CIV 2113 Fluid Mechanics I - University of Guyana - PDF

1 UNIVERSITY OF GUYANA FACULTY OF ENGINEERING & TECHNOLOGY DEPARTMENT OF CIVIL ENGINEERING CIV 2113: Fluid Mechanics I INTRODUCTION TO FLUID MECHANICS PRESENTATION 2 OUTLINE 1.0 Introduction 2.0 Units & Dimensions 3.0 Dimensional Equatio...

1 UNIVERSITY OF GUYANA FACULTY OF ENGINEERING & TECHNOLOGY DEPARTMENT OF CIVIL ENGINEERING CIV 2113: Fluid Mechanics I INTRODUCTION TO FLUID MECHANICS PRESENTATION 2 OUTLINE 1.0 Introduction 2.0 Units & Dimensions 3.0 Dimensional Equations 4.0 Differences Between Solids & Fluids 5.0 Liquids & Gases 6.0 Molecular Structure of Materials 7.0 The Continuum Concept of a Fluid References  1.0 INTRODUCTION 3 Fluid Mechanics, as the name indicates, is that branch of applied mechanics that is concerned with the statics and dynamics of liquids and gases. The analysis of the behaviour of fluids is based upon the fundamental laws of applied mechanics that relate to the conservation of mass- energy and the force-momentum equation, together with other concepts and equations. Frequently, Fluid Mechanics is concerned with the behavior of streams of fluid instead of individual bodies or particles.  1.0 INTRODUCTION 4 Fluids flow under the action of such forces, deforming continuously for as long as the force is applied. A fluid is unable to retain any unsupported shape; it flows under its own weight and takes the shape of any solid body with which it comes into contact. Deformation or change of shape is caused by shearing forces. Therefore, if shearing forces are acting in a fluid it will flow.  1.0 INTRODUCTION 5 A fluid is a substance which deforms continuously under the action of shearing forces, however small they may be. 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 upon which they act.  1.0 INTRODUCTION 6 Branches of Fluid Mechanics Fluid Mechanics is divided into three main categories: 1. Hydrostatics considers the forces acting on a fluid at rest. 2. Fluid kinematics is the study of the geometry of fluid motion. 3. Fluid dynamics considers the forces that cause acceleration of a fluid.  1.0 INTRODUCTION 7 Engineering Applications of Fluid Mechanics Fluid Mechanics is one of the primary engineering sciences that has important applications in many engineering disciplines. For example, aeronautical and aerospace engineers use Fluid Mechanics principles to study flight, and to design propulsion systems. Civil engineers use this subject to design drainage channels, water networks, sewer systems and water-resisting structures such as dams and levees.  1.0 INTRODUCTION 8 Engineering Applications of Fluid Mechanics Fluid mechanics is used by mechanical engineers to design pumps, compressors, turbines, process control systems, heating and air conditioning equipment, and to design wind turbines and solar heating devices. Chemical and petroleum engineers apply this subject to design equipment used for filtering, pumping, and mixing fluids.  1.0 INTRODUCTION 9 Engineering Applications of Fluid Mechanics Engineers in the electronics and computer industry use fluid mechanics, principles to design switches, screen displays, and data storage equipment. Apart from the engineering profession, the principles of fluid mechanics are also used in biomechanics, where it plays a vital role in the understanding of the circulatory, digestive, and respiratory systems and meteorology to study the motion and effects of tornadoes and hurricanes.  2.0 UNITS & DIMENSIONS 10 The SI system of units has seven basic units which are given below: Quantity Unit Symbol Length metre m Mass kilogramme kg Time second s Electric Current ampere A Thermodynamic Temperature kelvin K (Formerly °K) Luminous Intensity candela cd Amount of Substance mole mol  2.0 UNITS & DIMENSIONS 11 All other units are derived from these fundamental units, since the SI is a coherent system in which the product or quotient of any two unit quantities within the system is the unit of the resultant quantity. For example, the unit of velocity is obtained by dividing the unit of distance, the metre, by the unit of time, the second, and will therefore be metres per second. Over the years, the way in which the derived units are written has changed.  2.0 UNITS & DIMENSIONS 12 Until recently, two abbreviated forms of notation were in common use. For example, metre/second could be abbreviated to mΤs or m s−1 where, in the second example, a space separates the m and s. In recent years, there has been a strong movement towards a third form of notation, which has the benefit of clarity and the avoidance of ambiguity. The half-high dot (also known as the middle dot) is now widely used in scientific work in the construction of derived units. Using the half-high dot, metre/second is expressed as m ⋅ s−1.  2.0 UNITS & DIMENSIONS 13 The following are some common derived units: Quantity Unit Symbol Units Force Newton N kg ⋅ m ⋅ s−2 Pressure and Stress pascal Pa N ⋅ m−2 Work, Energy, Quantity of Heat joule J N⋅m Power watt W J ⋅ s−1 Frequency hertz Hz s−1 Plane Angle radian rad  2.0 UNITS & DIMENSIONS 14 The following are some units that are accepted for use with the SI: Name Quantity Symbol Value in SI Units minute time min 1 min = 60 s hour time h 1 h = 60 min = 3600 s day time d 1 d = 24 h = 86,400 s degree plane angle ° 1° = 𝜋Τ180 rad minute plane angle ′ 1′ = 1Τ60 ° = 𝜋Τ10800 rad second plane angle ′′ 1′′ = 1Τ60 ′ = 𝜋Τ648000 rad litre volume L 1 L = 1 dm3 = 10−3 m3 Metric ton or tonne mass t 1 t = 103 kg  2.0 UNITS & DIMENSIONS 15 Larger or smaller units are formed by adding a prefix to the basic unit. For example, one thousandth part of a metre is a millimetre while one thousand metres is a kilometre. The tables on the following slides display some common prefixes used.  2.0 UNITS & DIMENSIONS 16 The following prefixes for multiples of SI units are currently in use: Prefix Symbol Multiplying Factor deca da 10 hecto h 102 kilo k 103 mega M 106 giga G 109 tera T 1012  2.0 UNITS & DIMENSIONS 17 The following prefixes for sub-multiples of SI units are currently in use: Prefix Symbol Multiplying Factor deci d 10−1 centi c 10−2 milli m 10−3 micro μ 10−6 nano n 10−9 pico p 10−12 femto f 10−15  2.0 UNITS & DIMENSIONS 18 Care is needed in using the prefixes. The symbol for a prefix should always be written close to the symbol of the unit it qualifies, for example, kilometre km , megawatt MW , microsecond μs. Only one prefix at a time should be applied to a unit. The symbol ‘m’ represents both the basic unit ‘metre’ and for the prefix ‘milli’, so care is needed in using that. When a unit with a prefix is raised to a power, the exponent applies to the whole multiple and not just to the original unit.  2.0 UNITS & DIMENSIONS 19 Thus, 1 mm2 means 1 mm 2 = 10−3 mm 2 = 10−6 m2 , and not 1m m 2 = 10−3 m2. The symbols for units refer not only to the singular but also to the plural. For instance, the symbol for kilometres is km, not kms. Capital or lower case (small) letters are used strictly in accordance with the definitions, no matter what combination the letters may appear.  2.0 UNITS & DIMENSIONS 20 The units chosen for measurement do not affect the quantity measured. One kilogramme of water means exactly the same as 2.2046 lb of water. It is sometimes convenient to not use any particular system but to think in terms of mass, length, time, force, temperature, etc. In mechanics all quantities can be expressed in terms of mass 𝑴, length 𝑳 and time 𝑻.  2.0 UNITS & DIMENSIONS 21 Thus: 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑡𝑖𝑚𝑒 2 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐿 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 2 = 2 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑡𝑖𝑚𝑒 𝑇 Another unit in common use is the metric tonne = 103 kg = 2205 lb.  2.0 UNITS & DIMENSIONS 22 Similarly: 𝐹𝑜𝑟𝑐𝑒 = 𝑚𝑎𝑠𝑠 × 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝐹𝑜𝑟𝑐𝑒 = 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑚𝑎𝑠𝑠 × 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑀𝐿 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝐹𝑜𝑟𝑐𝑒 = 2 𝑇  2.0 UNITS & DIMENSIONS 23 Temperature Temperatures are conveniently expressed using the Celsius temperature scale. The symbol ℃ is used to express Celsius temperature. The Celsius temperature (symbol 𝒕) is related to the thermodynamic temperature (symbol 𝑇) by the equation: 𝑡 = 𝑇 − 𝑇0 Where 𝑇0 = 273.15 K by definition. For many purposes, 273.15 K can be rounded off to 273 without significant loss of accuracy.  2.0 UNITS & DIMENSIONS 24 Force The force unit, which is the Newton is defined from Newton’s second law as: 𝐹 = 𝑚𝑎 1 N = 1 kg × 1 m ⋅ s −2 Thus, 1 N force acting on a 1 kg mass will give the mass an acceleration of 1 m ⋅ s −2.  2.0 UNITS & DIMENSIONS 25 Gravitational Acceleration The weight 𝑊 and mass 𝑚 of a body are related by: 𝑊 = 𝑚𝑔 The quantity represented by the symbol 𝑔 is variously described as the gravitational acceleration, the acceleration of gravity, weight per unit mass, the acceleration of free fall, and other terms. As an acceleration, the units of 𝑔 are usually represented in the natural form as m ⋅ s−2.  2.0 UNITS & DIMENSIONS 26 Gravitational Acceleration However, it is sometimes convenient to express them in the alternative form N ⋅ kg −1 , a form which follows from the definition of the newton. Standard gravity is 9.807 m ⋅ s−2 (commonly approximated as 9.81 m ⋅ s−2 ) so that a 1 kg mass weighs 9.81 N under standard gravity.  2.0 UNITS & DIMENSIONS 27 Work The unit of work, the joule, is the work done when the point of application of a 1 N force is displaced through a distance of 1 m in the direction of the force. Thus: 1J=1N⋅m  2.0 UNITS & DIMENSIONS 28 Power The unit of power, the watt, is defined as a joule per second. Thus: 1 W = 1 J ⋅ s−1 = 1 N ⋅ m ⋅ s −1  2.0 UNITS & DIMENSIONS 29 Pressure and Stress One (1) pascal is the pressure induced by a force of 1 N acting on an area of 1 m2. The pascal, abbreviated Pa, is small for most purposes, and thus multiples are often used.  2.0 UNITS & DIMENSIONS 30 Volume In the measurement of fluids, the name ‘litre’ is commonly given to 10−3 m3. Both l and L are internationally accepted symbols for the litre. However, as the letter l is commonly mistaken for one, the symbol L is now recommended.  2.0 UNITS & DIMENSIONS 31 Angular Velocity The SI unit for the plane angle is the radian. Consequently, the SI unit for angular velocity is rad ⋅ s−1. Another measure of plane angle is the revolution, which is equal to 360°. However, the revolution is not part of the SI, nor is it a unit accepted for use with the SI. The revolution, abbreviated by rev, is easy to measure.  2.0 UNITS & DIMENSIONS 32 Angular Velocity The units reported in industry for rotational speed is in revΤs. In order to distinguish between the two sets of units, the symbol 𝜔 is used for angular velocity and the symbol 𝑁 when the units are revΤs. Thus, 𝑁 is related to 𝜔 by the expression: 𝜔 𝑁= 2𝜋  2.0 UNITS & DIMENSIONS 33 Conversion Factors In solving problems it is essential to keep to one system of units only. If the data are in different systems they should be converted immediately to the system selected. Consider the identity 1 inch ≡ 25.4 mm. The use of three lines (≡), instead of the two lines of the usual equal sign, indicates not simply that 1 inch equals or is equivalent to 25.4 mm, but that 1 inch is 25.4 mm.  2.0 UNITS & DIMENSIONS 34 Conversion Factors At all times and in all places, 1 inch and 25.4 mm are precisely the same. The identity may be rewritten as: 25.4 mm 1≡ 1 inch The above ratio is equal to unity and is known as a conversion factor. The conversion factor may also be used in reciprocal form when the desired result requires it.  3.0 DIMENSIONAL EQUATIONS 35 If an equation is to represent something which is physically real the terms on both sides must be of the same sort (for example, all forces) as well as both sides being numerically equal, otherwise the equation is meaningless. Every term must have the same dimensions so that like is compared with like. For example, the equation 𝑣 2 = 𝑢2 + 2𝑎𝑠 gives the final velocity 𝒗 of a body which started with an initial velocity 𝒖 and received an acceleration 𝒂 for a distance 𝒔.  3.0 DIMENSIONAL EQUATIONS 36 When dimensions are substituted for the quantities each term must have the same dimensions if the equation is true. The dimensions of the quantities are: 𝑣 = 𝐿𝑇 −1 , 𝑢 = 𝐿𝑇 −1 , 𝑎 = 𝐿𝑇 −2 Dimensions of 𝑣 2 are 𝐿𝑇 −1 2 = 𝐿2 𝑇 −2 Dimensions of 𝑢2 are 𝐿𝑇 −1 2 = 𝐿2 𝑇 −2 Dimensions of 2𝑎𝑠 are 𝐿𝑇 −2 × 𝐿 = 𝐿2 𝑇 −2  3.0 DIMENSIONAL EQUATIONS 37 All three terms have the same dimensions and the equation is dimensionally correct and could represent a real event. A check on dimensions will not show whether any pure numbers in the equation are correct since pure numbers are ratios and have the dimension of unity. 4.0 DIFFERENCES BETWEEN SOLIDS & FLUIDS 38 In everyday life, we recognize three states of matter: solid, liquid and gas. Although different in many respects, liquids and gases have a common characteristic in which they differ from solids: they are fluids, lacking the ability of solids to offer permanent resistance to a deforming force. For a solid, the strain is a function of the applied stress, provided that the elastic limit is not exceeded. For a fluid, the rate of strain is proportional to the applied stress. 4.0 DIFFERENCES BETWEEN SOLIDS & FLUIDS 39 The strain in a solid is independent of the time over which the force is applied and, if the elastic limit is not exceeded, the deformation disappears when the force is removed. A fluid continues to flow for as long as the force is applied and will not recover its original form when the force is removed.  5.0 LIQUIDS & GASES 40 Although liquids and gases both share the common characteristics of fluids, they have many distinctive characteristics of their own. A liquid is difficult to compress and, for many purposes, may be regarded as incompressible. A given mass of liquid occupies a fixed volume, irrespective of the size or shape of its container, and a free surface is formed if the volume of the container is greater than that of the liquid. A gas is comparatively easy to compress.  5.0 LIQUIDS & GASES 41 Changes of volume with pressure in a gas are large, cannot normally be neglected and are related to changes of temperature. A given mass of a gas has no fixed volume and will expand continuously unless restrained by a containing vessel. It will completely fill any vessel in which it is placed and, therefore, does not form a free surface.  6.0 MOLECULAR STRUCTURE OF MATERIALS 42 Solids, liquids and gases are all composed of molecules in continuous motion. However, the arrangement of these molecules, and the spaces between them, differ, giving rise to the characteristic properties of the three different states of matter. In solids, the molecules are densely and regularly packed and movement is slight, each molecule being restrained by its neighbours.  6.0 MOLECULAR STRUCTURE OF MATERIALS 43 In liquids, the structure is looser; individual molecules have greater freedom of movement and, although restrained to some degree by the surrounding molecules, can break away from this restraint, causing a change of structure. In gases, there is no formal structure, the spaces between molecules are large and the molecules can move freely. The molecules of a substance exert forces on each other which vary with their intermolecular distance.  7.0 THE CONTINUUM CONCEPT OF A FLUID 44 Although the properties of a fluid arise from its molecular structure, engineering problems are usually concerned with the bulk behaviour of fluids. The number of molecules involved is immense, and the separation between them is normally negligible by comparison with the distances involved in the practical situation being studied. Under these conditions, it is usual to consider a fluid as a continuum, that is, a hypothetical continuous substance.  7.0 THE CONTINUUM CONCEPT OF A FLUID 45 Quantities such as velocity and pressure can then be considered to be constant at any point, and changes due to molecular motion may be ignored. Variations in such quantities can also be assumed to take place smoothly, from point to point. In this course, fluids will be assumed to be continuous substances and, when the behaviour of a small element or particle of fluid is studied, it will be assumed that it contains so many molecules that it can be treated as part of this continuum.  REFERENCES 46 ✓ Caprani, D. C., 2006. Fluid Mechanics. Dublin: Dr. Colin Caprani. ✓ Douglas, J., 1986. Solving Problems in Fluid Mechanics: Volume I. Singapore: Longman Singapore Publishers Pte Ltd. ✓ Douglas, J. F., Gasiorek, J. M., Swaffield, J. A. & Jack, L. B., 2005. Fluid Mechanics. 5th ed. Essex: Pearson Education Limited. ✓ Evett, J. B. & Liu, C., 1989. 2500 Solved Problems in Fluid Mechanics and Hydraulics. New York: McGraw Hill Inc. ✓ Hibbeler, R. C., 2015. Fluid Mechanics. Essex: Pearson Education Limited.  47 QUESTIONS ? THE END