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, Planeamento Biofísico, 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.
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 available). 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, internet...).
Bibliography Fluid Mechanics, Frank White, McGraw-Hill, (or any other Fluid Mechanics Introduction book de introdução). Apontamentos de Mecânica dos Fluidos I (Mecânica). 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. 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 in 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:
Summary 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….