Fluid Mechanics

MATH24412 Fluid Mechanics 10 Semester 2 Undergraduate Level 2 Tom Shearer Unit coordinator 1 FHEQ level (Framework for Higher Education Qualifications) ' Middle part of Bachelors ' European Credit Transfer & Accumulation System Rating : 5.0 The primary aim of this course unit is to provide students with a first introduction to continuum mechanics in general and theoretical fluid mechanics in particular. The material provides the student with an essential background to many third and fourth level courses on physical applied mathematics.Fluid mechanics is concerned with understanding, and hence predicting, the properties (pressure, density, velocity etc.) of liquids and gases under external forces. This subject provides one of the major modern areas for the successful practical application of mathematics. Water, blood, air are all examples of fluids; of the many diverse fields where an understanding of the motion of fluids is important, one can mention oceanography and meteorology (in particular the dynamics of ocean circulation and weather forecasting), biological fluid dynamics (for example, blood flows through arteries), and aerodynamics.The main physical focus at the end of the course is to calculate the forces on a body moving in a fluid e.g. aeroplane wing; the same study also relates to the behaviour of balls in football, cricket and golf, and of boomerangs and frisbees.  The primary aim of this course unit is to provide students with a first introduction to continuum mechanics in general and theoretical fluid mechanics in particular. The material provides the student with an essential background to many third and fourth level courses on physical applied mathematics.Fluid mechanics is concerned with understanding, and hence predicting, the properties (pressure, density, velocity etc.) of liquids and gases under external forces. This subject provides one of the major modern areas for the successful practical application of mathematics. Water, blood, air are all examples of fluids; of the many diverse fields where an understanding of the motion of fluids is important, one can mention oceanography and meteorology (in particular the dynamics of ocean circulation and weather forecasting), biological fluid dynamics (for example, blood flows through arteries), and aerodynamics.The main physical focus at the end of the course is to calculate the forces on a body moving in a fluid e.g. aeroplane wing; the same study also relates to the behaviour of balls in football, cricket and golf, and of boomerangs and frisbees. This course aims to offer an introduction to the study of the motion of fluids (liquids and gases), in the important and widely-applicable case where the internal resistance of the fluid can be neglected.The course starts by looking at how to visualise fluid flows, before building in such important concepts as conservation of mass, and deriving the equation of motion for fluid under the action of different types of forces. Integrating the equation of motion then leads to Bernoulli’s Equation which has varied applications.After a short consideration of angular velocity in fluids, the course then considers flows which are two-dimensional, such as that past an aircraft wing section. A succession of interesting and powerful results follow, culminating in being able to calculate the lift on such a wing section.   On completion of this unit successful students will be able to:<ul><li>Derive and apply identities involving Grad, Div and Curl, the Material Derivative, the Divergence Theorem and Stokes' Theorem.</li><li>Solve for the streamlines, particle paths and streamlines of a given fluid flow.</li><li>Apply the governing equations of fluid mechanics, including the hydrostatic equilibrium equation, conservation of mass equation, Euler's equations of motion and Bernoulli's equation to model specific flows.</li><li>Identify the vorticity and the circulation in a fluid flow and exploit the simplifications resulting from irrotational motion.</li><li>Use the velocity potential and the complex potential to find the streamlines and velocities of irrotational, incompressible, two-dimensional fluid flows.</li><li>Apply the circle theorem and Blasius' theorem to find the forces on a body in a suitably simple flow.</li></ul> <ul> <li> Basic assumptions: Differences between fluids and solids. Differences between liquids and gases. Typical flow speeds and compressibility. Fluid particles and the continuum approximation.Lagrangian and Eulerian descriptions of a flow. Steady Flow.</li> <li> Vector calculus: Recap on Grad, Div and Curl, and the Divergence Theorem and Stokes’ Theorem. </li> <li> Visualising fluid flows: Streamlines, Stagnation points, Streaklines and Particle paths.</li> <li> Rates of change: The Material Derivative, and the acceleration of a fluid particle.</li> <li> Suffix Notation</li> <li> Modelling: Forces, Pressure and Hydrostatic Equilibrium. Conservation of Mass. Equations of Motion. Constitutive equations. Boundary Conditions.</li> <li> Energy and momentum: Bernoulli’s Equation for steady flow, and applications.</li> <li> Angular Velocity: Vorticity and Irrotational motion. Velocity potential. Laplace’s equation. Bernoulli’s Equation for irrotational flow.</li> <li> Two-dimensional motion: The stream-function and vorticity, in Cartesians and other co-ordinate systems. Equipotentials and streamlines. The complex potential and the complex velocity. (Some elementary complex analysis is discussed.) Some special 2-D flows. The Method of Images. Source in a uniform stream. Dipole in a uniform stream. The Circle Theorem and examples. Force on a cylinder. Blasius’ Theorem. The lift on a circular cylinder with circulation, and the lift on an aerofoil.</li></ul>  0 Other 20% 1 Written exam 80% <ul> <li> Coursework; Weighting within unit 20%</li> <li> End of semester examination; Weighting within unit 80%</li> </ul> Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour. MATH24420 Partial Differential Equations & Vector Calculus Co-Requisite Compulsory MATH24412 Co-Requisite: Students must be enrolled on MATH24420 in order to enrol onto MATH24412Students must have taken MATH11411, and MATH11422 or MATH11412 N All of these books, which are introductions to Fluid Mechanics, are to be found in Blue, Floor 2 532 or 532.5 or 532.6 in the John Rylands Library; there are many others in the same sections which may be worth browsing over (e.g. Schaum’s Outline Series).1. Lighthill, M.J.“An Informal Introduction to Theoretical Fluid Mechanics”Library: 532/L162. Prandtl, L. & Tietjens, O.G.“Fundamentals of Hydro- and Aeromechanics”Library: 532/P463. Paterson, A.R.“A first course in Fluid Dynamics”Library: 532.5/P64. Batchelor, G.K.“An Introduction to Fluid Dynamics”        (hard!)Library: 532.5/B425. Currie, I.G.“Fundamental Mechanics of Fluids”          (good but easy content) Library: 532/C366. Milne-Thomson, L.M.“Theoretical Aerodynamics” and “Theoretical Hydrodynamics”Library: 532.6/M24 and 532.5/M49 respectively7. Lamb, Sir H.“Hydrodynamics”Library: 532.5/L55Publishers: 1: Oxford University Press                2 & 6: Dover                3, 4 & 7: Cambridge University Press                5: New York, Marcel Dekker  Lectures 22 Practical classes & workshops 6 Tutorials 6 0 66