Solve Example


This result immediately rises the question of the meaning of the angle q, limitations of equation and its confirmation by experiment.

The angle q is known as the angle of contact, and it results from the surface tension phenomena of complex nature.

Surface Tension for a Droplet

0 xF

0 sFFF inout PP

0222 rrPrP inout s

rrPP outin s 22

rP

s2D

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Vapor pressure, Pv For boiling to occur, the equilibrium must be upset either

by

1. Raising the temperature to cause the vapor pressure to equal or exceed the total pressure applied at the free surface,

OR

2. By lowering the total pressure at the free surface until it is equal to or less than the vapor pressure.

The more volatile the liquid (e.g. alcohol, benzene), the higher its vapor pressure.

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Cavitation

If cavitation occurs in contact with a solid surface, very serious damage can result due to the very large force with which the liquid hits the surface.

Cavitations can affect the performance of hydraulic machinery such as pumps, turbines and propellers, and the impact of collapsing bubbles can cause local erosion of metal surface.

Cavitation

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