# L01_Fluid_Properties

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AR20385

Civil Engineering Hydraulics 1

Fluid Properties

1

UNIVERSITY OF BATH

DEPARTMENT OF ARCHITECTURE AND CIVIL ENGINEERING

CIVIL ENGINEERING HYDRAULICS 1

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Fluid Properties

1.

INTRODUCTION

1.1

What is Hydraulics?

Hydraulics is a
branch of

applied science and engineering which deals with the
mechanical properties
and the motion
of
fluids
. Hydraulics is based on
the
principles of
“fluid dyaic”
which is the field of study in which the
fundamental principles of general mechanics (conservat
ion of mass,
conservation of energy and Newton

s laws of motion) are applied to fluids.

1.2

What is a Fluid?

It is well known that there are three states of matter:

Solid

Liquid

Gas

Liquids and gases
are both

fluid
s
, but what is a fluid?

The following are
often quoted definitions of a fluid:

A substance that has no fixed shape.

A substance that yields easily to external pressure.

A continuous, amorphous substance whose molecules move freely past one
another.

A substance which has the tendency to assume the
shape of its container.

Howeve he defiiive defiiio of a fluid i “
a substance that
continuously

deforms
or flows
under an applied shear stress
.”

To explain this concept, consider two flat plates of infinite length placed a
distance
h

apart

as show
n in

Fig
. 1
. The lower plate is fixed while the upper
plate is allowed to move.

Let us now fill the gap in between the plates first with
a solid substance. If a shear force is applied to the upper plate the solid block
will deform a small amount as shown.

Line

ab

assumes a new position

ab
1

and
the upper plate is displaced by a distan
ce

bb
1
.

The deformation produced is
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Fluid Properties

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proportional to the applied shear stress

F/A
, where

A

is the area of the solid
surface in contact with the plate, however once this deformation has occurred,
no further deformation will occur unless the applied force is

changed.

Now let us fill the gap with a fluid, say water.

What happens when a shear
force is applied to the top plate?

We find that it moves continuously ie., point b
keeps moving and occupies positions

b1, b2, b3, b4

etc at different instants of
time.
The fluid block between the plates deforms and continues to deform as
long as the force is applied.

This experiment shows that a fluid at rest cannot
resist shear stress.

Figure
1
.

Deformation of (a) a solid, and (b) a fluid under a shear force

A liqui
d is a fluid which, when placed in an open container, takes the shape of
the container and has a free surface.

A gas is a fluid which, when placed in a closed container expands to fill the
container. A gas is much less dense than a liquid, i.e. it contain
s fewer
molecules per unit volume.

1.3

Civil Engine
e
ring Hydraulics

The fluid of primary interest to civil
engineers is water
,

and hydraul
ics is
involved in nearly all areas of Civil Engineering either directly or indirectly.
Areas of key interest include:

Water conservation and supply

Wastewater treatment

Harbour and river works

Flood defences

Coastal and ocean engineering

Irrigation, and

Drainage

In this subject we will predominantly focus on flow in pipes and open channels,
however subsequent courses
will examine hydrology, groundwater, coastal
engineering, hydropower and sanitation.

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2.

COMMON PHYSICAL PROPERTIES OF FLUIDS

2.1

Density

The density

(pronounced

roe

)
of a substance is the ratio of the mass of a given
amount of a substance (
m
) to the volume
it occupies (

):

(1)

Where the d
ensity

has SI units of kg/m
3
.

The density of water is typically taken to be 1000 kg/m
3

for engineering
purposes, however the exact density varies with temperature as shown in Fig. 2.

Figure 2.

Variation of the density of water with temperature

The density of water also varies with salinity
. For typical

seawater with a
salinity of 35% a density of 1025 kg/m
3

is generally used.

2.2

Relative Density or Specific Gravity

The ratio of the density of a
substance

to that of pure water at a specified
temperature (often 20�C).

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ࢊࢋ ࢙ ࢚

࢙࢛࢈࢙࢚ࢇ ࢉࢋ

࢝ࢇ࢚ࢋ࢘

Note that the relative density is a ratio of two densities and so is dimensionless.

Example:

What is the
relative density of air if the density of air is taken as 1.23kg/m
3
?

Solution
:

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ࢊࢋ ࢙ ࢚

ࢇ ࢘

࢝ࢇ࢚ࢋ࢘

૛૜
૚૙૙૙

૙૙૚૛૜

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2.3

Unit Weight

The unit weight

of a substance

is

the ratio of the weight of a given amount of a
substance (
W
) to the volume it occupies (

):

(2)

Where the unit weight

has SI units of N/m
3
.

2.4

Pressure

A fluid will always experience pressure as
a result of molecular interactions. A
fluid is made up of a vast number of molecules separated by empty space. The
molecules are in continual movement and the action of innumerable molecular
collisions within the fluid means that any part of the fluid mu
st experience
forces exerted on it by adjo
ining fluid or solid boundaries

(Fig. 3)
. If therefore,
part of a fluid is arbitrarily divided from the rest by an imaginary plane, forces
from the adjacent fluid will act at right angles to that plane

(Fig. 4)
.

Figure 3.

Pressure exerted by particle collisions inside a closed container. A
force is applied by the fluid on the container wall, and an equal and opposite
force is applied by the wall on the fluid.

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Figure 4.

Pressure forces acting at right angles t
o the plane of a fluid element.

The pressure at any point in a fluid
p

is equal to the normal force applied to a
surface element of fluid divided by its area:

(3)

Where the SI unit of pressure is Pascals, which is one Newton per square
metre
(N/m
2
).

Note however, that pressure is also given in atmospheres, where 1 atm
= 1.01325x10
5

Pa.

2.5

Measurement of Pressure

Pressure cannot be measured directly. All instruments that are used to
determine pressure in fact measure a difference in press
ure between two points.

There are two commonly used reference pressures that provide the starting
point of the measurement scale:

a)

Absolute Pressure,

where pressure is given relative to the pressure in a
perfect vacuum.

b)

Gauge Pressure

(or relative pressur
e)
,

where pressure is given
relative to the local atmospheric pressure.

Table 1.

Pressure measurements using absolute and gauge pressure.

Absolute
Pressure (kPa)

Gauge
Pressure (kPa)

Perfect Vacuum

0

-
101.3

Atmospheric Pressure at
Sea Level

101.3

0

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2.6

Compressibility

All
fluids are

compressible to some extent such that a change in pressure will
cause a change in the density of the substance. The compressibility of a fluid is
expressed by its
bulk modulus of elasticity, E
, which is the ratio of the incr
ease
in pressure (
dp
) to the resulting volumetric strain (
dV/V
):

(4)

Note that the sign in equation (4) denotes a negative volume change for a
positive pressure change. Equation (4)

can also be expressed in terms of the
change in
density of the substance (
d

):

(5)

For liquids
E

is much larger than for gases. Water has a bulk modulus of
elasticity
E

= 2.2x10
9

Pa or 2.2GPa at 20�C.

Example

Find the change in volume of one m
3

of water due to an increase in pressure o
f
405 kPa
.

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૚૙

૚૙

૙૙૙૚ ૝

The above example indicates that the change in density

of water

with an
increase in pressure
is very small. As such, water is typically treated as
incompressible

and the density is assumed to be constant. This greatly
simplifies the majority of Civil Engineering hydraulics problems.

2.7

Viscosity

The

viscosity

of a

fluid

is a measure of its

resistance

shear or

tensile stress

and occurs mainly
due to cohesive molecular forces
. For
liquids, it corresponds to the informal notion of "thickness". For example,

oil

has
a higher viscosity than

water
. Gases also have viscosity, though much lower
than that for liquids.

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To illustrate the effect of viscosity, consider two parallel, horizontal plates a
small distance Y apart with a fluid occupying the space between them (Fig. 5).
If the upper plate is moved in the x
-
direction with a
low
velocity
U

and the lower
plate is st
ationary, the molecules in contact with the upper plate will move with
it and those in contact with the lower plate will remain stationary.

Due to inter
-
molecular cohesion and the vertical transfer of momentum of the
rapidly fluctuating molecules, the
fluid between the plates

is subjected to a
forward drag from above and a restraining drag from belo
w. In effect, all of the
fluid undergoes a shearing deformation and offers some resistance to the
shearing motion.

Figure 5.
Velocity and shear stress dis
tribution for viscous motion between a
fixed plate and a parallel, moving plate.

The force
F

that moves the upper plate and the equal and opposite force
restraining the lower plate result in a constant shear stress throughout the fluid
equal to
F/A
, where
A

is the area of the plates.

The fluid velocity will vary with depth, with a velocity gradient of
du/dy
.
Newo’ law of vicoiy ae ha:

(6)

Where

(poouced ‘ew’)
is the
dynamic viscosity

of the fluid. Dynamic
viscosit
y is a fluid property and its SI units are Pa.s.

Fluids that conform to this relationship including air and water are called
Newtonian fluids. For non
-
Newtonian fluids, the viscosity is also dependent on
the rate of shear or the shear rate history. Examp
les of non
-
Newtonian fluids
include toothpaste, ketchup, paint, slurries and shampoo.

For the case in Fig. 5, since

and

are constant,
du/dy

is also constant and a
linear relationship exists between velocity and the distance from the lower
plate.

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For pr
actical purposes, viscosity is independent of pressure changes. The
viscosity of liquids, being primarily due to molecular cohesion, decreases with
increases in temperatures.

At 20�C, the dynamic viscosity of water is approximately 10
-
3

Pa.s.

At 15�C, the

dynamic viscosity of air at atmospheric pressure is 1.8x10
-
5

Pa.s
(about 1/56 the dynamic viscosity of water).

The
kinematic viscosity

(poouced ‘ew’)
of a fluid is the ratio of its
dynamic viscosity to its density:

(7)

The SI
units of kinematic viscosity are m
2
/s.

Example

The space between two parallel plane walls 5mm apart is filled with fuel oil
with a relative density of 0.86 and a dynamic viscosity of 0.0072 Pa.s.

A flat plate, 1.5 m long, 0.25 m wide and 1 mm thick is pull
ed in the direction of
its length, parallel to and midway between the walls at a velocity of 0.4 m/s.

Find the force required to pull the plate.

A:

଴଴ଶ

ʹͲͲ

Ͳ

ͲͲ ʹ

ʹͲͲ

ͳ

Note

there will be a shear force

on BOTH sides of the plate

ʹ

ʹ

ͳ

ͳ

Ͳ

ʹ

ͳ

Ͳ

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2.8

Surface Tension

The cohesive forces between liquid molecules at depth within a liquid are shared
with all neighbouring molecules so there is no net force on each molecule. Those
on
the surface however have no neighbouring molecules above and a
s a
consequence, these molecules cohere more strongly to the other molecules at the
surface.
The effect of this is similar to a membrane or skin covering the surface:
with the surface capable o
f sustaining a tensile force of

N/m.

The formation of
hi ‘uface fil’ which ake i oe difficul o ove a objec houh he
surface than when completely submerged.

Figure 6.
Intermolecular forces within a liquid.

Surface tension is the reason

that water forms into droplets rather than
Intermolecular forces

mean that a small un
-
enclosed volume of a
liquid

tend
s

to be
pulled into a spherical shape by the cohesive forces of the
surface layer.

Surface tension is typically measure
d in dynes/cm, though the SI units are
mN/m, where 1 dyn
e
/cm = 0.001 N/m.

2.9

At a junction of a solid, a liquid and a gas, such as where the free surface of a
liquid meets a solid boundary, the angle of contact of the liquid
-
gas interface
with the solid surface depends on the relative magnitudes of
the
liquid molecu
les to the solid molecules and the cohesion of the
liquid

molecules
to each other.

If the adhesion exceeds the cohesion, the solid surface is said to be completely
wetted and the contact angle is shown in Fig. 7a. This is the case with water
and a clean

glass surface.

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If the cohesion greatly exceeds the adhesion, there is negligible wetting and the
contact angle is shown in Fig. 7b. Mercury and clean glass form such a contact,
the angle

being approximately 135�.

Figure 7,
Wetted

and non
-
wetted

soli
d surfaces.

2.10

Capillarity

Surface tension and adhesion result in the phenomenon of capillary rise of
liquids in narrow tubes and passages in porous materials. If a tube is
sufficiently narrow and the liquid adhesion to the walls is sufficiently strong,
men
iscus will form and surface tension can support the weight of liquid in the
tube and draw the liquid up the tube

(Fig. 8)
. The height that the column
of
water
is lifted to
is known as the capillary rise
h

and

is given by:

(8)

where
r

is the radius of the tube and

is the contact angle between the fluid and
the wall.

Figure 8.
Capillary rise.

Note that if

is greater than 90�, as with mercury in a glass container, the
liquid will be depressed rather than lifted.

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CHEAT SHEE
T

1.

A

fluid i “a ubace ha coiuouly defo o flow ude a
applied hea e.”

2.

Density (kg/m
3
)

3.

Relative density (dimensionless)

࢙࢛࢈࢙࢚ࢇ ࢉࢋ

࢝ࢇ࢚ࢋ࢘

4.

Unit weight (
N/m
3
)

5.

The pressure at any point in a fl
uid
p

is equal to the normal force applied
to a surface eleme
nt of fluid divided by its area

6.

Water is typically assumed to be incompressible for engineering
hydraulics applications.

7.

Newo’ law of vicoiy ae ha fo a Newoia fluid

a.

Dynamic viscosity

has units of Pa.s

b.

Kinematic viscosity

has units of m
2
/s and

8.

Surface tension

has units of mN/m and is caused by a net inward
intermolecular force at the surface of a liquid.

9.

Surface tension and adhesion result i
n capillary rise, where

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FLUID PROPERTIES
TUTORIAL QUESTIONS

1.

Approximate the capillary rise of water (in contact with air where

=0.072 N/m) in a clean glass tube 1, 3, 5 and 7 mm in

assuming a
contact angle of 30�
.

[0.
0
127m;
0.
0
042m; 0.
0
025m; 0.
0
018m]

2.

What pressure must be applied to water to reduce its volume by 1%
(Assume a bulk modulus of elasticity of 2.17x10
6

kPa)
. [21700 kPa]

3.

The density of an oil is 850 kg/m
3
.

Find its relative density and kinematic
viscosity if the

dynamic viscosity is 5x10
-
3
?

[0.85; 5
.88
x10
-
6

m
2
s
-
1
]

4.

5.6 m
3

of oil weigh
s 46800N. Find its mass density

and relative density?

[852 kg/m
3
; 0.852]

5.

The space between two parallel plane walls 8mm apart is filled with fuel
oil with a relative density of

0.76 and a dynamic viscosity of 0.0072 Pa.s.

A flat plate, 1.0 m long, 0.2 m wide and 1 mm thick is pulled in the
direction of its length, parallel to and midway between the walls at a
velocity of 0.6 m/s.

Find the force required to
move

the plate.

[
0.49
2

N]

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6.

A weighted cylinder falls at a constant velocity of 50 mm/s inside a larger
diameter cylinder and there is a layer of oil between the two cylinder
walls (see figure).

The inner cylinder has a weight of 15N, length of 200 mm and an outer
diameter of

mm.

Calculate the

dynamic

viscosity of the oil.

[0.794 Pa s]

7.

The velocity distribution of a viscous fluid (dynamic viscosity

= 0.9
Ns/m
2
) flowing over a fixed plate is given by
u =
0.68
y

y
2

(
u

is velocity in
m/s and
y

is the distance from the plate in metres).

What are the shear stresses at the plate surface,
y

= 0 m and
y

= 0.34 m?

[0.612 N/m
2
;
0 N/m
2
]

8.

In a fluid, the velocity measured at a distance of 75 mm from the
boundary is 1.125
m/s. The fluid has dynamic viscosity of 0.048 Pa s and
a relative density of 0.913. What is the velocity gradient and shear stress
at the boundary assuming a linear velocity distribution?

[15 s
-
1
; 0.720 Pa
]

9.

A flat, cylindrical container
,

302mm in diam
eter and

4mm deep is filled
with a liquid with a kinematic viscosity of 1.19 � 10
-
3 m2/s and relative
density of 1.26. A circular disc 300mm in diameter is placed in contact
with the liquid surface and rotated at 30 revolutions per minute.
A
ssuming that

centrifugal effects are negligible, find the torque required
to rotate the disc

(Recall that torque,
T
=
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