MTH- 486: Fluid Mechanics Instructor: Dr. Fahad Munir Abbasi Assistant Professor Department of Mathematics Comsats Institute of Information Technology Islamabad, Pakistan Layout of this lecture Introduction of the instructor Introduction of the course Learning outcomes and recommended books Course evaluation and commonly asked questions Historical background and importance of fluid mechanics Some basic definitions Summary My introduction MSc. Mathematics in 2008, from Quaid-i-Azam University, Islamabad, Pakistan M.Phil. Mathematics in 2010, from Quaid-i-Azam University, Islamabad, Pakistan PhD Mathematics in 2014, from Quaid-i-Azam University, Islamabad, Pakistan cont... Presently, working as Assistant Professor at Department of Mathematics, CIIT, Islamabad, Pakistan. Specialization in Newtonian and non-Newtonian fluid mechanics, Peristaltic flows, Magnetohydrodynamics, Heat and mass transfer, Analytical and numerical solutions of D.Es HEC Indigenous scholar during my PhD, Presently 35 International publications to my credit with total impact factor > 38. Course introduction Course code: MTH-486 Course title: Fluid Mechanics Credit hours: 03 cont.… • Basic definitions and concepts Course Outlines (e.g. Different classifications of fluids; properties of fluids (like density, viscosity etc.); Eulerian and Langrangian approach; velocity of a fluid at a point, streamlines and pathlines; flow classifications; velocity potential; vorticity vector; local and particle rates of change; Sources, sinks; doublets; stream lines and stream functions etc.) cont.… • Fundamental governing Course Outlines laws and equations (e.g. Law of conservation of mass, law of conservations of momentum, energy conservation laws, The basic hydrostatic equation; The continuity equation, The Euler’s equation of motion, The Bernoulli's equation, The Navier-Stokes equation etc.) cont.… • Applications Course Outlines (e.g. in Poiseuille flow; Couette flow; Unsteady unidirectional low; sudden motion of a plane boundary in a fluid at rest; Flow due to an oscillatory boundary; i.e. Stokes problems etc. ) cont.… • Some fundamental theorems Course Outlines (e.g. Miline-Thomson circle theorem; Blasius’ theorem, Kelvin’ s minimum energy theorem; Uniqueness theorem etc. ) Learning outcomes What we are aiming for? • This first course in fluid mechanics familiarizes the student with the basic definitions, fundamental laws and equations and some of the basic techniques for the analysis of fluid behavior. • At the end of this course the students will have a good working knowledge of the basic concepts related to fluids mechanics, techniques and equations governing the fluid motion. Recommended books Books related to this course: • White, F. M., Introduction to Fluid Mechanics, 4th Edition, MacGraw-Hill, 2014. • Chorlton, F., Textbook of fluid Dynamics, D. Van Nostrand Co. Ltd. 1967. • Thomson, M., Theoretical Hydrodynamics, Macmillan Press, 1979. • Landau, L.D., and Lifshitz, E.M., Fluid Mehanics, Pergamon Press, 1966. • Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge University Press, 1969. Course evaluation • Quizzes Evaluation will include: • Assignments • Sessional exams • Terminal exam Commonly asked questions Why fluid mechanics? • Occurrence (from inter-cellular to interstellar space you will find fluid everywhere, it is in our veins, our arteries, within the body of all living beings, in our houses, at our offices, in the atmosphere, even in our tears) • Utility and applications (better understanding of the subject can prove very helpful in facilitating us in all forms of life) • Wide scope of research for students willing to pursue to MPhil and PhD. • It is very interesting and grasping subject. cont.… • The answer is “NO”. Is prior knowledge of physics mandatory? • Although better understanding of physics may prove fruitful in better understanding of the subject, yet it is not mandatory. • Since we will be perusing this course from mathematical point of view so it is really not difficult for those students who haven't studied physics at the intermediate level. • The mathematical results which we will derive will speak for their physical interpretation them self and we will explain things in a simple manner so that you can understand it well cont.… • A wise man once said that “there is no How to peruse this course in order to get best out of it? simpler way of doing mathematics than DOING IT”. • CRAMMING isn’t the, key PRACTICE is. • Try to learn mathematics instead of memorizing it. Last words, before the start • After every lecture, try to reproduce what you have learnt in your own words and do all the mathematics by yourself. Tips for our students: • Never feel hesitant to ask questions, there is not better time for you to learn fluid mechanics. • Feel free to communicate, we are here to help. MTH- 486: Fluid Mechanics Instructor: Dr. Fahad Munir Abbasi Starting from what you already know • Matter: Matter has many definitions, but the most common is that it is any substance which has mass and occupies Reminders space. All physical objects are composed of matter, in the form of atoms, which are in turn composed of protons, neutrons, and electrons. cont.… • States of Matter: Reminders Matter can exist in various states/phases: solid, liquid, gas, or plasma. Most substances can transition between these phases based on the amount of heat the material absorbs (or loses). • States of Matter: Gases, liquids and solids are all made up of microscopic particles, but the behaviors of these particles differ in the three phases. cont.… Fluid Mechanics Fluid + Mechanics cont.… Fluid • Anything that can flow can be named as fluid e.g. water, air, blood, urine, lava, paints, lubricants etc. • To be more precise “anything that deforms continuously and indefinitely under the action of applied stress is called Fluid ” • from the examples quoted above you can see that water is different from air, but since both can flow so both are fluids, but surely of different types (which we will soon study in detail). cont.… Mechanics • The branch of physics that deals with the action of forces on any substance. It further has two branches, namely, Statics and Dynamics. Statics: Study of the effects of forces on a body at rest; Dynamics: Study of the effects of forces on a body in motion. • Celestial mechanics, the motion of bodies in space: planets, comets, stars, galaxies, etc. • Soil mechanics, mechanical behavior of soils Historical background Some glimpses through history • Development of the subject of fluid mechanics started even before the birth of Christ. It took hundred of years and life time inputs of countless scientists to make this subject reach where it stands today. Out of those numerous researches some are mentioned here as an acknowledgement to their contribution to this field: • Archimedes (287-212 B. C.) for his famous Buoyancy principle. • Benito Castelli (1577-1644) for his work on continuity principle. cont.… Some glimpses through history • Blaise Pascal (1623-1662) for his work on pressure for static fluids. • Sir Issac Newton (1642-1727) for his resistance law for fluids. • Daniel Bernoulli (1700-1782) for his well known Bernoulli’s equation. • Leonhard Euler (1707-1783) for his famous Euler’s equation. • Navier (1785-1836) and Stokes (18191903) for their famous Navier-Stokes equations. • Osborne Reynolds (1842–1912) for his work on turbulent flows. cont.… Some glimpses through history • Ludwig Prandtl (1875 – 1953) with his boundary layer theory, not only answered many questions from the past but also opened new ways for the future research and developments in the field of fluid mechanics. • For further details you can concern the books mentioned in the introduction of this lecture. Importance of fluid mechanics Applications • Aeronautics and astronautics: aircraft and missile aerodynamics, designing and applications of satellites, control hydraulics etc. • Civil engineering: pipe and channel flows, surface and ground water hydrology, designing of wind and water structures, water and waste treatments etc. • Physics: treatment of corrosive and sensitive fluids, hydro and magnetohydrodynamics, super conductors, cont.… Applications • Astrophysics: interstellar gas dynamics, solar winds and comet tails etc. • Biology: blood flow, urine transport, breathing aids, surgical processes, heart-lung and dialysis machines, drug delivery systems etc. • Mechanical and Nuclear engineering: pumps and compressors, heat exchangers, heating and cooling systems etc. • Geophysics: Metrology, oceanography, geo-magnetism, glacier flows etc. • Any many others… Fluid mechanics: basic definitions Quantities we will look for: • IT is worth mentioning that state of the fluid can be completely determined with the aid of five quantities: Three components of the velocity V(x, y, z), the pressure P (x, y, z) and the density ρ (x, y, z). cont.… Basic laws and principles • • • • • Law of conservation of mass Newton’s second law (F = ma) Law of conservation of momentum First law of thermodynamics Second law of thermodynamics cont.… Velocity field • We define the velocity of the fluid at any point “c” as the instantaneous velocity of the fluid particle that is passing through that point at the given instant of time. At a given instant, the velocity field, V is a function of the space coordinates x, y and z. The velocity of the fluid at any point varies from instant to instant. Therefore, the complete representation of velocity field is given by V = V (x, y, z, t). • The velocity vector V can be written in terms of three scalar components. Denoting the components in the x, y and z directions as u, v and w, then V = u i + v j + w k . cont.… Density • The mass per unit volume (for a 3-D object) at a given temperature and pressure, or stress conditions and is expressed by the relation ρ = m/V Specific weight • Weight of a unit volume, γ = ρg Specific volume • Volume of a unit mass = 1/ρ cont.… Rate of discharge • The quantity of a fluid, flowing per second, through a section of a pipe or a channel, is known as the rate of discharge or simply “discharge”. In general it is denoted by Q. • Now consider a liquid flowing through a pipe. If “a” denotes the cross sectional area of the fluid and “V” the average velocity of the fluid, then discharge = Area X average velocity or Q = a X V. Summary of the lecture Introduction of this course and the instructor Historical background and importance of fluid mechanics Some important and basic definitions Today’s quote: A man is honored, proportional to the knowledge he has.