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### Transcript of · PDF file CIVE1400: Fluid Mechanics Section 1: Fluid Properties CIVE1400: Fluid Mechanics...

• CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties 1

LECTURE CONTENTS

Section 0: Introduction Section 1: Fluid Properties

Fluids vs. Solids Viscosity Newtonian Fluids Properties of Fluids

Section 2: Statics Hydrostatic pressure Manometry/Pressure measurement Hydrostatic forces on submerged surfaces

Section 3: Dynamics The continuity equation. The Bernoulli Equation. Application of Bernoulli equation. The momentum equation. Application of momentum equation.

Section 4: Real Fluids Boundary layer. Laminar flow in pipes.

Section 5: Dimensional Analysis An Intro to Dimensional analysis Similarity

CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties 2

What make fluid mechanics different to solid mechanics?

The nature of a fluid is different to that of a solid In fluids we deal with continuous

streams of fluid.

In solids we only consider individual elements.

In this section we will consider how we can classify the differences in nature

of fluids and solids.

What do we mean by nature of a fluid?

Fluids are clearly different to solids. But we must be specific.

We need some definable basic physical difference.

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• CIVE1400: Fluid Mechanics Section 1: Fluid Properties

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We know that fluids flow under the action of a force, and the solids don’t -

but solids do deform.

So we can say that

fluids lack the ability of solids to resist deformation.

fluids change shape as long as a force acts.

(These definitions include both gasses and liquids as fluids.)

CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties 4

What use can we make of these ideas?

In the analysis of fluids we often take small volumes (elements)

and examine the forces on these.

Take the rectangular element below.

What forces cause it to deform?

A B

C D

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• CIVE1400: Fluid Mechanics Section 1: Fluid Properties

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F

F C D

A’ B’

Forces acting along edges (faces), such as F, are know as shearing forces.

From this we arrive at the definition:

A Fluid is a substance which deforms continuously, or flows, when subjected to shearing forces.

This has the following implications for fluids at rest:

If a fluid is at rest there are NO shearing forces acting on it, and

any force must be acting perpendicular to the fluid

CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties 6

Fluids in motion

Consider a fluid flowing near a wall. - in a pipe for example -

Fluid next to the wall will have zero velocity.

The fluid “sticks” to the wall.

Moving away from the wall velocity increases to a maximum.

v

Plotting the velocity across the section gives “velocity profile”

Change in velocity with distance is

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• CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties 7

As fluids are usually near surfaces there is usually a velocity gradient.

Under normal conditions one fluid particle has a velocity different to its

neighbour.

Particles next to each other with different velocities exert forces on each other (due to intermolecular action ) ……

i.e. shear forces exist in a fluid moving close to a wall.

What if not near a wall?

v

No velocity gradient, no shear forces.

CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties 8

What use is this observation?

It would be useful if we could quantify this shearing force.

This may give us an understanding of what parameters govern the forces

different fluid exert on flow.

We will examine the force required to deform an element.

Consider this 3-d rectangular element, under the action of the force F.

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• CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties 9

F

F

A B

C D

a b

δy

δz

δx

under the action of the force F

F

F

A B

C D

a b

A’ B’

a’ b’

E

CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties10

A 2-d view may be clearer… F

F

B

C D

A’ B’

φ xE E’

y

The shearing force acts on the area

A z x

Shear stress, is the force per unit area: F A

The deformation which shear stress causes is measured by the angle , and is know as

shear strain.

Using these definitions we can amend our definition of a fluid:

In a fluid increases for as long as is applied - the fluid flows

In a solid shear strain, , is constant for a fixed

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• CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties11

It has been shown experimentally that the rate of shear strain is directly proportional to shear stress

time

Constant t

We can express this in terms of the cuboid.

If a particle at point E moves to point E’ in time t then: for small deformations

shear strain x y

rate of shear strain

(note that x t

u is the velocity of the particle at E) So

CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties12

Constant u

y u/y is the rate of change of velocity with distance,

in differential form this is du dy

The constant of proportionality is known as

the dynamic viscosity,

giving

du dy

which is know as Newton’s law of viscosity

A fluid which obeys this rule is know as a Newtonian Fluid

(sometimes also called real fluids)

Newtonian fluids have constant values of

Non-Newtonian FluidsJN TU

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• CIVE1400: Fluid Mechanics Section 1: Fluid Properties

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Some fluids do not have constant . They do not obey Newton’s Law of viscosity.

They do obey a similar relationship and can be placed into several clear categories

The general relationship is:

A B u y

n

where A, B and n are constants.

For Newtonian fluids A = 0, B = and n = 1

CIVE1400: Fluid Mechanics Section 1: Fluid Properties

CIVE1400: Fluid Mechanics Section 1: Fluid Properties14

This graph shows how changes for different fluids.

S h

ea r

st re

ss , τ

Rate of shear, δu/δy

Bingham plastic

plastic Pseudo plastic

Newtonian

Dilatant

Ideal, (τ=0)

Plastic: Shear stress must reach a certain minimum before flow commences.

Bingham plastic: As with the plastic above a minimum shear stress must be achieved. With this classification n = 1. An example is sewage sludge.

Pseudo-plastic: No minimum shear stress necessary and the viscosity decreases with rate of shear, e.g. colloidial substances like clay, milk and cement.

Dilatant substances; Viscosity increases with rate of shear e.g. quicksand.

Thixotropic substances: Viscosity decreases with length of time shear force is applied e.g. thixotropic jelly paints.

Rheopectic substances: Viscosity increases with length of time shear force is applied

Viscoelastic materials: Similar to Newtonian but if there is a sudden large change in shear they behave like plastic.JN TU

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• CIVE1400: Fluid Mechanics Section 1: Fluid Properties

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Liquids vs. Gasses

Liquids and gasses behave in much the same way

Some specific differences are

1. A liquid is difficult to compress and often regarded as being incompressible. A gas is easily to compress and usually treated as such - it changes volume with pressure.

2. A given mass of liquid occupies a given volume and will form a free surface A gas has no fixed volume, it changes volume to expand to fill the containing vessel. No free surface is formed.

Causes of Viscosity in Fluids

Viscosity in Gasses Mainly due to molecular exchange between layers Mathematical considerations of this momentum exchange can lead t