CV2601/G263, Fluid Mechanics
Dr Chiew Yee Meng,Lecturer/Course Coordinator
Office, N1-1b-53
Tel, 67905256
Email, cymchiew@ntu.edu.sg
Dr Shuy Eng Ban,Lecturer
Office, N1-1a-23
Tel, 67905326
Email, cshuyeb@ntu.edu.sg
TEXT
Munson,B R,Young,D F and Okiishi,T H,Fundamentals of Fluid
Mechanics,4th Edition,John Wiley & Son,2002
REFERENCES
? Shames IH,Mechanics of Fluids,3rd Edition,McGraw-Hill,1992.
? Potter MC and Wiggert DC,Mechanics of Fluids,Prentice Hall,1991,
Course Outline
Fluid Mechanics
12 hrs lectures,5 Tutorials,1 Quiz
We e k
No
Con t e n t L e c T u t Re f
1 De f in it io ns o f f lu id, F lu id pr oper ti e s
F lu id S tatics,B a si c e qu a t io n f or h y d r ost a ti c
pre ss ur e
1
2
C ha p,1
C ha p,2
2 P r e ss ure mea su r e m e nt, Manom e ter
c om put a ti ons
H y d r ost a ti c thrust on a pl a ne surf a c e,
Mag ni tu de a nd ce nt r e of pre ss ure
3
4
T1
C ha p,2
C ha p,2
3 H y d r ost a ti c thrust on c ur ve surf a c e s
B uo y a nc y, S tabil it y o f f l oa ti ng bod ies
5
6
T2
C ha p,2
C ha p,2
4 B a si c f lu id f lo w c onc e pt s,C lassi f ica ti on of
f lo w,S y st e m a nd c ont r ol vol um e, Co nt in ui t y
e quatio n
Ener g y e qu a ti on for st e a d y incomp r e ss ib le f lu id
f lo w
7
8
T3
C ha p,4
C ha p,5
C ha p,3
5
B e r nou ll i’s e qu a ti on a nd it s a ppl ica ti ons,
F lo w me a surin g d e vi c e s
9
10
T4
C ha p,3
C ha p,5
6 Mo m e nt um e quatio n f or st e a d y f lo w,
Appli c a ti ons of the mom e nt um e quatio n, F or c e s
on ob jec ts
11
12
Q1
C ha p,5
C ha p,5
Course Outline
Hydraulics
14 hrs lectures,5 Tutorials,1 Quiz
7 Dim e ns io nal ana l y si s
B ucki n g ham P i t heore m
13
14
T 5
C hap,7
C hap,7
8 S ig ni f ica nc e of c om m on di m e ns io nal g r ou ps
S im il i tu de a nd sca le m od e ls
15
16
T6
C hap,7
C hap,7
9 Conc e pt s o f Boun da r y La y e r, L a m in a r f lo w
betwee n p a r a ll e l p lates
L a m in a r a nd tu r bu lent f lo ws i n p ip e s
17
18
T7
C hap,8
C hap,8
10 En e r g y c on c e pt s i n p ip e f lo ws
Da r c y - W e is ba c h equ a ti on
19
20
T8
C hap,8
C hap,8
11
M oo d y di a g r a m
F r icti on a nd m in or lo ss e s
21
22
T9
C hap,8
C hap,8
12 P r in c ip les o f f lu id m a c hi nes,P e r f orm a n c e
c har a c ter is ti c s o f pu m ps,
S im il a r it y laws,S pe c ific sp e e d and m a c hi ne
s e lec ti on
23
24
Q2
C hap,12
C hap,12
13 S y st e m c ha r a c t e r is ti c s an d m a tchi ng,
C a vi tati on a nd NP S H,
P a r a ll e l and ser i e s o per a ti on s o f pu m ps,
25
26
T10
C hap,12
C hap,12
Assessment
? CA (2 quizes), Up to 30 %
? Final Examination, At least 70 %
? Do Not Skip Lectures
– Highlight Important Topics/Concepts/Equation
– Solve Additional Examples and Past Year
Questions
? 10 Tutorials,Start on Week 2
– Tutorials 1 to 4
– Quiz 1
– Tutorials 5 to 9
– Quiz 2
– Tutorial 10
Why Study Fluid Mechanics?
For Civil Engineers,
? Water Supply, Dams,reservoirs,Treatment and Distribution
network systems
? Drainage and Irrigation Systems, Open Channel Hydraulics
? Environmental Hydraulics, Sewerage Systems,Pollutant
Dispersion Modelling in Air and Water
For Mechanical Engineers,
? Aerodynamics – Airfoil design,Lift and Drag,CFD
? Rotodynamic Machinery – Pumps and Turbines
? Hydraulic/Pneumatic Control Systems
? Industrial Hydraulics, Piping Systems – Gas and Oil
Industries
What is Fluid?
Three States of Matter,
? Solid
? Fluids,
? Liquid
? Gas
Solids,
? Shear Strain = function of Stress
? Body recovers when stress is removed
Fluids,
? Deform continuously when subjected to shear stress
? Rate of Shear Strain = function of Shear stress
? Body does not recover when stress is removed
Differences Between Liquids and Gases
Liquids
? Practically incompressible
? Has a finite volume at given pressure and temperature
Gases
? Highly compressible
? Always expands to fill up container
?
What is Fluid Mechanics?
Application of principles of mechanics to
fluid motion,
? Conservation Laws, Mass,Energy,Momentum
? Newton’s Laws of Motion
? Thermodynamic laws for Gases
Main Areas,
? Fluid Statics, Study of Fluid at rest
? Fluid Kinematics, Study of Fluid motion without
considering forces
? Fluid Dynamics, Study relation between motion and
forces
? Hydraulics, Application of Fluid Mechanics to practical
problems
Dimensions and Units
Fluid characteristics (properties) can be described
qualitatively in terms of certain ‘Basic Dimensions’ or
‘Primary Quantities’,
? Length,L
? Mass,M
? Time,T
? Temperature,?
‘Secondary Quantities’ can be derived in terms of the
‘primary quantities’,e.g.
? Area,L2
? Velocity LT-1
? Density,ML-3
Units are standards for quantitative measurement, m,
s,kg
Forces on a Plane in Fluid
Resultant force FR on a plane can be resolved into,
? perpendicular component FN
? tangential component FT
? Pressure P is defined as:
P = Normal force per unit area = FN / A
? Shear stress ? is defined as:
? = Tangential force per unit area = FT / A
? In Fluid statics,FT = 0,only P exists
FR
FN
FT
Plane Area A
Fluid Properties,Definitions
Density ? = mass/ volume,kg/m3
Specific Volume Vs = 1/?,m3/kg
Specific weight ? = ? g,N/m3
Specific gravity (Relative density) s = ?/?w
Density of water as function of temperature
Viscosity
Shear Deformation/Strain
F
F
y
x = u, t
u
Deformed shape after time t
Original shape at time t = 0
?
F = shear force on top surface of area A
Shear stress ?
= F / A
Shear strain ?
= angular deformation
= x / y = (u.t) / y
Rate of shear strain
= ? / t = (u.t)/(y.t) = u / y
= velocity gradient in direction perpendicular to u (y-
direction)
Newton’s Law of Viscosity
For most fluids,shear stress is proportional to
rate of angular strain,
? ? (?u / ?y)
= ? (?u / ?y)
= ? (u / y)
(Where velocity variation in y direction is linear)
? = property of fluid known as dynamic or absolute
viscosity of fluid,kg/(m.s) or N.s/m2
? = Kinematic viscosity = ? /?,m2/s
Law applies to Newtonian Fluids in laminar
motion
Newtonian Fluids
For Newtonian fluids,
Linear relation
between shear stress
and rate of shearing
strain passing through
origin
Non-Newtonian
fluids
Viscosity of Common Fluids
Linear Velocity Gradient
u/y = V / SV
Stationary plate
S
Moving plate
Moving
piston V
S u/y = V / S
Stationary cylinder
S
R
Rotating
piston
? rad/s
V=R?
u/y = R.? / S
Stationary cylinder
V
S
Moving plate
Moving plate
W u/y = (W+V)/S
Non- Linear Velocity Gradient
y U
Velocity gradient at distance y varies,
?u / ?y = f (y)
Boundary layer flow
Example
1 m/s
F =?
?= 75 mm
Sleeve,? = 75.5 mm
Lubricant ? = 0.19 kg/m.s
L = 150 mm
Velocity gradient u/y = 1/0.00025 = 4000 m/s/m
? = ? (U/y)
= 0.19 x 4000
= 760 N/m2
F = ?,A
= 760 x ?(0.075)0.15
=26.9 N
Piston
Newton’s Law of Viscosity
For Newtonian Fluids in laminar
motion,
? = ? (?u / ?y)
= ? (u / y)
(Where velocity variation is linear)
? = dynamic or absolute viscosity of
fluid,kg/(m.s) or N.s/m2
Example
W = ? g Vol = 1100 (9.81) 0.23 = 86.33 N
? = ? U/y = 0.08 (Vt)/0.000005
F = ?A = 640 Vt = W Sin 10o
Vt = 0.0234 m/s
0.2x0.2x0.2 m cube
? = 1100 kg/m3
10o
Vt =?
0.005 mm gap,oil of ? = 0.08 Ns/m2
W
W Sin ? F
Pressure
Pressure P = normal force per unit area,N/m2
?1 Pascal (Pa) = 1 N/m2
?1 bar = 100,000 N/m2 = 100 kPa
Relative pressures are measured,
?Gauge pressure Pgauge
= pressure above atmospheric Patm
?Absolute pressure Pabs
= Pressure above vacuum
= Pgauge + Patm
Patm ? 1 bar ? 100,000 Pa abs
Vacuum = 0 Pa abs
Compressibility of Liquid
New Pressure P = Po + ?P
New volume V = Vo - ?V
?P ? (- ?V/Vo)
= - K (?V/Vo)
= K (??/ ?o)
where ??is the increase in density and ?o is the original density
K = Bulk modulus of elasticity of liquid N/m2
C = Compressibility C = 1/K
Original pressure Po
Increase by ?P
Original volume Vo
Decrease by ?V
Compressibility of Liquid
oo
V
V
V
dVd
V
dV
V
dV
V
m
dV
V
m
d
V
m
?
??
?
??
??????
?
?
?
?
?
??
?
2
Example
Calculate density of sea water at 200 m below
surface,given density at surface is 1025 kg/m3,
Kwater = 2.3x109 N/m2
(Ans, 1025.896 kg/m3)
Solution,
At 200 m below surface,
?P = ? g h = 1025 x 9.81 x 200
= K ?? / ?
?? = 0.896 kg/m3
?’ = ? + ?? = 1025.896 kg/m
Surface Tension
Liquid surface behaves like an elastic membrane under
tension
Surface tension ? = tension force per m of membrane,N/m
Water droplet of radius R,
Increase in Pressure due to surface tension ?P = 2?/R
?
Liquid jet of radius R,
Increase in Pressure due to surface tension ?P = ?/R
Capillary Rise/Depression
Cohesion,
?Attraction between liquid molecules
Adhesion
?Attraction between liquid molecules and solid
boundary
Capillary rise/depression
h = ( 4? Cos ? ) / ( ? g D)
? = Contact angle
Dr Chiew Yee Meng,Lecturer/Course Coordinator
Office, N1-1b-53
Tel, 67905256
Email, cymchiew@ntu.edu.sg
Dr Shuy Eng Ban,Lecturer
Office, N1-1a-23
Tel, 67905326
Email, cshuyeb@ntu.edu.sg
TEXT
Munson,B R,Young,D F and Okiishi,T H,Fundamentals of Fluid
Mechanics,4th Edition,John Wiley & Son,2002
REFERENCES
? Shames IH,Mechanics of Fluids,3rd Edition,McGraw-Hill,1992.
? Potter MC and Wiggert DC,Mechanics of Fluids,Prentice Hall,1991,
Course Outline
Fluid Mechanics
12 hrs lectures,5 Tutorials,1 Quiz
We e k
No
Con t e n t L e c T u t Re f
1 De f in it io ns o f f lu id, F lu id pr oper ti e s
F lu id S tatics,B a si c e qu a t io n f or h y d r ost a ti c
pre ss ur e
1
2
C ha p,1
C ha p,2
2 P r e ss ure mea su r e m e nt, Manom e ter
c om put a ti ons
H y d r ost a ti c thrust on a pl a ne surf a c e,
Mag ni tu de a nd ce nt r e of pre ss ure
3
4
T1
C ha p,2
C ha p,2
3 H y d r ost a ti c thrust on c ur ve surf a c e s
B uo y a nc y, S tabil it y o f f l oa ti ng bod ies
5
6
T2
C ha p,2
C ha p,2
4 B a si c f lu id f lo w c onc e pt s,C lassi f ica ti on of
f lo w,S y st e m a nd c ont r ol vol um e, Co nt in ui t y
e quatio n
Ener g y e qu a ti on for st e a d y incomp r e ss ib le f lu id
f lo w
7
8
T3
C ha p,4
C ha p,5
C ha p,3
5
B e r nou ll i’s e qu a ti on a nd it s a ppl ica ti ons,
F lo w me a surin g d e vi c e s
9
10
T4
C ha p,3
C ha p,5
6 Mo m e nt um e quatio n f or st e a d y f lo w,
Appli c a ti ons of the mom e nt um e quatio n, F or c e s
on ob jec ts
11
12
Q1
C ha p,5
C ha p,5
Course Outline
Hydraulics
14 hrs lectures,5 Tutorials,1 Quiz
7 Dim e ns io nal ana l y si s
B ucki n g ham P i t heore m
13
14
T 5
C hap,7
C hap,7
8 S ig ni f ica nc e of c om m on di m e ns io nal g r ou ps
S im il i tu de a nd sca le m od e ls
15
16
T6
C hap,7
C hap,7
9 Conc e pt s o f Boun da r y La y e r, L a m in a r f lo w
betwee n p a r a ll e l p lates
L a m in a r a nd tu r bu lent f lo ws i n p ip e s
17
18
T7
C hap,8
C hap,8
10 En e r g y c on c e pt s i n p ip e f lo ws
Da r c y - W e is ba c h equ a ti on
19
20
T8
C hap,8
C hap,8
11
M oo d y di a g r a m
F r icti on a nd m in or lo ss e s
21
22
T9
C hap,8
C hap,8
12 P r in c ip les o f f lu id m a c hi nes,P e r f orm a n c e
c har a c ter is ti c s o f pu m ps,
S im il a r it y laws,S pe c ific sp e e d and m a c hi ne
s e lec ti on
23
24
Q2
C hap,12
C hap,12
13 S y st e m c ha r a c t e r is ti c s an d m a tchi ng,
C a vi tati on a nd NP S H,
P a r a ll e l and ser i e s o per a ti on s o f pu m ps,
25
26
T10
C hap,12
C hap,12
Assessment
? CA (2 quizes), Up to 30 %
? Final Examination, At least 70 %
? Do Not Skip Lectures
– Highlight Important Topics/Concepts/Equation
– Solve Additional Examples and Past Year
Questions
? 10 Tutorials,Start on Week 2
– Tutorials 1 to 4
– Quiz 1
– Tutorials 5 to 9
– Quiz 2
– Tutorial 10
Why Study Fluid Mechanics?
For Civil Engineers,
? Water Supply, Dams,reservoirs,Treatment and Distribution
network systems
? Drainage and Irrigation Systems, Open Channel Hydraulics
? Environmental Hydraulics, Sewerage Systems,Pollutant
Dispersion Modelling in Air and Water
For Mechanical Engineers,
? Aerodynamics – Airfoil design,Lift and Drag,CFD
? Rotodynamic Machinery – Pumps and Turbines
? Hydraulic/Pneumatic Control Systems
? Industrial Hydraulics, Piping Systems – Gas and Oil
Industries
What is Fluid?
Three States of Matter,
? Solid
? Fluids,
? Liquid
? Gas
Solids,
? Shear Strain = function of Stress
? Body recovers when stress is removed
Fluids,
? Deform continuously when subjected to shear stress
? Rate of Shear Strain = function of Shear stress
? Body does not recover when stress is removed
Differences Between Liquids and Gases
Liquids
? Practically incompressible
? Has a finite volume at given pressure and temperature
Gases
? Highly compressible
? Always expands to fill up container
?
What is Fluid Mechanics?
Application of principles of mechanics to
fluid motion,
? Conservation Laws, Mass,Energy,Momentum
? Newton’s Laws of Motion
? Thermodynamic laws for Gases
Main Areas,
? Fluid Statics, Study of Fluid at rest
? Fluid Kinematics, Study of Fluid motion without
considering forces
? Fluid Dynamics, Study relation between motion and
forces
? Hydraulics, Application of Fluid Mechanics to practical
problems
Dimensions and Units
Fluid characteristics (properties) can be described
qualitatively in terms of certain ‘Basic Dimensions’ or
‘Primary Quantities’,
? Length,L
? Mass,M
? Time,T
? Temperature,?
‘Secondary Quantities’ can be derived in terms of the
‘primary quantities’,e.g.
? Area,L2
? Velocity LT-1
? Density,ML-3
Units are standards for quantitative measurement, m,
s,kg
Forces on a Plane in Fluid
Resultant force FR on a plane can be resolved into,
? perpendicular component FN
? tangential component FT
? Pressure P is defined as:
P = Normal force per unit area = FN / A
? Shear stress ? is defined as:
? = Tangential force per unit area = FT / A
? In Fluid statics,FT = 0,only P exists
FR
FN
FT
Plane Area A
Fluid Properties,Definitions
Density ? = mass/ volume,kg/m3
Specific Volume Vs = 1/?,m3/kg
Specific weight ? = ? g,N/m3
Specific gravity (Relative density) s = ?/?w
Density of water as function of temperature
Viscosity
Shear Deformation/Strain
F
F
y
x = u, t
u
Deformed shape after time t
Original shape at time t = 0
?
F = shear force on top surface of area A
Shear stress ?
= F / A
Shear strain ?
= angular deformation
= x / y = (u.t) / y
Rate of shear strain
= ? / t = (u.t)/(y.t) = u / y
= velocity gradient in direction perpendicular to u (y-
direction)
Newton’s Law of Viscosity
For most fluids,shear stress is proportional to
rate of angular strain,
? ? (?u / ?y)
= ? (?u / ?y)
= ? (u / y)
(Where velocity variation in y direction is linear)
? = property of fluid known as dynamic or absolute
viscosity of fluid,kg/(m.s) or N.s/m2
? = Kinematic viscosity = ? /?,m2/s
Law applies to Newtonian Fluids in laminar
motion
Newtonian Fluids
For Newtonian fluids,
Linear relation
between shear stress
and rate of shearing
strain passing through
origin
Non-Newtonian
fluids
Viscosity of Common Fluids
Linear Velocity Gradient
u/y = V / SV
Stationary plate
S
Moving plate
Moving
piston V
S u/y = V / S
Stationary cylinder
S
R
Rotating
piston
? rad/s
V=R?
u/y = R.? / S
Stationary cylinder
V
S
Moving plate
Moving plate
W u/y = (W+V)/S
Non- Linear Velocity Gradient
y U
Velocity gradient at distance y varies,
?u / ?y = f (y)
Boundary layer flow
Example
1 m/s
F =?
?= 75 mm
Sleeve,? = 75.5 mm
Lubricant ? = 0.19 kg/m.s
L = 150 mm
Velocity gradient u/y = 1/0.00025 = 4000 m/s/m
? = ? (U/y)
= 0.19 x 4000
= 760 N/m2
F = ?,A
= 760 x ?(0.075)0.15
=26.9 N
Piston
Newton’s Law of Viscosity
For Newtonian Fluids in laminar
motion,
? = ? (?u / ?y)
= ? (u / y)
(Where velocity variation is linear)
? = dynamic or absolute viscosity of
fluid,kg/(m.s) or N.s/m2
Example
W = ? g Vol = 1100 (9.81) 0.23 = 86.33 N
? = ? U/y = 0.08 (Vt)/0.000005
F = ?A = 640 Vt = W Sin 10o
Vt = 0.0234 m/s
0.2x0.2x0.2 m cube
? = 1100 kg/m3
10o
Vt =?
0.005 mm gap,oil of ? = 0.08 Ns/m2
W
W Sin ? F
Pressure
Pressure P = normal force per unit area,N/m2
?1 Pascal (Pa) = 1 N/m2
?1 bar = 100,000 N/m2 = 100 kPa
Relative pressures are measured,
?Gauge pressure Pgauge
= pressure above atmospheric Patm
?Absolute pressure Pabs
= Pressure above vacuum
= Pgauge + Patm
Patm ? 1 bar ? 100,000 Pa abs
Vacuum = 0 Pa abs
Compressibility of Liquid
New Pressure P = Po + ?P
New volume V = Vo - ?V
?P ? (- ?V/Vo)
= - K (?V/Vo)
= K (??/ ?o)
where ??is the increase in density and ?o is the original density
K = Bulk modulus of elasticity of liquid N/m2
C = Compressibility C = 1/K
Original pressure Po
Increase by ?P
Original volume Vo
Decrease by ?V
Compressibility of Liquid
oo
V
V
V
dVd
V
dV
V
dV
V
m
dV
V
m
d
V
m
?
??
?
??
??????
?
?
?
?
?
??
?
2
Example
Calculate density of sea water at 200 m below
surface,given density at surface is 1025 kg/m3,
Kwater = 2.3x109 N/m2
(Ans, 1025.896 kg/m3)
Solution,
At 200 m below surface,
?P = ? g h = 1025 x 9.81 x 200
= K ?? / ?
?? = 0.896 kg/m3
?’ = ? + ?? = 1025.896 kg/m
Surface Tension
Liquid surface behaves like an elastic membrane under
tension
Surface tension ? = tension force per m of membrane,N/m
Water droplet of radius R,
Increase in Pressure due to surface tension ?P = 2?/R
?
Liquid jet of radius R,
Increase in Pressure due to surface tension ?P = ?/R
Capillary Rise/Depression
Cohesion,
?Attraction between liquid molecules
Adhesion
?Attraction between liquid molecules and solid
boundary
Capillary rise/depression
h = ( 4? Cos ? ) / ( ? g D)
? = Contact angle
