Lecture 1 Ramiro Neves,

Teachers Ramiro Neves, ext. 1397, telem – Lígia Pinto Offices: Pavilhão de Mecânica I, 1º floor

Where to use Fluid Mechanics? About everywhere...

Thessaloniki NATO ARW (19-24 April 2005) Boussinesq Model Douro Estuary mouth: West and SW Waves

Dia mundial da água, Cascais, D Overland flow Precipitation variable in time & space 3D Porous Media 1D Drainage network Integrated Basin Modelling

Integrated Basin Modeling Rain Intensity Flow Production: ●2 Different Soils ●Infiltration ●Overland Flow

Dia mundial da água, Cascais, 2007 Integrated Basin Modeling Rain Intensity Sediment transport: ●2 Catchments ●1 Reservoir

Classical Problems

Reduction of air resistance

Flow in a artery Flow around a leaf

Balloon fish Low mobility high toxicity....

Even the bacteria know the importance of fluids mechanics

Difficulties? 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 Sciences

Set of courses downstream MFA Transferência de Energia e Massa Hidráulica Ambiental Hidrologia Ambiental e Recursos Hídricos Física da Atmosfera e do Oceano Ecologia.... Modelação Ambiental Gestão Integrada de Bacias Hidrográficas

Requirements Physics: Forces, Newton law and acceleration, kinetic energy, momentum, fluxes. Mathematics: derivative, integral, divergence, gradient, vector internal and external products.

Knowledge to acquire Understanding of fluid mechanics equations and processes that determine the fluid motion Control of the advection and diffusion concepts and of the evolution equation concept essential for the downstream courses

MFA practical part A computational component is added to the classical exercises with 3 objectives: –To show that Fluid Mechanics goes much beyond simple analytical solutions; –To help students to enhance their programing skills; –To replace the classical laboratory lectures (essential before computational methods were available) This component will be consolidated with a group home work programmed using (preferentially) VBA – VBA is part of the MS Office, is object oriented and useful for a wide range of engineering issues (database, internet...).

Bibliography Fluid Mechanics, Frank White, McGraw-Hill, (or any other Fluid Mechanics Introduction book) Apontamentos de Mecânica dos Fluidos I (Mecânica) Texts about specific subjects Lectures’ PP T

Students Knowledge Assessment Tests/Exam (75%) Computational exercise report (25%) Exams: (14 and 31 January 2015)

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 in solids not. 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

Elementary Volume Is large enough to include a large number of molecules and small enough to have uniform properties. If the Elementary volume is of the order of the size of a molecule, the density value is uncertain due to the number of molecules it contains. If it is too large the density can change from one region to another.

Continuum Hypothesis The elementary 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 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: 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 :

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 at a point) we can compute the discharge across an area integrating the velocity along the whole area: Defining a specific property as its value per unit of volume (when the volume tends to zero) And the flux of M across a surface is: We can say that the flux of M across an elementary surface is: If c is uniform along the surface then:

Summary We know what is fluid Mechanics and what is 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….