1. FLUID MECHANICS
Department of Nuclear Engineering and Fluid Mechanics
University College of Engineering
University of Basque Country (EHU/UPV)
Vitoria-Gasteiz
Instructor: Iñigo Errasti Arrieta

2. CONTENTS
LESSON 1. INTRODUCTION
LESSON 2. FLUID STATICS
LESSON 3. FLUID KINEMATICS
LESSON 4. FLUID DYNAMICS
LESSON 5. THE ENERGY EQUATION
LESSON 6. APPLICATIONS OF BERNOULLI EQUATION
LESSON 7. LINEAR MOMENTUM THEOREM
LESSON 8. DIMENSIONAL ANALYSIS AND SIMILITUDE
LESSON 9. INCOMPRESSIBLE VISCOUS FLOW
LESSON 10. ENERGY LOSSES IN PIPES
LESSON 11. STEADY-STATE FLOW IN PIPES
LESSON 12. TRANSIENT REGIMES IN PIPES
LESSON 13. FLOW THROUGH OPEN CHANNELS
LESSON 14. PUMPS AND TURBINES

3. LESSON 1. INTRODUCTION TO FLUID MECHANICS
1. Field of application of Fluid Mechanics
2. Brief history of Fluid Mechanics
3. Fluid as a continuum. Fluid definition
4. Dimensions and Units
5. Operators
6. Physical properties of fluids

4. 1. Field of application of Fluid Mechanics
“Fluid Mechanics”, definition
Physical phenomena in nature
Engineering
Other aspects in common life
Main branches:
• Statics
• Kinematics
• Dynamics
• Aerodynamics
• Computational Fluid Dynamics (CFD)

5. 1. Field of application of Fluid Mechanics
Weather & climate
Vehicles
Environment

6. 1. Field of application of Fluid Mechanics
Physiology and medicine
Sports & Recreation

7. 2. Brief history of Fluid Mechanics
Archimedes
Mariotte, Torricelli, Pascal, Castelli
Newton, Bernoulli, Euler, D’Alembert
Chezy, Navier, Coriolis, Darcy
Pouiseuille, Hagen, Reynolds, Stokes
Froude, Francis, Pelton, Herschel
Thomson, Kelvin, Rayleigh, Lamb
Prandtl, von Karman, Blasius
Taylor, Kolmogorov, Nikuradse

8. 2. Brief history of Fluid Mechanics
Archimedes Newton Leibniz Bernoulli Euler
(287-212 BC) (1642-1727) (1646-1716) (1667-1748) (1707-1783)
Navier Stokes Reynolds Prandtl Taylor Kolmogorov
(1785-1836) (1819-1903) (1842-1912) (1875-1953) (1886-1975) (1903-1987)

9. 3. Fluid as a continuum. Definition of fluid
Definition of fluid
Comparison to solid
States of matter (liquid and gas)
Modelling the fluid as a continuum

10. 3. Fluid as a continuum. Definition of fluid
Comparison to solid
Time t0 Time t1 Time t2
- Deformation of solid: F
r
F
Ф1 r
F
Ф2 = Ф1 r
F
invariable with time τ=
Solid Solid Solid
A
τ : Shear stress
Time t0 Time t1 Time t2
F: Shear force
- Deformation of fluid:
r Ф2 > Ф1 r A: Contact area
r Ф1
F continuous with time
F F
Fluid Fluid Fluid
Figure 1.1. Deformation of solids and fluids
Figure 1.2b. Molecules are at relatively fixed positions in a solid. Figure 1.2a. Unlike a liquid, a gas does not
Groups of molecules move about each other in the liquid phase. form a free surface and it expands to fill the
Molecules move around at random in the gas phase (Cengel-Cimbala) entire available space (Cengel-Cimbala)

11. 3. Fluid as a continuum. Definition of fluid
The model of the continuum
- Differential volume range for analysing the fluid as a continuum
- Definition of the density in a fluid as a continuum
Figure 1.3. Differential volume in a fluid Figure 1.4. Density calculated in function of
region with varying density (from White) the differential volume (from White)
δm
ρ = lim
δV →δV * δ 0V

12. 4. Dimensions and Units
Dimensions, magnitudes and units
Primary dimensions
Secondary dimensions
SI and English systems
Conversion ratios
Some SI units: English units:
mass: kg pound-mass (lbm)
length: meter foot (ft)
time: second second (s)
force: newton pound-force (lbf)
work: joule British thermal unit (btu)

13. 4. Units
Prefixes, dimensions and unity conversion factors
Table 1.1. Prefixes
Table 1.2. Units in the SI and US system
Table 1.3. Unity conversion factors

14. 5. Operators
• Gradient of a scalar function f:
r r ∂f ( x, y, z ) r ∂f ( x, y, z ) r ∂f ( x, y, z ) r
gradf ( x, y, z ) = ∇f = i+ j+ k
∂x ∂y ∂z
r
• Divergence of a vector function f :
r r r ∂f x ( x, y , z ) ∂f y ( x, y, z ) ∂f z ( x, y, z )
divf ( x, y, z ) = ∇f = + +
∂x ∂y ∂z
r
• Curl of a vector function f :
r r r
i j k
r r r r ∂ ∂ ∂
cu rlf ( x, y, z ) = ∇xf =
∂x ∂y ∂z
f x ( x, y , z ) f y ( x, y , z ) f z ( x, y , z )

15. 6. Physical properties of fluids
1. Density, specific weight and specific gravity
2. Pressure
3. Ideal gas equation of state
4. Compressibility
5. Viscosity
6. Vapour pressure. Saturation pressure. Cavitation
7. Surface tension and capillary effect

16. 6.1. Density and specific weight
Density (definition, dimensions, units)
Specific weight (definition, dimensions, units)
Relative density or specific gravity (definition)

17. 6.1. Density, specific weight and specific gravity
• Density (definition)
m dm
ρ= ρ= dV
V dV dm
Mass per unit volume (kg/m3) V
Figure 1.5a. Concept of density of a fluid
1
• Specific volume (definition) Vs =
ρ
• Specific weight (definition) γ = ρg
ρ
SG =
• Specific gravity or relative density (definition)
ρH O 2

18. 6.1. Density, specific weight and specific gravity
Figure 1.5b. Approximate physical properties of common liquids at atmospheric pressure

19. 6.2. Pressure
Definition (2 forms)
Dimensions. Units
Pressure level with different references

20. 6.2. Pressure. Units
• Definition (2 forms)
Atmosphere Ratio of normal force (Fn)
to area at a point
Fn
A
Fn
P=
A
Figure 1.6. Pressure in plane A
r
dA ΔFn dFn
r P = lim =
dFn ΔA→0 ΔA dA
1 Pa= 1 Nw m-2
A 1 baria = 1 dyne cm-2
V
1 atm = 760 mmHg=1.013 x 105 Pa = 10.33 mwc
Figure 1.7. General concept of pressure = 2116 lbf ft-2
1 psi = 6895 Pa

21. 6.2. Pressure
• Pressure references
Absolute pressure (pabs): pressure relative to
absolute zero, absolute vacuum, p = 0 Pa.
pabs > patm ; pgauge > 0
(overpressure) Gauge pressure (pgauge): pressure relative to
the local atmospheric pressure.
pabs = patm ; pgauge = 0
pabs = p gage + patm
pabs < patm ; pgauge < 0
• Example 1:
(vacuum, suction)
A gage pressure of 50 kPa recorded in a location
where the atmospheric pressure is 100 kPa is
pabs = 0 expressed as either
p= 50 kPa gage or p=150 kPa abs
• Example 2:
Figure 1.8. Pressure references

22. 6.3. Ideal gas equation of state
• Definition of the equation of state
• Equation of state for an ideal gas
pV = nRT R = 8.314 J / Kmol K Universal gas constant
• Other forms of the equation of state
⎡R⎤
p = ρR * T R* = ⎢ ⎥
⎣M ⎦
(J/kg K) M: molecular weight (Kg/Kmol) of the gas
p = γR ' T ⎡ R ⎤
R' = ⎢ ⎥
⎣ Mg ⎦
(m/K)

23. 6.4. Compressibility and elasticity
Parameters:
dp
E =−
• Bulk modulus of elasticity E: (dV / V ) 1
E=
(dV / V ) α
• Compressibility α: α=−
dp
F
A
p=F/A ; V
Increase in F
(p + dp) ; (V + dV)
V Negative value (decrease)
Figure 1.9. Decrease in volume by an increase in pressure

24. 6.4. Compressibility and elasticity
Table 1.4 Values of bulk modulus of elasticity for some liquids
Liquid E (GPa)
Water 2,07
Ethanol 1,21
Benzene 1,03
Carbon tetrachloride 1,10
Mercury 26,20
Ideal gases: E = kp
Newton's formula for speed of sound c2 =
dp E
=
(Newton – Laplace equation): dρ ρ

25. 6.5. Viscosity
1. Viscosity: Dynamic and kinematic
2. Newton’s law of viscosity
3. Rheological diagram
4. Dependence on pressure and temperature
5. Viscometers

26. 6.5. Viscosity
Definition
Physical phenomena causing viscosity:
• Intermolecular cohesion:
r
r U2
U
r 2 τ
U1 r
U1
Figure 1.10. Influence of intermolecular cohesion on viscosity
● Collisional exchange of momentum:
r r
U2 U2
τ
r
U1 r
U1
Figure 1.11. Influence of momentum exchange on viscosity

27. 6.5. Newton’s law of viscosity
• Velocity profile, shear stress, deformation:
τ τ
y U + dU U + dU
ds
dθ
dV dy
dV dV
U U
μ: Dynamic viscosity
τ τ
U=0
τ F: Shear force
Time (t) Time (t + dt) A: Contact area
Figure 1.2 Deformation of a fluid element
F dθ dU
● Shear stress: τ= ● Deformation rate: =
dt dy
A
dθ dU dθ dU
• Newton’s law of viscosity: τ=μ =μ F = μA = μA
dt dy dt dy
• Inviscid flow hypothesis (ideal fluid)
μ =0 τ =0

28. 6.5. Rheological diagram
Newtonian fluid dθ dU μ: Dynamic viscosity
τ=μ =μ
Pseudo plastic fluid dt dy
Dilatant fluid
Ideal plastic
Ideal fluid
Elastic solid
Ideal plastic (Bingham) Dilatant τ = k ( gradU ) , n > 1
n
τ (N/m2) r 1
gradU = (τ − τ 0 )
μ
Newtonian
Plastic
Elastic Pseudo plastic τ = k ( gradU )n , n < 1
solid
Ideal fluid
dθ dU −1
= (s )
dt dy
Figure 1.13 Rheological diagram

29. 6.5. Dynamic and kinematic viscosity. Unities
Dynamic viscosity (absolute viscosity)
dθ dU
• Newton’s law of viscosity: τ=μ =μ
dt dy
SI system: : 1 Pa s = 1 Poiseuille = 1 Nw m-2s
CGS system: 1 poise = 1 dyne cm-2s
Kinematic viscosity: μ
ν=
ρ
SI system: : 1 m2s-1
CGS system: 1 cm2s-1 = 1 stoke

30. 6.5. Dependence on pressure and temperature
Effect of temperature
μ (Pa·s) μ↓ as T↑ ν↓ decreases when T↑
- Liquid:
ρ ≈ Cte (incompressible)
Gas
- Gas: μ ↑ con T↑
ν↑↑ increases when T↑
Liquid ρ ↓ con T↑ ( p = γR’T )
T (K) Sutherland correlation for viscosity of gases
Figure 1.14 Dynamic viscosity of gases and liquids as a function of temperature
Effect of pressure
μ ≈ Cte μ ≈ Cte (∆p not excessive)
- Liquid: ν ≈ Cte - Gas: ν↓ decreases when p↑
ρ ≈ Cte (incompressible) ρ ↑ as p↑

31. 6.5. Dependence on pressure and temperature
(Figures 1.15, 1.16 from White)

32. 6.5. Viscometers
a) b)
Engler viscometer
Redwood viscometer
Brookfield viscometer
Falling sphere viscometer
c) d)
Figures 1.17 Schematic of a) Engler, b) Redwood, c) Brookfield and d) falling sphere-type viscometers

33. 6.6. Vapour pressure and saturation pressure
Reviewing the concepts of vapour pressure and saturation pressure
Figure 1.18. Phase diagram

34. 6.6. Vapour pressure and saturation pressure
Cavitation:
Process of formation of the vapour of liquid when it is subjected to reduced
pressure at constant ambient temperature
Gaseous and vaporous cavitation
Cavitation number:
p − p sat p↓ , T p↑, T
Ca = p, T boiling implosion
1
ρU 2
2
p < psat (T) p > psat (T)
Figure 1.19. Schematic of the cavitation process

35. 6.6. Vapour pressure and saturation pressure
Figure 1.20. Consequence of cavitation damage on Figure 1.21. Consequence of cavitation damage
an impeller of a pump on an impeller of a pump. Detail
Figure 1.22. Consequence of cavitation damage on Figure 1.23. Consequence of cavitation damage on
an impeller of a pump. Profile view an impeller of a pump. Profile view

36. 6.7. Surface tension and capillary effect
Surface tension
Figure 1.24. Forces acting on a liquid at the
surface and deep inside (Cengel-Cimbala).
F
σ=
l
Capillary rise
2σ
h= cos φ
ρgR Figure 1.25. Capillary rise and fall of water
and mercury in a small diameter glass tube
(Cengel-Cimbala).
Figure 1.26. Forces acting on a liquid column that has risen in a tube.