UCL CIVIL ENVIRONMENTAL and GEOMATIC ENGINEERING DEPARTMENT
CEGE1009 MECHANISMS
1st YEAR FLUID MECHANICS
INTRODUCTION TO FLUID MECHANICS
Objectives:
*
Context
- examples where Civil & Environmental Engineers need to understand fluids
*
About fluids
- why they are different from solids
- why they are important for other scientists and society in general
*
Hydrostatics
- the study of forces exerted by stationary fluids
- Pascal's Law
- Archimedes
- buoyancy and flotation
- absolute and gauge pressure
- Ideal gases
- forces on submerged structures
- manometers
*
The Continuity principle
*
Frictionless flows
- Bernoulli's equation (static, dynamic and stagnation pressures, Cp)
- Pitot tubes and flow measuring devices (orifice, venturi, etc.)
- Coefficients of discharge
Communications:
*
Notes will be distributed during lectures. However, if you lose them, they should be available
through Moodle.
Reference Texts:
*
Massey, B. S. (2006) Mechanics of fluids. Revised by John Ward-Smith. 8th edition.
Taylor and Francis
Library location: ENGINEERING GH 5 MAS
*
Hamill L. (2001) Understanding Hydraulics. 2nd ed. Palgrave
Library location: ENGINEERING CP 50 HAM
*
Douglas, D.F., Gasiorek, J.M., Swaffield, J.A. and Jack, L.J. (2005) Fluid Mechanics, 5th edition.
Prentice Hall:
Library location: ENGINEERING GH 5 DOU
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1.
Introduction:
i) Lecturers:
Prof Richard Simons [Ground floor Mezzanine GM13]
Dr Eugeny Buldakov [GM02]
ii) Methods:
Lectures – two per week
Problem classes – one hour per week
Lecture notes
Problem sheets
If you have difficulty with notes, question sheets, coursework, or past
exam papers, ask the relevant lecturer for help. If they are not in their
room when you call, send an e-mail and ask for an appointment.
iii) Assessment:
End of Session examination – Mechanisms (with Soils)
Coursework [laboratory reports]
iv) Semantics:
The subject of Fluid Mechanics applies to any gas or liquid. The term
"Hydraulics" is sometimes (mis)used to describe the same topic but
should, strictly, be applied to the study of liquids only.
v) Units:
Answers to numerical calculations should (almost) always have UNITS
written after them.
If asked how far it is from Watford to UCL, the answer 24 would not be
very helpful; 24 miles would make more sense. It is also important to
stick to a consistent set of units: for this course, you are advised to
use only kilogram, metre, and second units. This is recommended
because it is so easy to mix units during a long calculation and thereby
end up with an incorrect solution. [Remember that if you had $20 and
£15 in your wallet, you would not simply add them together and say you
had £35 - you would convert the $20 into £ first. Do the same in all
engineering calculations.]
vi) Dimensions:
All additive terms in an equation must have the same units, or the
equation is meaningless. This is a very useful way of checking if you
have remembered an equation correctly. This technique is known as
"dimensional homogeneity" and will be explained later.
vii) Accuracy:
An observed value is described as "accurate" if it lies close to the true
value. [Do not confuse with "precision".]
viii) Precision:
An observed value is described as "precise" if it is expressed to a large
number of decimal places or significant figures. Quoting many decimal
places does not necessarily make for a correct or accurate value. In fact,
one of the commonest failings of undergraduates when preparing
laboratory reports is to write down a long string of digits from their
calculators despite only having measured the original quantity to 5%.
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1.1
About fluids - and why Civil Engineers need to understand them:
Fluids (liquids and gases) differ from solids in that they cannot offer a permanent resistance to any
deforming force - flowing under the action of such forces for as long as they are applied, however
small they are. A fluid is unable to retain any unsupported shape, flowing under its own selfweight and taking up the shape of any solid body into which it comes into contact.
x


u
h
u
h
Solids:
Under the action of shear:
shear stress t
and strain x/h
are related by a
“modulus of rigidity”
Fluids:
Continuous deformation
for velocity u at surface,
with velocity gradient u/h.
Shear stress and velocity
gradient are related by the
"coefficient of viscosity".
Liquids can form a free surface in a container whose volume is greater than the volume of the
liquid; in contrast, gases can expand to occupy a container completely.
Liquids are generally considered incompressible. Gases are compressible - but are generally
considered incompressible if flowing significantly slower than the speed of sound in the gas.
The study of Fluid Mechanics is widely applicable in such areas as the aerodynamic design of
aircraft, ships and motor cars; heating and ventilation systems; meteorology; astronomy; blood
flow in medicine; chemical engineering; oceanography; coastal and estuarine morphology; ports
and harbours; and gas flow in turbines and engines.
For Civil Engineers in particular, Fluid Mechanics is important for those dealing with: the
capacity of pipelines carrying water, gas, oil and sewage; wind forces and structural vibration in
buildings, cables and masts; wave loading on oil rigs and piers; fluid loading on submerged
structures; canals; reservoirs; flood control in rivers; stream gauging; flood routing; irrigation;
dams, spillways and hydroelectric schemes; groundwater flow to wells; highways drainage; water
treatment plants; and coastal defence with sea walls and breakwaters.
The subject develops the laws of Conservation of Mass, of Energy, and of Momentum.
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1.2
Fluid properties
For the purposes of this course, all fluids are assumed to be continuous media, and molecular scale
variations are ignored. This allows the description: "at a point" to be meaningful in describing
local variations due to the fluid behaviour rather than to any molecular distribution. Hence a
"particle of fluid" or a "small element of fluid" is assumed to contain sufficient molecules to give
an average description of the fluid behaviour. Remember that 10-15 m3 of air contains roughly 3 x
1010 molecules at STP.
_
mean density 
mass of given quantity of substance/volume occupied by this quantity
[kg / m3 ]
density
at point
relative
density


specific volume v
[m3 / kg ]
1.3
density / some reference density (normally taken as water at 40 C)
1 / density
[kg / m.s ]
measure of fluid resistance to an externally applied shear stress 
du
for laminar flow,  =  /dz where z is normal to the flow direction
_
mean pressure p
[Pa = N/m2 ]
force acting perpendicular to plane surface in fluid / area of surface
i.e. the normal stress on that plane
pressure at point
limit of mean pressure as area of plane surface tends to zero
absolute pressure
[bar = 105 Pa]
magnitude of pressure relative to vacuum reference pressure of zero
gauge pressure
magnitude of pressure relative to atmospheric pressure pa
Perfect gas
a gas that obeys the relationship: p =  R T
R
the Gas Constant in joule/Kg K, taken as 287 joule/KgK for air
[note: p, T are absolute values]
viscosity

limit of mean density as volume tends to zero,
assuming molecular uniformity/homogeneity
Pascal's Law
Consider the forces acting on a very small wedge-shaped element of static fluid, sides (dx, dy, dz),
with base angle , the slope looking down the y-axis, and the z-axis vertical. Normal stress 
acts on the sloping face, z acts on the base, and y acts on the vertical end - all at the centres of
the faces. Normal forces on each face are calculated from the product of the normal stress and the
area, and the other force acting is the body force (weight) due to gravity. There is no shear as the
fluid is stationary.
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z


y
z
mg
x
y

y
z
x
For vertical equilibrium of the element, the upward force of the base must exactly balance the
downthrust on the sloping face and the weight of the element:
 z  x  y = (    x  y sec ) cos +
 z   
and
1
 g x yz
2
1
 g z
2
This implies that the difference between z and  is negligible when dz is sufficiently small.
z = 
And in the limit,
Similarly, for horizontal equilibrium of the element parallel with the y-axis:
 y  x  z = (    x  z cosec ) sin 
giving
y = 
Having thus shown that the normal stress  on a small element of fluid acting in an arbitrary
direction is equal to both y and x, and noting that the orientation of the wedge is also arbitrary,
it is clear that the magnitude of the normal stress is the same in ALL directions provided the fluid
is static and there is no shear. In other words, the pressure (normal stress) at a point in a fluid acts
equally and simultaneously in all directions and is a scalar quantity.
1.4
Partial differentiation
Let the quantity q depend on two variables.
q = f (x , y) could be plotted as a surface.
The change in q, denoted by dq1 which arises on moving a small increment dx in the x direction
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only (i.e. keeping y constant) is now written:
 q1 =
q
q
x , where
is the partial derivative of q
x
x
(with respect to x at constant y), that is the gradient of the function at a constant value of y.
Similarly, the change in q, denoted dq2 which arises on moving a small increment dy in the y
direction only (i.e. keeping x constant) is now written:
 q2 =
q
q
y , where
is the partial derivative of q
y
y
(with respect to y at x constant), that is the gradient of the function at a constant value of x.
The total change produced by the two increments is then:
q =  q1 +  q 2 =
q
q
x +
y
x
y
with a similar result if q depends on three variables.
2.
Fluid Statics
2.1
Archimedes principle
The upthrust (or buoyancy) force on an object partially or totally immersed in a fluid is equal to
the weight of the fluid displaced by the object: FB =  V g
Note that this applies both to liquids and to gases,
although in gases FB is often treated as negligible, as
the density is relatively small.
FB
The upthrust must act at the centre of mass of the fluid
displaced, that is, the centroid of volume V if the fluid
is of constant density
2.2
Fundamentals
The pressure variation in a fluid at rest (or with no relative
movement between particles) is described as:
Volume V
p = f (x, y, z)
But what form does this function take?
Consider the equilibrium of small cylindrical elements of fluid
with axes lying parallel to the three orthogonal axes:
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For the horizontal elements, there is no net axial force:
p
x)a  0 and
x
p
pa  ( p  y )a  0
y
pa  ( p 
z
a
y
z
leading to the conclusion that p is independent of x and y,
pressure is constant on any horizontal surface, and isobars
are horizontal in a gravitational field:
p
= 0;
x
a
p
= 0
y
a
For vertical equilibrium,
p
pa  ( p 
z )a  gza = 0
z
p
p
+ g = 0
or
=g
z
z
that is, p decreases with increasing height z.
x
y
x
P δa
δx
(P+∂P/∂x.δx)δa
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CEGE1009 Mechanisms (Fluids): TUTORIAL SHEET 1
Density of water  = 1000 kg / m3
1.
A steel sphere for which  = 7.8 has a diameter of 30 mm. Find its mass, its weight in air and its
weight in water.
[Answer: ??, 1.082 N, 0.943 N.]
2.
A circular disc of glass ( = 2.2) is 3 mm thick, it has a diameter of 50 mm and a uniform layer of
cork ( = 0.15) covers one face. Find the minimum thickness of the cork layer which will prevent
the disc from sinking in water. If the cork thickness is increased to 10 mm, how much cork will
protrude above the water surface?
[Answer: 4.24 mm, 4.9 mm.]
3.
Determine the position adopted by a square slab of material of relative density  = 0.9 when
floating completely immersed in a vessel containing two liquids which do not mix and whose
relative densities are 1.2 and 0.8
[Answer: Liquid interface lies 25% of slab thickness above the slab base.]
4.
An object weighs 10 N in air and 8 N when immersed in water. What is its relative density?
[Answer: 5]
5.
A closed rectangular pontoon of dimensions 12 m x 8 m x 4 m high has a mass of 84,000 kg. Find
the depth of immersion when launched in sea water ( = 1.029). The pontoon is moored to the
seabed by four vertical cables of equal length, one at each corner. Find the load in the cables
required to produce a depth of immersion of 3 m. What is the maximum volume of oil ( = 0.9)
which can be stored in the pontoon without influencing the depth of immersion?
[Answer: 0.85 m, 0.52 MN, 235.6 m3
6.
Water in an 8 m wide canal is 2.5 m deep where it passes over a 12 m span aquaduct bridge.
Estimate the hydrostatic force on the bridge i) when a 30 Mg barge is at centre-span on the
aquaduct, and ii) after the barge has moved off the aquaduct.
[Answer: ???, ???]
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