Basic fluid Mechanics Lecture Notes

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Fluid Mechanics Fluid Mechanics 2nd Year Civil & Structural Engineering Semester 2 2006/7 Dr. Colin Caprani Chartered Engineer 1 Dr. C. Caprani Fluid Mechanics 1. Introduction 1.1 Course Outline Goals The goal is that you will: 1. Have fundamental knowledge of fluids: a. compressible and incompressible; b. their properties, basic dimensions and units; 2. Know the fundamental laws of mechanics as applied to fluids. 3. Understand the limitations of theoretical analysis and the determination of correction factors, friction factors, etc from experiments. 4. Be capable of applying the relevant theory to solve problems. 7 Dr. C. Caprani Fluid Mechanics Syllabus Basics: • Definition of a fluid: concept of ideal and real fluids, both compressible and incompressible. • Properties of fluids and their variation with temperature and pressure and the dimensions of these properties. Hydrostatics: • The variation of pressure with depth of liquid. • The measurement of pressure and forces on immersed surfaces. Hydrodynamics: • Description of various types of fluid flow; laminar and turbulent flow; Reynolds’s number, critical Reynolds’s number for pipe flow. • Conservation of energy and Bernoulli’s theorem. Simple applications of the continuity and momentum equations. • Flow measurement e.g. Venturi meter, orifice plate, Pitot tube, notches and weirs. • Hagen-Poiseuille equation: its use and application. • Concept of major and minor losses in pipe flow, shear stress, friction factor, and friction head loss in pipe flow. • Darcy-Weisbach equation, hydraulic gradient and total energy lines. Series and parallel pipe flow. • Flow under varying head. • Chezy equation (theoretical and empirical) for flow in an open channel. • Practical application of fluid mechanics in civil engineering. 8 Dr. C. Caprani Fluid Mechanics 1.2 Programme Lectures There are 4 hours of lectures per week. One of these will be considered as a tutorial class – to be confirmed. The lectures are: • Monday, 11:00-12:00, Rm. 209 and 17:00-18:00, Rm 134; • Wednesday, to be confirmed. Assessment The marks awarded for this subject are assigned as follows: • 80% for end-of-semester examination; • 20% for laboratory work and reports. 9 Dr. C. Caprani Fluid Mechanics 1.3 Reading Material Lecture Notes The notes that you will take in class will cover the basic outline of the necessary ideas. It will be essential to do some extra reading for this subject. Obviously only topics covered in the notes will be examined. However, it often aids understanding to hear/read different ways of explaining the same topic. Books Books on Fluid Mechanics are kept in Section 532 of the library. However, any of these books should help you understand fluid mechanics: • Douglas, J.F., Swaffield, J.A., Gasiorek, J.M. and Jack, L.B. (2005), Fluid Mechanics, 5th Edn., Prentice Hall. • Massey, B. and Ward-Smith, J. (2005), Mechanics of Fluids, 8th Edn., Routledge. • Chadwick, A., Morfett, J. and Borthwick, M. (2004), Hydraulics in Civil and Environmental Engineering, 4th Edn., E & FN Spon. • Douglas, J.F. and Mathews, R.D. (1996), Solving Problems in Fluid Mechanics, Vols. I and II, 3rd Edn., Longman. The Web There are many sites that can help you with this subject. In particular there are pictures and movies that will aid your understanding of the physical processes behind the theories. If you find a good site, please let me know and we will develop a list for the class. 10 Dr. C. Caprani Fluid Mechanics 1.4 Fluid Mechanics in Civil/Structural Engineering Every civil/structural engineering graduate needs to have a thorough understanding of fluids. This is more obvious for civil engineers but is equally valid for structural engineers: • Drainage for developments; • Attenuation of surface water for city centre sites; • Sea and river (flood) defences; • Water distribution/sewerage (sanitation) networks; • Hydraulic design of water/sewage treatment works; • Dams; • Irrigation; • Pumps and Turbines; • Water retaining structures. • Flow of air in / around buildings; • Bridge piers in rivers; • Ground-water flow. As these mostly involve water, we will mostly examine fluid mechanics with this in mind. Remember: it is estimated that drainage and sewage systems – as designed by civil engineers – have saved more lives than all of medical science. Fluid mechanics is integral to our work. 11 Dr. C. Caprani Fluid Mechanics 2. Introduction to Fluids 2.1 Background and Definition Background • There are three states of matter: solids, liquids and gases. • Both liquids and gases are classified as fluids. • Fluids do not resist a change in shape. Therefore fluids assume the shape of the container they occupy. • Liquids may be considered to have a fixed volume and therefore can have a free surface. Liquids are almost incompressible. • Conversely, gases are easily compressed and will expand to fill a container they occupy. • We will usually be interested in liquids, either at rest or in motion. Liquid showing free surface Gas filling volume Behaviour of fluids in containers 12 Dr. C. Caprani Fluid Mechanics Definition The strict definition of a fluid is: A fluid is a substance which conforms continuously under the action of shearing forces. To understand this, remind ourselves of what a shear force is: Application and effect of shear force on a book Definition Applied to Static Fluids According to this definition, if we apply a shear force to a fluid it will deform and take up a state in which no shear force exists. Therefore, we can say: If a fluid is at rest there can be no shearing forces acting and therefore all forces in the fluid must be perpendicular to the planes in which they act. Note here that we specify that the fluid must be at rest. This is because, it is found experimentally that fluids in motion can have slight resistance to shear force. This is the source of viscosity. 13 Dr. C. Caprani Fluid Mechanics Definition Applied to Fluids in Motion For example, consider the fluid shown flowing along a fixed surface. At the surface there will be little movement of the fluid (it will ‘stick’ to the surface), whilst further away from the surface the fluid flows faster (has greater velocity): If one layer of is moving faster than another layer of fluid, there must be shear forces acting between them. For example, if we have fluid in contact with a conveyor belt that is moving we will get the behaviour shown: Ideal fluid Real (Viscous) Fluid When fluid is in motion, any difference in velocity between adjacent layers has the same effect as the conveyor belt does. Therefore, to represent real fluids in motion we must consider the action of shear forces. 14 Dr. C. Caprani Fluid Mechanics Consider the small element of fluid shown, which is subject to shear force and has a dimension s into the page. The force F acts over an area A = BC×s. Hence we have a shear stress applied: Force Stress = Area F τ = A Any stress causes a deformation, or strain, and a shear stress causes a shear strain. This shear strain is measured by the angle φ . Remember that a fluid continuously deforms when under the action of shear. This is different to a solid: a solid has a single value of φ for each value of τ . So the longer a shear stress is applied to a fluid, the more shear strain occurs. However, what is known from experiments is that the rate of shear strain (shear strain per unit time) is related to the shear stress: Shear stress ∝ Rate of shear strain Shear stress=× Constant Rate of shear strain 15 Dr. C. Caprani Fluid Mechanics We need to know the rate of shear strain. From the diagram, the shear strain is: x φ = y If we suppose that the particle of fluid at E moves a distance x in time t, then, using SR = θ for small angles, the rate of shear strain is: ⎛⎞ ∆φ x x 1 = t=⋅ ⎜⎟ ∆tt y y ⎝⎠ u = y Where u is the velocity of the fluid. This term is also the change in velocity with height. When we consider infinitesimally small changes in height we can write this in differential form, du dy . Therefore we have: du τ=× constant dy This constant is a property of the fluid called its dynamic viscosity (dynamic because the fluid is in motion, and viscosity because it is resisting shear stress). It is denoted µ which then gives us: Newton’s Law of Viscosity: du τµ = dy 16 Dr. C. Caprani Fluid Mechanics Generalized Laws of Viscosity We have derived a law for the behaviour of fluids – that of Newtonian fluids. However, experiments show that there are non-Newtonian fluids that follow a generalized law of viscosity: n ⎛⎞ du τ=+ AB ⎜⎟ dy ⎝⎠ Where A, B and n are constants found experimentally. When plotted these fluids show much different behaviour to a Newtonian fluid: Behaviour of Fluids and Solids 17 Dr. C. Caprani Fluid Mechanics In this graph the Newtonian fluid is represent by a straight line, the slope of which is µ . Some of the other fluids are: • Plastic: Shear stress must reach a certain minimum before flow commences. • Pseudo-plastic: No minimum shear stress necessary and the viscosity decreases with rate of shear, e.g. substances like clay, milk and cement. • Dilatant substances; Viscosity increases with rate of shear, e.g. quicksand. • Viscoelastic materials: Similar to Newtonian but if there is a sudden large change in shear they behave like plastic. • Solids: Real solids do have a slight change of shear strain with time, whereas ideal solids (those we idealise for our theories) do not. Lastly, we also consider the ideal fluid. This is a fluid which is assumed to have no viscosity and is very useful for developing theoretical solutions. It helps achieve some practically useful solutions. 18 Dr. C. Caprani Fluid Mechanics 2.2 Units Fluid mechanics deals with the measurement of many variables of many different types of units. Hence we need to be very careful to be consistent. Dimensions and Base Units The dimension of a measure is independent of any particular system of units. For example, velocity may be in metres per second or miles per hour, but dimensionally, −1 it is always length per time, or LT =LT . The dimensions of the relevant base units of the Système International (SI) system are: Unit-Free SI Units Dimension Symbol Unit Symbol Mass M kilogram kg Length L metre m Time T second s Temperature kelvin K θ Derived Units From these we have some relevant derived units (shown on the next page). Checking the dimensions or units of an equation is very useful to minimize errors. For example, if when calculating a force and you find a pressure then you know you’ve made a mistake. 19 Dr. C. Caprani Fluid Mechanics SI Unit Quantity Dimension Derived Base −1 −1 Velocity m/s ms LT 2 −2 −2 Acceleration m/s ms LT −2 −2 Force Newton, N kg m s MLT Pressure Pascal, Pa -1 −2 -1 2 kg m s ML T 2 Stress N/m 3 -3 -3 kg m Density kg/m ML 3 -2 −2 -2 −2 kg m s Specific weight N/m ML T Relative density Ratio Ratio Ratio 2 -1 −1 -1 −1 kg m s Viscosity Ns/m ML T Joule, J 22 − 22 − Energy (work) kg m s ML T Nm Watt, W 23 − 23 − kg m s Power Nm/s ML T 2 Note: The acceleration due to gravity will always be taken as 9.81 m/s . 20 Dr. C. Caprani Fluid Mechanics SI Prefixes SI units use prefixes to reduce the number of digits required to display a quantity. The prefixes and multiples are: Prefix Name Prefix Unit Multiple 12 Tera T 10 9 Giga G 10 6 Mega M 10 3 Kilo k 10 2 Hecto h 10 1 Deka da 10 -1 Deci d 10 -2 Centi c 10 -3 Milli m 10 -6 µ Micro 10 -9 Nano n 10 -12 Pico p 10 Be very particular about units and prefixes. For example: • kN means kilo-Newton, 1000 Newtons; • Kn is the symbol for knots – an imperial measure of speed; • KN has no meaning; • kn means kilo-nano – essentially meaningless. Further Reading • Sections 1.6 to 1.10 of Fluid Mechanics by Cengel & Cimbala. 21 Dr. C. Caprani Fluid Mechanics 2.3 Properties Further Reading Here we consider only the relevant properties of fluids for our purposes. Find out about surface tension and capillary action elsewhere. Note that capillary action only features in pipes of ≤ 10 mm diameter. Mass Density The mass per unit volume of a substance, usually denoted as ρ . Typical values are: 3 • Water: 1000 kg/m ; 3 • Mercury: 13546 kg/m ; 3 • Air: 1.23 kg/m ; 3 • Paraffin: 800 kg/m . Specific Weight The weight of a unit volume a substance, usually denoted as γ . Essentially density times the acceleration due to gravity: γ = ρg Relative Density (Specific Gravity) A dimensionless measure of the density of a substance with reference to the density of some standard substance, usually water at 4°C: density of substance relative density = density of water specific weight of substance = specific weight of water ργ ss == ργ ww 22 Dr. C. Caprani Fluid Mechanics Bulk Modulus In analogy with solids, the bulk modulus is the modulus of elasticity for a fluid. It is the ratio of the change in unit pressure to the corresponding volume change per unit volume, expressed as: Change in Volume Chnage in pressure = Original Volume Bulk Modulus −dV dp = VK Hence: dp KV =− dV In which the negative sign indicates that the volume reduces as the pressure increases. The bulk modulus changes with the pressure and density of the fluid, but for liquids can be considered constant for normal usage. Typical values are: 3 • Water: 2.05 GN/m ; 3 • Oil: 1.62 GN/m . The units are the same as those of stress or pressure. Viscosity The viscosity of a fluid determines the amount of resistance to shear force. Viscosities of liquids decrease as temperature increases and are usually not affected by pressure changes. From Newton’s Law of Viscosity: τ shear stress µ== du dy rate of shear strain 2 Hence the units of viscosity are Pa ⋅s or Ns ⋅ m . This measure of viscosity is known as dynamic viscosity and some typical values are given: 23 Dr. C. Caprani Fluid Mechanics 24 Dr. C. Caprani Fluid Mechanics Problems - Properties 3 a) If 6 m of oil weighs 47 kN, find its specific weight, density, and relative density. 3 3 (Ans. 7.833 kN/m , 798 kg/m , 0.800) b) At a certain depth in the ocean, the pressure is 80 MPa. Assume that the specific 3 weight at the surface is 10 kN/m and the average bulk modulus is 2.340 GPa. Find: a) the change in specific volume between the surface and the large depth; b) the specific volume at the depth, and; c) the specific weight at the depth. -4 3 -4 3 3 (Ans. -0.335×10 m /kg, 9.475×10 m /kg, 10.35 kN/m ) c) A 100 mm deep stream of water is flowing over a boundary. It is considered to have zero velocity at the boundary and 1.5 m/s at the free surface. Assuming a linear velocity profile, what is the shear stress in the water? 2 (Ans. 0.0195 N/m ) d) The viscosity of a fluid is to be measured using a viscometer constructed of two 750 mm long concentric cylinders. The outer diameter of the inner cylinder is 150 mm and the gap between the two cylinders is 1.2 mm. The inner cylinder is rotated at 200 rpm and the torque is measured to be 10 Nm. a) Derive a generals expression for the viscosity of a fluid using this type of viscometer, and; b) Determine the viscosity of the fluid for the experiment above. -4 2 (Ans. 6 × 10 Ns/m ) 25 Dr. C. Caprani

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