1 FLUIDS AND THEIR PROPERTIES

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1 FLUID MECHANICS CONTENTS CHAPTER DESCRIPTION PAGE NO 1 FLUIDS AND THEIR PROPERTIES PART A NOTES 1.1 Introduction 1.2 Fluids 1.3 Newton s Law of Viscosity 1.4 The Continuum Concept of a Fluid 1.5 Types of Fluid Ideal Fluid Real Fluid Compressible Incompressible Fluid 1.6 Physical Properties Density Mass Density Specific Weight Specific Gravity Viscosity Dynamic Viscosity Kinematic Viscosity Surface Tension Capillary Vapor Pressure Compressibility and Bulk Modulus 1.7 Difference Between Solids And Fluids 1.8 Types of Fluid Behavior Newtonian Fluids Non Newtonian Fluids Plastic Fluids Dilatants Fluids Bingham Plastic Fluids Pseudo Plastic Fluids 1.9 Time Dependent Fluids Thixotropic Fluids Rheopectic Fluids 1.10 Temperature Dependency of Viscosity Liquids 1.10 Gases CAREER AVENUES/2015/FM 2

2 2 DIMENSIONAL ANALYSIS AND SCALE UP 3 FLUID STATICS 4 FLUID KINEMATICS 2.1 Introduction 2.2 Dimensions And Units 2.3 Dimensional Homogeneity 2.4 Rayleigh s Method 2.5 Buckingham s π Theorem 3.1 Pressure 3.2 Pascal s Law For Pressure At a Point 3.3 Hydrostatic Law 3.4 Classification of Pressure 3.5 Pressure Measuring Devices Piezo Meter Barometer Manometer U Tube Manometer Tilted Manometer U- Tube Manometer With One Leg Enlarged Pressure Gauge 3.6 Buoyancy Centre of Buoyancy Type of Equilibrium of Floating Bodies Stable Equilibrium Un-stable Equilibrium Neutral Equilibrium 4.1 Introduction 4.2 Types of Fluid Flow Steady And Unsteady Flow Uniform And Non Uniform Flow One, Two And Three Dimensional Flow Rotational And Irrotational Flow Laminar And Turbulent Flow Compressible And Incompressible Flow 4.3 Types of Flow Lines Path Line Stream Line Stream Tube Streak Line CAREER AVENUES/2015/FM 3

3 4.4 Continuity Equation Continuity Equation in Cartesian Coordinates 4.5 Velocity Potential 4.6 Stream Function Properties of Stream Function Cauchy Riemann Equations 4.7 Relation Between Stream Function And Velocity Potential 5 FLUID DYNAMICS 6 FLOW OF INCOMPRESSIBLE FLUID IN PIPES 5.1 Introduction 5.2 Types of Heads of Liquid in Motion Potential Head Velocity Head Pressure Head 5.3 Bernoulli s Equation 5.4 Euler s Equation of Motion 5.5 Bernoulli s Equation For Real Fluid 6.1 Flow Regimes 6.2 Navier Stokes Equation of Motion 6.3 Relation Between Shear Stress And Pressure Gradient 6.4 Flow of Viscous Fluid in Circular Pipes Hagen Poiseuille Equation 6.5 Laminar Uni Directional Flow Between Stationary Parallel Plates 6.6 Laminar Uni Directional Flow Between Parallel Plates Having Relative Motion 6.7 Flow Losses in Pipes 6.8 Darcy Equation For Head Loss Due to Friction 6.9 Empirical Correlations For Coefficient of Friction 6.10 Minor Head Losses Sudden Enlargement Sudden Contraction 6.11 Pipes in Series And in Parallel 6.12 Concept of Equivalent Pipe 6.13 Prandtl Mixing Length Theory For Shear Stresses in Turbulent Flow CAREER AVENUES/2015/FM 4

4 7 BOUNDARY LAYER THEORY 8 FLOW MEASUREMENT 9 FLOW PAST IMMERGED BODIES 7.1 Boundary Layer Theory 7.2 Boundary Layer Characteristics Boundary Layer Thickness Displacement Thickness Momentum Thickness Energy Thickness Energy Loss 7.3 Laminar Boundary Layer 7.4 Turbulent Boundary Layer 8.1 Introduction 8.2 Units of Flow 8.3 Measurement of Flow 8.4 Mechanical Flow Meters Variable Head Flow Meters Flow of Incompressible Fluid in Pipe Ratio Discharge Coefficient Flow Coefficient Orifice Flow Meter Venturi Meter Pitot Tube Variable Area Flow Meter Types of Variable Area Flow Meter Basic Equation 9.1 Introduction 9.2 Lift And Drag Concept 9.3 Drag Coefficient 9.4 Stokes Law Terminal Velocity Applications of Stokes Law 9.5 Stagnation Point CAREER AVENUES/2015/FM 5

5 10 FLOW THROUGH POROUS MEDIA 11 TRANSPORTATION OF FLUIDS 10.1 Description of Porous Media 10.2 Hydraulic Diameter 10.3 Porous Medium Friction Factor 10.4 Porous Medium Reynolds Number 10.5 Friction in Flow Through Beds of Solids Kozney Carman Equation Burke Plummer Equation Ergun Equation 10.6 Packed Column 10.7 Fluidization Minimum Fluidization Velocity 11.1 Pumping Equipments of Liquids 11.2 Positive Displacement Pumps 11.3 Centrifugal Pumps Characteristics of Centrifugal Pumps Specific Speed Operating Characteristics Affinity Laws For Pumps 11.4 Cavitations And NPSH Vapor Lock And Cavitations NPSH 12 LEVEL 1 13 LEVEL 2 PART B MULTIPLE CHOICE QUESTIONS PART C ASSIGNMENT 14 UNSOLVED QUESTIONS CAREER AVENUES/2015/FM 6

6 PART A: NOTES CAREER AVENUES/2015/FM 7

7 F FLUID MECHANICS FLUIDS AND THEIR CHAPTE ER 1 PROPERTIES 1.1 INTRODUCTION Fluid Mechanics, as the name indicates, is that branch of applied mechanics whichh is concerned with the statics and dynamics of liquids and gases. The analysis of the fluid is based upon the fundamental laws of applied mechanics whichh relates to the conservationn of mass, energy and the force momentum equation. There are, however, two major aspectss of fluid mechanics which differ from the solid body mechanics: (i) The nature and properties of the fluid itself, which are very different from those of a solid. (ii) Instead of dealing with individual bodies or elements of known mass, we are frequently concerned with the behavior of a continuous stream of fluid, without beginning or end. 1.2 FLUIDS We normally recognize three states of matter: solid; liquid and gas. However, liquid and gas are both fluids: in contrast to solids they lack the ability to resist deformation. Because a fluid cannot resist the deformation force, it moves, it flows under the action of the force. Its shape will change continuously as long as the force is applied. A solid can resist a deformation force while at rest, this force may cause some displacement but the solid does not continue to move indefinitely. The deformation is caused by shearing forces which act tangentially to a surface. Referring to the figure below, we see the force F acting tangentially on a rectangular (solid lined) element ABDC. This is a shearing force and produces the (dashed lined) rhombus element A B DC. We can say AA Fluid is a substance which deforms continuously or flows, when subjected to shearing forces. and conversely this definition implies the very important point that: CAREER AVENUES/2015/ FM 8

8 F FLUID MECHANICS If a fluid is at rest there are no shearing forces acting. All forces must be perpendicular to the planes which they are acting. When a fluid is in motion shear stresses are developed if the particles of the fluid move relative to one another. When this happens adjacent particles have different velocities. If fluid velocity is the same at every point then theree is no shear stress produced: the particles have zero relativee velocity. Consider the flow in a pipe in whichh water is flowing. At the pipe wall the velocity of the water will be zero. The velocity will increase as we move toward the centre of the pipe. This change in velocity across the direction of flow is known as velocity profile and shown graphically in the figure below: Because particles of fluid next to each other are moving with different velocities theree are shear forces in the moving fluid i.e. shear forces are normally present in a moving fluid. On the other hand, if a fluid is a long way from the boundary and all the particles are travelling with the same velocity, the velocity profile would look something like this: And there will be no shear forces present as alll particles have zero relativee velocity. In practice we are concerned with flow past solid boundaries; aero planes, cars, pipe walls, river channels etc. and shear forces will be present. 1.3 NEWTON S LAW OF VISCOSITY Let consider a 3d rectangular element of fluid, like that in the figure below. The shearing force F acts on the area on the top of the element. This area is given by A = δ s δ x. We can thus calculate the shear stress which is equal to force per unit area i.e. shear stress, F A CAREER AVENUES/2015/ FM 9

9 F FLUID MECHANICS The deformation which this shear stress causes is measured by the size of the angle Φ and is known as shear strain. It has been found experimentally thatt the rate of shear stress (shear stress per unit time, / time) is directly proportional to the shear stress. If the particle at point E (in the above figure) moves under the shear stress to point E and it takes time t to get there, it has moved the distance x. For small deformations we can write x shear strain, y rate of shear strain t x x 1. ty t y u y Where x u t the velocity of the particlee at E. Using the experimental result that shear stress is proportional to rate of shear strain then is u constant y The term u is the change in velocity with y, or the velocity y gradient, and may bee written in the differential form du dy. The constant of proportionality is du known as the dynamic viscosity, µ, of the fluid, giving dy This is known as Newton s law of viscosity. 1.4 THE CONTINUUM CONCEPT OF A FLUID The behavior of individual molecules comprising a fluid determines the observed properties of the fluid and for an absolutely complete analysis; the fluid should be studied at the molecular scale. The behavior of any one molecule is highly complex, continuously varying and May indeed is very different from neighboringg molecules at any instant of time. The problems CAREER AVENUES/2015/ FM 10

10 F FLUID MECHANICS normally encountered by engineers do not require knowledge and prediction of behavior at the molecular level but on the properties of the fluid mass that may result. Thus the interest is more on the average rather than the individual responses of the molecules comprising the fluid. At a microscopicc level, a fluid consistss of molecules with a lot of space in between. For our analysis, we do not consider the actual conglomeration of separate molecules, but instead assume that the fluid is a continuum, that is a continuous distribution of matter with no empty space. The sketch below illustrates this. However, we are interested in the property of the fluid particle at P and therefore we regard P as being a smear of matter (represented as a solid filled circle in the figure) with no space. 1.5 TYPES OF FLUID A fluid may be classified as follows: IDEAL FLUID An ideal fluid does not possess properties like viscosity, surface tension and compressibility. Such a fluid will not offer any resistance to displacement of surfaces in contact that is = 0. Ideal fluid is only an imaginary fluid REAL FLUID A real fluid possesses viscosity, surface tension and compressibility. Such a fluid will always resist displacement that is 0. Real fluid is an actual fluid which exists. CAREER AVENUES/2015/ FM 11

11 1.5.3 COMPRESSIBLE INCOMPRESSIBLE FLUID If the volume of an element of fluid is independent of its pressure and temperature, the fluid is said to be incompressible. If the volume of an element of fluid changes with its pressure and temperature, the fluid is said to be compressible. Although, all fluids are compressible - even water - their density will change as pressure changes. Under steady conditions, and provided that the changes in pressure are small, it is usually possible to simplify analysis of the flow by assuming it is incompressible and has constant density. In practical problems, the liquid are assumed to be incompressible and gases are said to be compressible. 1.6 PHYSICAL PROPERTIES The most common properties of fluids are as follows: DENSITY The density of a substance is the quantity of matter contained in a unit volume of the substance. It can be expressed in three different ways MASS DENSITY ( ) Mass Density is defined as the mass of substance per unit volume. Units: kg / m 3 Dimensions: ML 3 Typical values: Water = 1000 kg m 3, Mercury = kg m 3, Air = 1.23 kg m 3. (at P = N m -2, T = K) SPECIFIC WEIGHT (w) Specific Weight is defined as the weight per unit volume. or The force exerted by gravity, g, upon a unit volume of the substance. The Relationship between g and ω can be determined by Newton s 2 nd Law, since Weight per unit volume = mass per unit volume g w. g Units: N / m 3 CAREER AVENUES/2015/FM 12

12 Dimensions: ML 2 T -2. Typical values: Water = 9814 N m 3, Mercury = N m 3, Air = N m SPECIFIC GRAVITY (s) Specific Gravity is defined as the ratio of mass density of a substance to some standard mass density. For solids and liquids this standard mass density is the maximum mass density for water (which occurs at 4 0 C) at atmospheric pressure. s w Units: None, since a ratio is a pure number. Dimensions: 1. Typical values: Water = 1, Mercury = 13.5, Air = VISCOSITY Viscosity is the property of a fluid, due to cohesion and interaction between molecules, which offers resistance to sheer deformation. Different fluids deform at different rates under the same shear stress. Fluid with a high viscosity such as syrup deforms more slowly than fluid with a low viscosity such as water. All fluids are viscous; Newtonian Fluids obey the linear relationship, du given by Newton s law of viscosity.. dy Where is the shear stress; has Units: N m 2 ; kg m -1 s -2 Dimensions: ML -1 T -2 du is the velocity gradient or rate of shear strain, and has dy Units: radian s -1 Dimensions: T -1 µ is the coefficient of dynamic viscosity CAREER AVENUES/2015/FM 13

13 COEFFICIENT OF VISCOSITY ( μ ) The Coefficient of Dynamic Viscosity is defined as the shear force, per unit area, (or shear stress ), required to drag one layer of fluid with unit velocity past another layer a unit distance away. du Force Velocity Force Time Mass dy Area Dis tance Area Area Length Units: N s m 2 or, kg m -1 s -1 Typical values: Water = kg m -1 s -1, Air = kg m -1 s -1, Mercury =1.552 kg m -1 s KINEMATIC VISCOSITY ( ) Kinematic Viscosity is defined as the ratio of dynamic viscosity to mass density. Units: m 2 s -1 ( is often expressed in Stokes, St, where 10 4 St = 1 m 2 s -1 ) Dimensions: L 2 T -1. Typical values: Water = m 2 s -1, Air = m 2 s -1, Mercury = m 2 s SURFACE TENSION ( ) Surface tension is defined as a tension force acting on the surface of a liquid in contact with a gas or on the surface between two immiscible liquid, such that the contact surface behaves like a membrane under tension. It is the tendency of the surface of a liquid to behave like a stretched elastic membrane. Units: N m -1 Dimensions: M T -2 The forces of attraction binding molecules to one another give rise to cohesion, the tendency of the liquid to remain as one assemblage of particles rather than to behave as a gas and fill the entire space within which it is confined. On the other hand, forces between the molecules of a fluid and the molecules of a solid boundary give rise to adhesion between the fluid and the boundary. CAREER AVENUES/2015/FM 14

14 F FLUID MECHANICS It is the interplay of these two forces that determine whether the liquid will wet the solid surface of the container. If the adhesive forces are greater than the cohesive forces, then the liquid will wet the surface. if the cohesive forces are greater than the adhesive forces, then the liquid will not wet the surface CAPILLARY Capillarity phenomenon is defined as rise and falll of the liquid surface in a tube relative to the adjacent general level of liquid depends upon the specific weight of the liquid, diameter of tube and surface tension. The water column in the sketch below rises to a height h such thatt the weight of the column is balanced by the resultant surface tensionn forces acting at θ to the vertical at the contact with the tube. And from equilibrium of forces 4 Cos h gd VAPOR PRESSURE At the surface of a liquid, molecules are leaving and re-entering the liquid mass. The activity of the molecules at the surface creates a vapor pressure, which is a measure of the rate at which the molecules leave the surface. When the vapor pressure of the liquid is equal to the partial pressure of the molecules from the liquid which are in the gas above the surface, the number of molecules leaving is equal to the number entering. At this equilibrium condition, the vapor pressure is known as the saturation pressure. Units: N m -2 (same as atmospheric pressure) Dimensions: ML -1 T -2. Typical values: Water = 760 mmm Hg = bar = 1 atm (at T = C) COMPRESSIBILITY AND BULK MODULUS ( ) The degreee of compressibility of a substance is characterized by the bulk modulus of elasticity. P VV CAREER AVENUES/2015/ FM 15

15 F FLUID MECHANICS Where, P represents the small increased in pressure applied to the substance that causes a decrease of the volume by V from its original volume of V. The negative sign in the definition to ensure that the value of is always positive. Units: N m -2 Dimensions: ML -1 T -2. Typical values: For Water = N m -2, For Air = N m DIFFERENCE BETWEEN SOLIDSS AND FLUIDS The differences between the behaviors of solidss and fluids under an applied force are as follows: i. For a solid, the strain is a function of the applied stress, providing that the elastic limit is not exceeded. For a fluid, the rate of strain is proportional to the applied stress. ii. 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 as long as the force is applied and will not recover its original form when the force is removed. 1.8 TYPES OF FLUID BEHAVIOR When the measured values of shear stress or viscosity are plotted versus shear rate, various types of behavior may be observed depending upon the fluid properties, as shown in figure CAREER AVENUES/2015/ FM 16

16 1.8.1 NEWTONIAN FLUIDS If the shear stress versus shear rate plot is a straight line through the origin (or a straight line with a slope of unity on a log-log plot), the fluid is said du to be Newtonian: dy Where is the coefficient of viscosity NON NEWTONIAN FLUIDS If the shear stress versus shear rate plot is not a straight line, the fluid is said to be Non Newtonian: du A B dy Where A, B and n are constants. For Newtonian fluids A = 0, B = µ and n = 1. n PLASTIC FLUIDS A fluid, in which the shear stress is more than the yield value is known as the plastic fluid. i.e. Shear stress must reach a certain minimum before flow commences DILATANT FLUIDS The fluids having no minimum shear stress and the Viscosity increases with rate of shear, and the value of n>1. e.g. quicksand BINGHAM PLASTIC FLUIDS A minimum value of shear stress must be achieved before the fluid may start flow classification n = 1. An example is sewage sludge PSEUDO PLASTIC FLUIDS The fluids having no minimum shear stress and the viscosity decreases with rate of shear and the value of n < 1, e.g. colloidal substances like clay, milk and cement. 1.9 TIME DEPENDENT FLUID There are some fluids in which the viscosity depends upon the duration of shear. They can be classified as follows: CAREER AVENUES/2015/FM 17

17 1.9.1 THIXOTROPIC FLUIDS The fluids in which viscosity decreases with time for which shear force is applied e.g. jelly paints RHEOPECTIC FLUIDS The fluids in which viscosity increases with time for which shear force is applied e.g. gypsum suspension in water 1.10 TEMPERATURE DEPENDENCY OF VISCOSITY The viscosity of liquids and gases are varies with temperature as follows: LIQUIDS The viscosity of liquids varies with temperature as follows: GASES A exp( B / T) The viscosity of gases varies with temperature as follows: B AT Where, A and B are the constants and T is the temperature. Example 1.1 Calculate the specific weight, specific volume and specific gravity of a liquid having volume of 6 m 3 and weight of 44 kn. Solution: Specific weight, w : Weight of liquid 44 w kn / m Volume of liquid 6 Specific mass, : w / 3 kg m g 9.81 Specific Volume, v : v m 3 / kg Specific gravity, S : wliquid S w 9.81 water 3 CAREER AVENUES/2015/FM 18

18 Example 1.2 The velocity distribution for flow over a flat plate is given by u = 2y y 2 where u is the velocity in m / s at a distance y meters above the plate. Determine the velocity gradient and shear stress at the boundary and 1.5 m from it. Given Data: Dynamic viscosity of fluid is 0.9 N.s / m 2 Solution: u = 2y y 2 du 2 2y dy Velocity gradient, du dy : du 1 At the boundary: At y 0, 2s dy y0 At 0.15 m from the oundary: du 1 At y 0.15, 1.7s dy y0.15 Shear stress, : At the boundary: du N / m y0 dy y0 At 0.15 m from the oundary: du N / m y0.15 dy y0.15 Example 1.3 A 400 mm diameter shaft is rotating at 200 r.p.m. in a bearing of length 120 mm. if the thickness of oil film is 1.5 mm and the dynamic viscosity of the oil is 0.7 N.s / m 2, determine: (i) Torque required to overcome the friction in bearing; (ii) Power utilizes in overcoming the viscous resistance. Solution: Diameter of the shaft, d 400mm 0.4m Speed of the shaft, N 200rpm Thickness of the oil film, t 1.5mm mm Length of the bearing, l 120mm 0.12m 2 Viscosity, 0.7 Ns. / m Tangential velocity of the shaft dn = 4.19 m / s CAREER AVENUES/2015/FM 19

19 () i Torque required, T : du We know, dy where du change of velocity u m / s dy t m N / m Shear Force, F shear stress area F dl F F N Viscous torque F d N ( ii) Power utilised, P : 2 N P T, watts P kw 60 Example 1.4 A clean tube of diameter 2.5 mm is immersed in a liquid with a coefficient of surface tension = 0.4 N / m. the angle of contact of the liquid with the glass can be assumed to be The density of the liquid = kg / m 3. What would be the level of the level in the tube relative to the free surface of the liquid inside the tube? Solution: Given : d 2.5 mm; 0.4/ m, 135, kg / m The liquid level in the tube rises (or falls) due to capillary. The capillary rise (or fall), 4 Cos h wd Cos135 h h 3.39 mm CAREER AVENUES/2015/FM 20

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