1. BASIC CONCEPTS
1.1 INTRODUCTION
1.2 FLUID DEFINITION
1.3 FLOW PROPERTIES
1.4 HYDROSTATIC FORCES ON SURFACES
1.5 VELOCITY AND ACCELERATON FIELD
1.6 FLOW DESCRIPTION
1.7 FLOW CLASSIFICATION
1.8 PROPOSED EXERCICES
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1.1 INTRODUCTION
The aim of the subject of Fluid Mechanics is to study the interaction of
moving fluids with the enviroment: force genreation, los of energy...
This course can be complemented with others:
•
Computational Fluid Dynamics – Bachelor’s degree in Industrials Tecnologies
•
Distribution Systems in Pipes- Bachelor’s degree in Industrials Tecnologies
•
Project 1 and 2– Study of forces around foies/ Aerodynamic improvement of a car
•
Aerodynamics - Bachelor’s in Industrials Tecnologies
•
Aerodynamics – Master’s in Automotive
•
Hydraulic Machines– Master’s in Industrial Engineering
•
Hydraulic and Marine Energy - MSc SENSE - Smart Electrical Networks and Systems
•
Renewable Energies– Double Master’s degree in Industrial and Nuclear Engineering
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1.2 FLUID DEFINITION
STATES OF
MATTER
SOLID: It can withstand a tangencial strain, experiencing statitc deformation.
FLUID: It cannot resist a tangential strain  and is continuosly deformed over
time.
LIQUIDS
•
Molecules are closely grouped
•
Huge cohesive forces.
•
Tend to keep the volume.
•
Form a free surface in a gravitational field.
•
Molecules are separated from each other.
•
Choesive forces negligible.
•
No volume defined.
•
Do not form a free surface
FLUID TYPES
GASES
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1.2 FLUID DEFINITION
Fluid as a continuos medium
HYPOTHESIS
ADVANTATGES
CHECK
Fluid is distributed continuosly in space.
Its properties can be considered as variables in space and differential
calculation can be used for analysis.
Consider fluid density:
m
V 0 V
  lim
If ΔV -> 0 there is a límit volume ε below which
molecular variations can be important (ρ varies
discontinuosly).
 = 10-9 mm3 << usual physical dimensions
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1.3 FLUIDDYNAMICS PROPERTIES
• Flow properties
c
velocity
p
pressure

density
T
temperature
• Work, heat and energy properties
û
internal energy
h
enthalpy
s
entropy
cp, cv specific heat
• Friction and heat conduction properties (transport coefficients)

viscosity
k
thermal conductivity
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1.3.1.- PRESSURE
•
Normal force (compression) at a point in a fluid.
•
Pressure gradient  Fluid movement
pabsolute = patmosferic + pmanometric
Fn
A0 A
p  lim
101300 Pa
1,013 bar
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•
Vapor pressure: pressure at which líquid boils and is in balance with its own
vapor.
Pv (H2O, 20 ˚C) = 2337 Pa
•
Pv (H2O, 100 ˚C) = 101300 Pa
Cavitation : occrus when the pressure of a líquid approaches the vapor pressure ,
at constant temperature, and vapor bubbles appear in the líquid.
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1.3.2.- TEMPERATURE.
•
Related to the internal energy level of the fluid
•
Temperature gradient  heat transfer
•
Absolute temperature (Kelvin): K = ˚C + 273.16
1.3.3.- DENSITY.
• Mass per volume unit

m
lim
V  0 V
• Relative density d = fluid / referència
References: air (1.2 Kg/m3) or water (1000 Kg/m3 )
• Specific weight: weight per volume unit
  g
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1.3.4.- VISCOSITY.
Quantitative measurement of the resistance of a fluid to deform (flow).
Determines the rate of deformation of the fluid when a tangential strain  is
applied.
du· dt
dy

u= du
d
tgd 
dx
u=0
Speed profile
dudt d du
;

dy
dt dy
d du


dt
dy
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
Plastic fluid: it needs to exceed a threshold shear strain
level to start flowing. E.g.: toothpaste...
Dilatant fluid: It worsens its fluidity with shear rate. E.g.: cornstarch
Newtonian fluid: the shear strain applied is
proportional to the shear rate. E.g.: air, water,
most of oils...
Pseudoplastic fluid: The more shear strain
applied, the less viscous it becomes E.g.: Ketchup
du/dy
Newtonian fluids
• μ = Dynamic viscosity
du
 
dy
Units
𝑺𝑰 ∶
𝑲𝒈
𝒎·𝒔
𝑪𝑮𝑺 ∶
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𝒈
𝒄𝒎·𝒔
𝑴
𝑳·𝑻
𝑷𝒂 · 𝒔
𝑷𝒐𝒊𝒔𝒆
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The fluids viscosity depends on pressure and temperature:
• The fluids viscosity increases weakly with pressure.
• The viscosity of líquids decreases with temperature meanwhile it
increases in gases.
Viscosity generation mechanisms
Kinematic viscosity:



Units
𝑳𝟐
𝑻
𝒎𝟐
𝑺𝑰 ∶
𝒔
𝑪𝑮𝑺 ∶
𝒄𝒎𝟐
𝒔
𝑺𝒕𝒐𝒌𝒆
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1.3.5.- SURFACE TENSION.
It appears in the interface between two inmiscible fluids due to the
forces of attraction between the molecules found in the interphase.
  the contact angle between the interface and a solid surface.
  Surface tension coeficient.
The effect of surface tension, , and the contact angle, can
cause the líquid to rise or drop inide a capillary tube.
Depending on the affinity of the líquid with the sòlid:
•
Non surfactant behaviour: characterized by small angle
vàlues. The fluid would rise through in a capillary duct.
•
Surfactant behaviour: characterized by large angle values.
The fluid would go down in a capillary duct.
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1.4 HYDROSTATIC FORCES ON SURFACES
In a fluid without movement there are no deformations and therefore, there are
no shear straints .
The pressure at any point in a fluid without movement is a scalar, independent
of the direction (Law of Pascal).
The pressure at one point of a constant density fluid is:
𝑃 ℎ = 𝑃0 + 𝛾ℎ
being:
- h the difference in height between the two points.
- 𝛾=·g the specific weight
Forces on surfaces:
𝐹𝑛 =
𝑃 𝑑𝐴
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The horitzontal force on a submerged surface is:
𝐹𝐻 = 𝜌𝑔
À𝑟𝑒𝑎
𝑦𝑑𝐴𝑥 = 𝜌𝑔𝑌𝐶𝐷𝐺 𝐴𝑥 𝐴𝑥
being: - 𝑌𝐶𝐷𝐺 𝐴𝑥 : the depth of the projected surface center of gravity
- 𝐴𝑥 the projected surface in X direction
The point of application of the horitzontal force will be the center of gravity of the pressure
diagram on the projected surface.
The vertical force on a submerged surface is:
𝐹𝑉 = 𝜌𝑔
À𝑟𝑒𝑎
𝑦𝑑𝐴𝑦 = 𝜌𝑔 𝑉
essent: - 𝑉: 𝑓𝑙𝑢𝑖𝑑 𝑣𝑜𝑙𝑢𝑚𝑒 𝑎𝑏𝑜𝑣𝑒 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒
- 𝐴𝑦 the projected surface in Y direction
The point of application of the vertical force will be the center of gravity
of the volume of fuid above the Surface.
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1.5 VELOCITY AND ACCELERATION FIELDS
Reference systems:
• Lagrange: Follows the individual particles in motion. In fluid mechanics it is
complicated given the large number of particles involved in the flow.
• Euler: Consider the flow property fields. It identifies points in space and oberves
the properties of particles that pass through each point.
One of the most important property of a flow is the velocity field that is defined in a Cartesiar
and Eulerian reference system with the velocity vector vector:
z
C(x,y,z,t) = u(x,y,z,t)·i + v(x,y,z,t) ·j + w(x,y,z,t) ·k
dc
c(t)
c(t+dt)
y
x
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• Acceleration: is the total derivative of the velocity vector over time.
𝑎=
𝑑𝑐
𝑑𝑡
=
𝑑𝑢
𝑖
𝑑𝑡
+
𝑑𝑣
𝑗
𝑑𝑡
+
𝑑𝑤
𝑘
𝑑𝑡
Knowing:
𝑑𝑢
𝑑𝑡
=
𝜕𝑢
𝜕𝑡
+
𝜕𝑢 𝑑𝑥
𝜕𝑥 𝑑𝑡
+
𝜕𝑢 𝑑𝑦
𝜕𝑦 𝑑𝑡
+
𝜕𝑢 𝑑𝑧
𝜕𝑧 𝑑𝑡
𝑑𝑥
=𝑢
𝑑𝑡
𝑑𝑣
𝑑𝑡
=
𝜕𝑣
𝜕𝑡
+
𝜕𝑣 𝑑𝑥
𝜕𝑥 𝑑𝑡
+
𝜕𝑣 𝑑𝑦
𝜕𝑦 𝑑𝑡
+
𝜕𝑣 𝑑𝑧
𝜕𝑧 𝑑𝑡
𝑑𝑦
=𝑣
𝑑𝑡
𝑑𝑤
𝑑𝑡
=
𝜕𝑤
𝜕𝑡
𝑎=
𝑑𝑐
𝑑𝑡
+
=
𝜕𝑤 𝑑𝑥
𝜕𝑥 𝑑𝑡
𝜕𝑐
𝜕𝑡
+ 𝑢
+
𝜕𝑐
𝜕𝑥
𝜕𝑤 𝑑𝑦
𝜕𝑦 𝑑𝑡
+ 𝑣
+
𝜕𝑐
𝜕𝑦
𝜕𝑤 𝑑𝑧
𝜕𝑧 𝑑𝑡
+ 𝑤
𝑑𝑧
=𝑤
𝑑𝑡
𝜕𝑐
𝜕𝑧
=
𝜕𝑐
𝜕𝑡
+ 𝑐·𝛻 𝑐
Being:
•
Local acceleration: variation of vwelocity over time.
• Convective acceleration: variation of velocity due to geometry changes.
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1.6 FLOW DESCRIPTION
In order to study flows there are a number of analytical and experimental tools
detailed below.
1.6.1.- STREAM LINE: it is a line that at a given instant is tangential to the
velocity vector at any point.
It is a mathematical concept, it can not be generated experimentally.
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1.6.2.- PATHLINE: Path followed by a particle during a period of time.
Trajectories of particles under a water free surface with waves.
1.6.3.- STREAK LINE: geometric locations of particles that in consecutive
instants of time passed through a certain point.
Streak lines generated by the continuos
emission of smoke behind a cylinder.
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In stationary regime
𝜕
𝜕𝑡
= 0 Streaklines, pathlines and streamlines coincide.
1.6.3.- FLUID LINE: set of fluid particles that in a given instant formed a
straight line.
Fluid line example
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1.6.4.- EXAMPLES OF FLOW VISUALIZATION.
Flow around a cylinder.
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Streamlines around rotating objects.
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NUMERICAL SIMULATION
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EXPERIMENTATION
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1.7 FLOW CLASSIFICATION
1.7.1.- ONE, TWO, AND THREE-DIMENSIONAL FLOWS.
• In general the flows are three-dimensional:
u = u(x,y,z,t)
v = v(x,y,z,t)
w = w(x,y,z,t)
• TWO-DIMENSIONAL FLOW: the velocity vectors depend only on two spatial
variables. E.g.: a flat flow where C = C (x,y)
• ONE-DIMENSIONAL FLOW: C = C(r) E.g. Flows in straight tubes of constant
crossection or between parallel plates.
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1.7.2.- DEVELOPED FLOW.
• Velocity profiles do not vary with respect to the spatial coordinate in the direction
of the flow (they occur far from entries or geometry changes)
D
Entry lenght
Le
𝑙𝑒
𝐷
= f(Re)
le
≈ 0,058. R e Laminar flow
D
le
≈ 4,4. R1e
D
6
Turbulent flow
1.7.3.- UNIFORM FLOW.
• Velocity and other properties remain constant in each crossection.
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1.7.4.- VISCOUS OR INVISCID FLOWS.
Depending on whether the effects of viscosity are important or they can be neglected.
•VISCOUS FLOWS: these are the internal flows and the flows close to the surface of
objects (boundary layer). Viscous effects cause energy losses.
Inviscid
flow
•
Boundary
layer
Boundary
layer
thickness
Inviscid FLOWS: They are mainly external flows.
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1.7.5.- UNSTEADY (TRANSIENT) AND STEADY FLOW (STATIONARY).
•
Depending on whether the properties of the flow depend on time or not.
Unsteady
Steady
1.7.6.- COMPRESSIBLE AND INCOMPRESSIBLE FLOW
• The internal energy does not interact with other energies. The energy equation is
decoupled and it is not necessary to solve the complete system of equations.
•Mach number: it is dimensionless parameter that gives an idea of the relative
importance between the elastic forces and the inertia forces of the fluid.
𝑐𝑓𝑙𝑢𝑖𝑑
𝑀=
𝑐𝑠𝑜𝑢𝑛𝑑
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• M < 0.3  incompressible flow ⇒
𝑑𝜌
• M > 0.3  compressible flow
𝑑𝜌
⇒
𝑑𝑡
=0
𝑑𝑡
≠0
Mach’s number also gives rise to another flow classification.
• M < 1  subsonic flow
• M > 1  supersonic flow
If the flow velocity is greater than the velocity of information propagation inside the
fluid (approximately the speedof the sound) discontinuities will appear in the medium
(shock waves).
Condensation cloud produced by falling pressure witthin shock wave.
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NASA PICTURES OBTAINED
IN 2019
https://www.nasa.gov/centers/
armstrong/features/supersonic
-shockwave-interaction.html
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1.7.7.- LAMINAR AND TURBULENT FLOWS (Internal flows)
•
Reynold’s number measures the relative importance between inertia forces and viscous
forces.
Re =
𝐶𝐿
𝜈
=
𝜌𝐶𝐿
𝜇
Where L is a characteristic lenght, function of the type of problema:
•
Internal flows: circular ducts L= Diameter
•
External flows: L= 𝐴 (Frontal projection area); airfoils L = cloud legnht
• Mr. Reynolds found that there was a velocity which he called critical, in which regime change
ocurred. This Critical Re was found it in a range between 2000 and 6000.
• Lows Re  Laminar Flow. (Re < Re critical). Slow and viscous movement, inertia effects
negligible. Soft variations. There is no significant mix between the particles close to each other.
• High Re  Turbulent Flow. (Re > Re critical). Movement with strong fluctuations. Fluid
moviment varies irregularly, properties have random variations over time and space (described
by statistical indicators).
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LAMINAR FLOW
TURBULENT FLOW
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1.7.8.- ROTATIONAL OR IRROTATIONAL FLOW.
•
In the irrotational flow, the angular speed of the fluid is zero.
𝑣𝑜𝑟𝑡𝑖𝑐𝑖𝑡𝑦
Irrotational flow
𝑖
𝜕
𝜉 = 𝑟𝑜𝑡 𝑐 = 𝛻 ∧ 𝑐 =
𝜕𝑥
𝑢
𝑗
𝜕
𝜕𝑦
𝑣
𝑘
𝜕
𝜕𝑧
𝑤
Rotational flow
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1.8.- PROPOSED EXERCICES.
Exam swap
1
P_170407
P_171102
P_171102
P_180403
R_170704
QUESTION
TOPIC
13
7
11
15
1
1
1
1
1
1
THEORY /
PROBLEM
T
T
T
T
T
Exam swap
QUESTION
1
F_170123
10
P_161102
6
P_170407
8
P_171102
4
R_180704
9
TOPIC
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1
1
1
1
THEORY/P
ROBLEM
P
P
P
P
P
33
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