Aula Teórica 1&2
Ramiro Neves, 1397
[email protected]
• Ramiro Neves, ext. 1397, 917224732
• [email protected]
• David Brito, (Visual Basic)
• [email protected]
• Offices: Pavilhão de Mecânica I, 1º floor.
Where to use Fluid Mechanics?
• About Everywhere.....
Boussinesq Model
Douro Estuary mouth. West
and SW Waves
Thessaloniki NATO ARW (19-24 April
Integrated Basin Modelling
1D Drainage network
Q   Q 2
x  A
 H
Q2 n2 
A2 Rh4 / 3 
 x
2D Overland flow
A.Rh2 / 3 H / x
3D Porous Media
 h z 
  K ( h) 
 xi xi 
Dia mundial da água, Cascais, 2007
Variable in Time
& Space
Integrated Basin Modeling
Flow Production
• 2 Different Soils
• Infiltration
• Overland Flow
Rain Intensity
Integrated Basin Modeling
• 2 Catchments
• 1 Reservoir
Rain Intensity
Dia mundial da água, Cascais, 2007
Classical Problems
Reduction of air resistance
Flow in a artery and
around a leaf.
Baloon fish
Low mobility high toxicity.....
Até as Bactérias conhecem a
importância da Mecânica dos Fluidos
• The formalism....
Difficulties are apparent because:
• Fluid Mechanics requires a limited number of
physical concepts.
• Mathematical operators are mostly
derivatives, gradients and divergences.
• This course is an excellent opportunity to
consolidate basic concepts of Engineering
Set of courses downstream MFA
Transferência de Energia e Massa.
Hidráulica Ambiental,
Hidrologia Ambiental e Recursos Hídricos,
Física da Atmosfera e do Oceano,
Modelação Ambiental,
Planeamento Biofísico,
Gestão Integrada de Bacias Hidrográficas.
• Physics: Forces, Newton law and acceleration,
kinetic energy, momentum, fluxes.
• Mathematics: derivative, integral, divergence,
gradient, vector internal and external
Conhecimentos a aquirir
• Compreensão das equações da mecânica dos
fluidos e dos processos que determinam o
movimento do fluido.
• Domínio dos conceitos de advecção e de
difusão e do conceito de equação de evolução
essenciais para as disciplinas a jusante.
MFA practical part
• A computational component is added to the classical
exercises with 3 objectives:
1. To show that Fluid Mechanics goes much beyond simple
analytical solutions;
2. To help students to enhance their programing skills.
3. To replace the classical laboratory lectures (laboratories
were essential before computational methods were
• This component will be consolidated with a group
home work programmed using – preferentially - VBA. It
is part of the MS Office is object oriented and useful
for a wide range of engineering issues (database,
• Fluid Mechanics, Frank White, McGraw-Hill,
(or any other Fluid Mechanics Introduction
book de introdução).
• Apontamentos de Mecânica dos Fluidos I
• Texts about specific subjects,
• Lectures’ PPT.
Students Knowledge Assessment
• Tests/Exam (75%),
• Report on the computational exercise (25%)
What is a fluid?
• Is formed by molecules...
– That move, as in any other type of matter, above 0 kelvin.
– The difference between a fluid and a solid is that in the
fluid the molecules can change their relative positions
allowing them to get the shape of the containers.
– Fluids can be liquids or gases
• In gases molecules have free relative movement.
• In liquids molecules form groups with relative free
movement (allowing them to get the shape of the
container) which dimension depends on temperature
(influencing their viscosity).
Why is Fluid Mechanics distinct from
Solid Mechanics?
• In a fluid each molecule (or group of molecules) have
relative movement freedom and not in solids. The
consequence is that tangential stress deforms the
fluids. Or in other words, if there is tangential stress
there is movement.
• Normal stress compress the fluid, that can remain at
rest. Tangential shear moves the fluid in layers creating
velocity gradients.
Shear is proportional to
the rate of deformation.
Elemental Volume
• Is large enough to maintain the number of molecules, although
they move and small enough to have uniform properties.
Continuum Hypothesis
• The elemental volume is much larger than 10 nm
• Necessary because we cannot assess the
movement of individual molecules (too many and
the Heisenberg principle) .
• But they move individually....
– The unknown molecule movement will be dealt as
diffusion in the equations.
• When do we have velocity in a fluid?
– When there is net mass transport across a surface.
• What is velocity?
What is the velocity?
• Velocity is the flux of volume per unit of area.
• The Velocity is defined at a point and thus is the flow per unit of
area, when the area tends to zero.
un 
• A surface can have 3 orientations in a tridimensional space and thus
velocity can have up to 3 components.
• The velocity component in one direction is the internal product of
the velocity vector by the unitary vector along that direction. Using
the surface normal one can write :
un 
 u .n 
Discharge / Advective Flux
Knowing the 3 Velocity components and knowing that the
velocity is the discharge per unit of area when the area tends to
zero ( The velocity is defined in a point) we can compute the
discharge across an area integrating the velocity along the whole
 dQ  u.n dA  Q   u.n dA  Q   u j n j dA
Defining a specific property as its value per unit of volume,
(when the volume tends to zero)
We can say that the flux
of M across an
elementary surface is:
And the flux of M across
a surface is:
dM dVol
 cdQ  cu.n dA
dVol dt
m   cu.n dA
dm 
We know what is fluid Mechanics and what for.
We know what is a fluid,
We know what is velocity and the advective flux.
We know that Fluid Mechanics aims to study
flows and thus to know the velocity distributions.
• To compute fluxes we also need to know specific
properties distributions….

Aula Teórica 1