fluid mechanics equations
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Lecture 1 Introduction Mecânica de Fluidos Ambiental 2015/2016 Contacts • Lígia Pinto • • • • phone: 218419433; extensão: 3433 e-mail: ligia.pinto@tecnico.ulisboa.pt doubts, questions... : Thursday from 10am to 12am office: Pavilhão de Mecânica I - MARETEC, 1st floor • Ramiro Neves • extensão 1397 • e-mail: ramiro.neves@tecnico.ulisboa.pt Mecânica de Fluidos Ambiental 2015/2016 Bibliography • Fluid Mechanics, Frank M. White, McGraw-Hill • Notes of Fluid mechanics I (Professor Ramiro Neves) • Texts about specific subjects • Lectures slides Mecânica de Fluidos Ambiental 2015/2016 Students Knowledge Assessment • 2 Tests or Exam (75%) • Exam dates • 12/01/2016 (8:00 AM) ; 30/01/2016 (11:30 AM) • test dates • 21/10/2015 (2:30PM) • Computational work and report (25%) • Deadline - 31/12/2015 • Homework problems (maximum 1.5 points) • • • • • • 23/09/2015 30/09/2015 07/10/2015 14/10/2015 21/10/2015 28/10/2015 04/11/2015 11/11/2015 18/11/2015 25/11/2015 02/12/2015 09/12/2015 Mecânica de Fluidos Ambiental 2015/2016 Computational work • Computational component is added to: • show that Fluid Mechanics goes much beyond simple analytical solutions; • enhance students programing skills; • replace the classical laboratory lectures (essential before computational methods were available); • This component will be consolidated with a group homework programmed using VBA: • VBA is part of the MS Office, is object oriented and useful for a wide range of engineering issues Mecânica de Fluidos Ambiental 2015/2016 Keys to mastering Fluid Mechanics • Learning the fundamentals: read and understand the text • Study material to be covered before it is covered in class • Working many problems • Only by working many problems can you truly understand the basic principles and how to apply them • Study sample problems until you can solve them “closed book” • Do the homework problems you have been assigned Mecânica de Fluídos Ambiental 2015/2016 Requirements • Physics: Forces, Newton law and acceleration, kinetic energy, momentum, fluxes • Mathematics: derivative, integral, divergence, gradient, vector internal and external products Mecânica de Fluídos Ambiental 2015/2016 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 Mecânica de Fluídos Ambiental 2015/2016 Scope of fluid mechanics • Fluids: gases and liquids, air and water most prevalent in daily experience • Examples: • • • • • • Flow in pipes and channels Air resistance or drag Projectile motion Lubrication Irrigation Meteorology Mecânica de Fluídos Ambiental 2015/2016 -- air and blood in body -- wind loading -- jets, shock waves -- combustion -- sedimentation -- oceanography Scope of fluid mechanics • Where to use fluids mechanics? • About everywhere... • Used in the design of: • • • • • • Water supply system Dam spillways Shock absorbers, brakes Ships, submarines Aircrafts, rockets Windmills, turbines Mecânica de Fluídos Ambiental 2015/2016 -- waste water treatment -- valves, flow meters -- automatic transmissions -- breakwaters, marinas -- computer disk drives -- pumps, HVAC systems Fluids... Fluids... everywhere • In rivers and streams Mecânica de Fluídos Ambiental 2015/2016 Fluids... Fluids... everywhere • A tornado ... an atmospheric vortex Mecânica de Fluídos Ambiental 2015/2016 Fluids... Fluids... everywhere • Air to breathe Mecânica de Fluídos Ambiental 2015/2016 Fluids... Fluids... everywhere • Mixing milk in coffee Mecânica de Fluídos Ambiental 2015/2016 Fluids... Fluids... everywhere • Water or gas in conduits ... Mecânica de Fluídos Ambiental 2015/2016 Fluids... Fluids... everywhere • Flow of air arround cars Mecânica de Fluídos Ambiental 2015/2016 Fluids... Fluids... everywhere • Flow in a artery http://cnx.org/contents/03841c4c-9e9a-4822-95b2-12273c843a4e@3/Blood-Flow-Blood-Pressure-andMecânica de Fluídos Ambiental 2015/2016 Fluids... Fluids... everywhere • Flow around a leaf Mecânica de Fluídos Ambiental 2015/2016 • Even the bacteria know the importance of fluids mechanics Mecânica de Fluídos Ambiental 2015/2016 Difficulties? • The formalism... • Momentum equation Mecânica de Fluídos Ambiental 2015/2016 Difficulties are apparent • 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 Mecânica de Fluídos Ambiental 2015/2016 What is a fluid? • Is formed by molecules... • that move above 0 kelvin; • two classes of fluids, liquids and gases: • in gases molecules have free relative movement • in liquids molecules form groups with relative free movement (allowing to get the shape of the container) which dimension depends on temperature (influencing their viscosity) • 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; Mecânica de Fluídos Ambiental 2015/2016 Distiction between fluids and solids • A fluid is defined as a substance that deforms continuously when acted by a shear stress of any magnitude. • A shear stress (force per unit area) is created whenever a tangencial force acts on a surface • A shear stress aplied to a fluid will result in motion of that fluid • A solid can resist a shear stress by a static deformation Mecânica de Fluídos Ambiental 2015/2016 Elementary volume • Is a volume 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. Mecânica de Fluídos Ambiental 2015/2016 Continuum Hypothesis • Elementary volume is much larger than 10-9 mm3 • Is necessary because we cannot assess the movement of individual molecules (too many) • 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. Mecânica de Fluídos Ambiental 2015/2016 What is the velocity? • Velocity is the flux of volume per unit of area • At a given point, i, velocity is defined as a flow per unit of area, and when the A0: 𝑑𝑄 𝑣𝑖 = 𝑑𝐴𝑖 • In a 3D space, a surface can have 3 orientations and thus velocity can have up to 3 components • In 1 direction, the velocity component is given by the internal product of the velocity vector by the unitary vector along that direction • Using the surface normal 𝑛 we can write: 𝑑𝑄 𝑑𝑄 𝑣𝑛 = = 𝑣. 𝑛 𝑑𝐴 𝑑𝐴 Mecânica de Fluídos Ambiental 2015/2016 Discharge / Advective Flux • Knowing: • the 3 velocity components • the velocity is the discharge per unit of area when A0 • we can compute the discharge across an area integrating the velocity along the whole area 𝑑𝑣𝑜𝑙 𝑑𝑡 Mecânica de Fluídos Ambiental 2015/2016 = 𝑑𝑄 = 𝑣 ∙ 𝑛 𝑑𝐴 𝑄= 𝐴 𝑣∙ Discharge / Advective Flux • Defining a specific property (c) as its value (M) per unit of volume (when the volume tends to zero): 𝑐= 𝑑𝑀 𝑑𝑣𝑜𝑙 • the flux of M across an elementary surface is given by: 𝑑𝑚 = 𝑑𝑀 𝑑𝑡 = 𝑑𝑀 𝑑𝑣𝑜𝑙 𝑑𝑣𝑜𝑙 𝑑𝑡 = 𝑐 𝑑𝑄 = 𝑐 𝑣. 𝑛 𝑑𝐴 • the flux of M across a surface is given by: 𝑚= 𝐴 𝑐 𝑣. 𝑛 𝑑𝐴 • If c is uniform along the surface then: 𝑚=𝑐 Mecânica de Fluídos Ambiental 2015/2016 𝐴 𝑣. 𝑛 𝑑𝐴 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…. Mecânica de Fluídos Ambiental 2015/2016
