# CH.1 Overview of Fluid Mechanics/22 MARKS. 1.1 Fluid Fundamentals.

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1 Content : 1.1 Fluid Fundamentals. 08 Marks Classification of Fluid, Properties of fluids like Specific Weight, Specific gravity, Surface tension, Capillarity, Viscosity. Specification of hydraulic oil Pascal s law. Types of fluid flow- Steady, unsteady, rotational, irrational, laminar, turbulent, one, two and three dimensional flow, Uniform and non uniform flow. (Definitions and applications only) Pressure Measurement. Concept of atmospheric pressure, gauge pressure, vacuum pressure, absolute Pressure. Pressure Gauges - Piezometer tube, simple and differential manometer, micro manometer. (Theoretical Treatment only, No Analytical treatment / Problems on Manometers.) Bourdon tube pressure gauge. 1.2 Hydrodynamics. 14 Marks Basic principles of fluid flow Law of continuity and its applications. Energy possessed by the liquid in motion. Bernoulli's theorem and its applications such as Venturimeter, Orifice meter and pitot tube. (Analytical treatment with derivation for measurement of discharge is expected). Hydraulic coefficients Concept of Vena Contracta. Coefficient of contraction, coefficient of velocity, coefficient of discharge, Coefficient of resistance. Relation between the hydraulic coefficient Page 1

2 1.1 Fluid Fundamentals :- (Introduction) Read Once. Fluid :It is a substance as (liquid / gas) that is capable of flowing (or) that changes it shape at a steady rate when acted upon by a force tending to change its shape. It has no definite shape of its own. It takes the shape of container in which it is kept. Generally fluids are classified into three categories. i) Liquid :- Water, Oil etc. It is a in compressible. ii) Gaseous :- O2, H2, N2, etc. It is compressible. iii) Vapours :- When the liquid is heated upto certain temperature then it converts into vapour which is also compressible. Simillar to gases. Fluid Mechanics :It is the branch of science which deals with study of behavior of fluids at rest as well as in motion. a) Fluid statics (or) Hydrostatics : When the fluid is at rest (or) in a static condition i.e. not moving then the study of fluid is called as fluid statics (or) Hydrostaics. Eg : When it is stored in tank, container, Drum, etc. b) Fluid Kinematics (or) Hydro kinematics : when fluid is in motion i.e. When it is moving then study of fluid without considering external forces on fluid is called as fluid kinematics (or) hydrokinematics. Eg : When the fluid is running through pipe. Channels : Canals etc. C) Fluid Dynamics (or) Hydrodynamics : It is the study of fluid in motion with considering different forces acting on it then it is called as fluid dynamics (or) hydrodynamics. Eg : Pump discharge. Page 2

3 Classification of Fluids :1. Ideal Fluid :The fluid which does not show the property like viscosity & compressibility then fluid is called as Ideas fluid. It is an imaginary fluid such type of fluid does not exist in real world. 2. Real Fluid :A fluid which shows the property like viscosity, compressibility then such type of fluid is called as real fluid. Generally all fluids are real fluid in world. 3. Newtonian fluid :The fluids which obeys Newton`s law of viscosity. (or) A fluid whose viscosity does not change with the rate of deformation (or) shear strain is known as Newtonian fluid eg : water, air, thin motor oil. Graphically it is represented by straight line on shear stress v/s shear strain graph. 4. Non Newtonian fluid :A fluid which does not obey Newton s law of viscosity. (or) A fluid whose viscosity chages with the rate of deformation (or) shear strain is known as Non Newtonian fluid. Eg : Most Viscous fluid. Eg : Non drip paint, Polymer solution, Blood, Solid suspension. Graphically it is represented by a Curve on shear stress v/s Shear strain graph. 5. Ideal Plastic fluid :A fluid in which shear stress is larger than yield value of stress and is directly proportional to rate of shear strain (i.e. rate of deformation) (or) velocity gradient. Newton s law of Viscosity :It states that, The shear stress on a layer of fluid is directly proportional to the rate of shear strain. Page 3

4 Properties of Fluids :1.Density (Mass Density) : It is the ratio of mass of the fluid to its volume. It is denoted by symbol of Rho. S.I. Unit is Kg/m3 2.Specific volume :It is the reciprocal of density of fluid. (Or) It is the ratio of volume of fluid to its mass. S.I. Unit is m3/kg 3.Specific weight: Specific Weight of a fluid is the ratio between the weight of a fluid to its volume. Or weight per unit volume of a fluid is called specific weight. It is denoted by w. S. I. unit is N/m3 4.Specific gravity: It is defined as the ratio of the weight density (density) of a fluid to the weight density (density) of a standard fluid. It is denoted by S. It is a unit less quantity. 5.Surface Tension: - It is the force required to maintain unit called length of the film in equilibrium condition. Or It is the property of fluid which is defined as the tensile force acting on the surface of a liquid in contact with a gas or on the surface between two immiscible liquids such that the contact surface behaves like a membrane under tension. Unit:-N/m Phenomenon of Surface tension :Let us consider the two molecules of liquid at points A and B. Molecule at point A is equally attracted from all sides since it has molecules from all sides and therefore the forces acting at this point are in equilibrium condition. However at point B, there is no liquid molecule at above side and consequently there is a net downward force on the surface of liquid is normal to the liquid surface, due to this a special layer seems to form on a liquid at the surface, which is in tension and small loads can be supported over it e.g. a small needle placed gently upon the water will not sink but will be supported by the tension at the water surface. 6.Capillarity : :- It is defined as a phenomenon of rise or fall of a liquid surface in a small tube relative to the adjacent general level of liquid when the tube is held vertically in the liquid. The rise of liquid surface is known as capillary rise while the fall of the liquid surface is known as capillary fall or depression. Phenomenon of Capillarity:- I) when the liquid molecules possess relatively greater affinity for solid molecules or, in other words, liquid has adhesion greater than cohesion then it will wet the solid surface in contact and will tend to rise at the point of contact. This results concave upwards and the angle of contact θ which is less than 900. This is also known as Capillary Rise. II) If the liquid has less attraction for solid molecules or, in other words, Cohesion predominates, then liquid will not have tendency to wet the solid surface in contact and this will result in depression of liquid at that point in the concave downward Page 4

5 shape and at the angle θ more than 90 mercury. 0 e.g. glass tube is inserted inside the This phenomenon of rise or fall of liquid surface relative to the adjacent general level of liquid is known as Capillarity. 7.Viscosity Or Dynamic viscosity : It is the property of fluid which offers resistance to the moment of one layer of fluid over another adjacent layer of fluid. S. I. unit is N-S/m2 When the property is related to moving liquid then it is called as dynamic viscosity. Kinematic Viscosity :It is the ratio of absolute viscosity in (N-s/m2) to the density of liquid in (Kg/m3) Mathematically V = There is no specific unit of kinematic viscosity. Pascal s law : It states that The intensity of pressure at any point in a fluid at rest is same in all directions. In other words when a certain pressure is applied at any point in fluid at rest the pressure is equally transmitted in all directions and to every other point in the fluid. px = py = pz where, px = intensity of pressure in x direction; py = intensity of pressure in y direction; pz= intensity of pressure in z direction. Applications:- Hydraulic press, Hydraulic brakes, Hydraulic jack, hydraulic lift. Page 5

6 Explanation : Figure This law discovered by French scientist Blasie pascal. Now a days which is a very useful in Industrial fluid power sector. This law provides a phenomenon of force multiplication using multiple areas. Here small force applied on one side or smaller piston i.e. 10 N that of obtained from bigger piston end is 100 N. this is due to Pascal s law. According to definition pressure at small end is P1 = 10/0.1 = 100 N/m2 P = F/A But intensity of pressure at different areas are same like P1 = P2 = P3 Hence force obtained at outlet. F2 = P3 X A2 =100 X 1= 100N. Force obtained at bigger piston is 10 times more & displacement rate at smaller piston is 10 times more than that of at bigger end. Page 6

7 List different types of fluid flow. Types of Flows are: Steady and Unsteady flow. Uniform and Non Uniform flow. Laminar and Turbulent flow. Rotational and irrotational flow. Compressible and incompressible flows. One, Two, Three dimensional flows. 1) Steady flow : The flow is said to be steady when the flow characteristics, such as velocity, density, pressure and temperature do not change with time. Example : Flow of water through tap. 2) Unsteady flow: The flow is unsteady if the velocity and other hydraulic characteristics change with respect to time. Example : Flow of water through conical pipe. 3) Uniform flow : The flow is said to be uniform when the velocity and other characteristics are constant in a particular reach. Example : 4) Non uniform flow : The flow is non uniform when the flow characteristics change at various points along the path. Example : 5) Laminar flow: The flow in which each liquid particle has definite path and the path of individual particles do not cross each other is called as stream line flow. Example: flow of river having large banks, flow of tap water, flow of water through cannel, flow of thick oil through tube. 6) Turbulent flow: Flow in which each liquid particle does not have a definite path, and the paths of individual particles also cross each other is called turbulent flow. Example: flow of river during flood, flow of water after opening valve. 7) Compressible flow : The flow is said to be Compressible when the volume of the fluid and density of fluid changes during flow. Example: Flow of air, gas through pipe. Page 7

8 8) Incompressible flow : The flow is said to be incompressible when the volume of fluid and its density does not changes. Example: Flow of water, oil through pipe. 9) Rotational flow : A flow, in which the fluid particles also rotate about their own axis while flowing, is called a rotational flow. Example: Natural cyclones, cyclones in water during flood. 10)Irrotational flow : A flow in which the fluid particles do not rotates about their own axis and retain their original orientations while flowing, is called a irrotational flow. Example: Steady and continues flow of water 11)One dimensional flow :- When the various characteristics of flowing fluid are functions of only one of the three co-ordinate directions and time t. i.e. these may not vary only in one direction, then the flow is said to be one dimensional flow. Or a flow in which streamlines of its moving particles may be represented by straight line is called one dimensional flow. Example: Not available in actual practice. 12)Two dimensional flow :- When the various characteristics of flowing fluid are functions of any two of the three co-ordinate directions and time t. i.e. these may vary in any two of the three directions then the flow is said to be two dimensional flow. Or a flow whose streamlines may be represented by a curve is called two dimensional flow. Example: Flow of water through pipe. 13)Three dimensional flow :- when the various characteristics of flowing fluid such as velocity, pressure, density, temperature etc. are function of space and time. i.e. these may vary with co-ordinate (x,y,z) & time, such fluids are said to be three dimensional flow. Example: Page 8

9 Concept of Pressure : Atmospheric Pressure: At the earth surface, the pressure due to the weight of air above the earth surface is called as atmospheric pressure. Gauge Pressure: If the pressure is measured above the atmospheric pressure it is called as gauge pressure.(positive pressure) Vacuum Pressure: If the pressure is measured below the atmospheric pressure it is called as Vacuum pressure.(negative pressure) Absolute Pressure: Absolute Pressure is defined as the pressure which is measure with reference to Absolute zero pressure.(it may be above or below atmospheric pressure) Relation between pressure 1.Gauge pressure = Absolute pressure Atmospheric pressure 2.Vacuum pressure = Atmospheric pressure Absolute pressure Graphical representation Pressure Gauges : Types of manometers : a)simple manometers : 1.Piezometer tube 2.U-Tube manometer 3.Single column manometer.(micro/sensitive manometer) b)differential manometers : 1.U-tube differential manometer Page 9

10 2.Inverted U-tube differential manometer Simple manometers : (NOTE : According to weight age add analytical treatment for manometer.) 1.Piezometer tube :It is a simplest form of manometer used for measurement of gauge pressure specially for low and moderate pressure. Construction : It consist of small diameter transparent glass tube which is bend at lower end in 90o There is a calibrated scale on the surface of glass tube, having one of its end connected to a point where pressure is to be measured and other end remains open to atmosphere. Working : When water is flowing through pipe which is shown in figure, water will rush in glass tube called as Piezometer tube. Due to which water level in tube rises which is shown by letter h in glass tube.ie in the term of pressure head directly. Pressure head in the pipe = h m of water. Intensity of pressure = w x h N/m2 Where w is specific weight of water. Limitations of Piezometer tube: 1.Piezometers can measure gauge pressures only. It is not suitable for measuring negative pressures. 2. Piezometers cannot be employed when large pressures in the lighter liquids are to be measured since this would require very long tubes, which cannot be handled conveniently. Page 10

11 2.Simple U tube manometer : Page 11

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14 3.Micro manometer : For only figure and construction refer following article It is sensitive manometer of a modified version of U tube manometer & used for precise measurement of pressure. With micro manometer we can measure the slight changes of pressure in pipe. These slight changes are difficult to measure with the help of ordinary U- tube manometer. The left limb is connected to the pipe and right limb is open to atmosphere. Right limb is either vertical or inclined. The high pressure liquid in pipe will push the heavy liquid in basin downward this causes liquid in right limb to rise considerably. That means we can magnify the slight pressure difference at point A, which can be easily measured by right limb. figure OR For getting expression of pressure refer following article Page 14

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20 1.U tube differential manometer : Page 20

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22 2.Inverted U-tube differential manometer construction and working of inverted U tube differential manometer An inverted differential manometer is used for measuring difference of low pressure, where accuracy is the prime consideration. It consists of an inverted U tube, containing light liquid. One end is connected to A and other is connected to B. Let us assume that the pressure at point A is more than that at point B. Let us take Z-Z as the datum line in this case h1=height of liquid in the left limb below Z-Z h2=reading of the manometer h3= Height of liquid in the right limb in cm S1 and S3=Specific gravities of liquids in the left limb and liquid in the right limb respectively. ha= pressure head in pipe A hb=pressure head in pipe B with reference to the Fig ha S1h1 = hb - S2h2 - S3h3 ha - hb = S1h1 - S2h2 - S3h3 m of water Pressure difference between pipe A and B is w x (ha - hb) Page 22

23 Mechanical type Bourdon s tube pressure gauge : It is a device which is used for the measurement of high pressure (ie +ve) which is above atmospheric as well as pressure below the atmospheric (ie.- ve) pressure. Construction : The device consist of metallic tube, generally this cross section is elliptical. One end of the tube is closed and another is fitted to the pipe where pressure is to be measured. The dial and the pointer fitted over the mechanism. Working : As flowing fluid under pressure enters the tube, the tube tends to be straightening. This causes the free end of the tube to move which is connected to pinion and sector arrangement. The pointer deflect on the calibrated scale, which directly indicates pressure in the term of N/m2 Application : 1.To check the pressure of air in tyre tube, in compressor. 2.To check the pressure of steam in boiler. 3.To check the pressure of liquid in pipe. Page 23

24 1.2 Hydrodynamics : 14 Marks Rate of flow (or) Discharge :The quantity of fluid flowing per second through a section of pipe (or) channel is known as rate of flow (or) discharge. It is denoted by letter Q. Now consider liquid flowing through pipe whose crossectional area is a with velocity v then the discharge is given as. Q = a.v Where a= Corssectional area of pipe at section (1)-(1) in m2 V= Avg velocity of flow in pipe in m/sec. Discharge = Q= a.v = m2 m/sec = m3/sec (or) litre /sec. i) when the fluid is incompressible (eg. water) then discharge is expressed in term of volume of fluid flowing per second through pipe (or) channel. Unit of Q is m3/sec (or) litre/sec ii) When the fluid is compressible (eg. Air, gas) then discharge is expressed in term of weight of fluid flowing per second through pipe (or) channel. Unit of Q is Kgf/sec (or) Newton /sec. Page 24

25 Law of Continuity :- Continuity equation is also based upon principle of conversation of mass For a fluid flowing through the pipe at all the cross-section, the quantity of fluid flowing per second is constant. Let, V1= Average velocity of fluid at section a-a P1= Density of fluid at section at section a-a A1= Area of pipe at section a-a And V2, P2, A2, are corresponding valve at section b-b Then rate of flow at section a-a (mass of liquid flowing per unit time) = P1A1V1 Similarly, rate of flow of section b-b = P2A2V2 According to the principle of conservation of mass i.e. mass can t be created not be destroyed. The total quantity of fluid passing through section a-a and b-b is same. P1A1V1 = P2A2V2 Above equation is applicable for both compressible and incompressible fluid. If considering liquids, P1= P2 A1V1=A2V2 Page 25

26 Further AV= Q Where Q is volume of liquid flowing through any section per unit time or volume rate of flow of liquid which is known as discharge. It is expressed in terms of m 3/sec. or lit/sec. Applications :1. steady and unsteady flow 2. uniform and non uniform flow 3. compressible and incompressible flow State the law of continuity. Water flows through a pipe of diameter 1.6m with a velocity of 3m/s Find the rate discharge though pipe. Law of continuity: It states that if an incompressible liquid is continuously flowing through a pipe or a channel whose cross sectional area may or may not be constant then quantity of liquid passing through it per second is same at all sections. Given data : Diameter of pipe (d) = 1.6 m Velocity of flow (v) = 3 m/s Area of pipe A = /4 d2 = /4 (1.6) 2 = m2 By continuity question, Q=A v = = 6.03 m3/s Application of continuity equation :Branching of Pipe :We know that in day to day life for diverting a flow of water any other place. We make a branches of pipes which is shown in fig. Page 26

27 In fig, Single pipe having crossectional area A ; Which divide into two equal parts. (or) branches. Then Total Discharge (or) = Q = Q1 + Q2 Flow rate AV = A1 V1 + A2 V2 Where V = Velocity of flow in pipe (A) V1 = Velocity of flow in pipe (B) V2 = Velocity of flow in pipe (C) Bernoulli s theorem : This theorem states that whenever there is a continuous flow of liquid, the total energy at every section remains the same provided that there is no loss or addition of the energy. Mathematically, P/w + v2 /2g + Z = constant Page 27

28 Where, P/w = pressure energy V2 /2g = kinetic energy Z = potential energy Assumption: 1) The fluid is ideal. 2) The flow is steady. 3) The flow is incompressible. 4) The flow is irrotational. Limitations of Bernoulli s Theorem :1. We know from Bernoulli s Theorem velocity of liquid at every section is remains constant but in actual practice velocity changes at every section because velocity in pipe at centre is maximum while it is minimum at wall side due to friction between liquid surface and pipe surface. 2. According to Bernoulli s Theorem there is no addition of external forces except gravitational force but in actual practice due to friction there is addition of external forces. 3. According to Bernoulli s Theorem there is no loss of energy but practically kinetic energy is converted into heat energy. 4. If liquid is passing through a curved path then the energy due to centrifugal force come into picture. Applications of Bernoulli s Theorem List of applications:- 1) Venturi meter 2) Orifice meter 3) Pitot tube 4) Rota meter 5) Nozzle meter or Flow nozzle 6) Elbow meter or Pipe bend meter Page 28

29 Venturimeter :- A Venturimeter consists of a converging cone, a throat section and diverging cone, all combined in one unit. As the flow takes place in the converging cone, velocity increases, and there is a fall in the pressure according to the Bernoulli s equation. Consider the arrangement shown in the figure where the fluid passes from point 1 (inlet) to point 2 (throat) and the manometer is fixed between them. Appling Bernoulli s equation to points 1 and 2 with datum at this axis, considering horizontal venturimeter, Z1=Z2 Construction: it is device used for measuring the rate of flow or a discharge of a fluid flowing through a close pipe or channel. It consist of three parts: 1. Short convergent cone 2. Throat 3. Long divergent cone 1.Short convergent cone: The inlet section of Venturimeter is called as a convergent cone. The diameter of convergent cone is equivalent of diameter of pipe.(d1). In convergent section the diameter decreases to diameter to (d2). Here velocity increases and pressure decreases. The other end of convergent section is attached to throat. In convergent section the pressure measuring device i. e. piezometer is connected which gives a pressure head in cm or m. the angle suspended by a convergent section with throat is Throat: It is small constant diameter pipe. In which there is no fall or increase in pressure and velocity. At the upper end of throat, pressure measuring device is connected to measure the pressure head in terms of cm or m of liquid. The pressure difference between two pizometer is measured and rate of discharge is calculated. 3.Long divergent cone: It is used to regain original pressure which is smaller in convergent section. In divergent cone due to increase in diameter there is increase in pressure. However if decrease in velocity of flow in divergent section is allowed to Page 29

30 take place rapidly in small length, then the flowing fluid will not remain in contact with the boundary of diverging flow passage, flow will separate from walls and eddies are formed. Therefore length of divergent section has more than convergent section and it is kept 2 to 3 times that of convergent section. Application: 1. Lubricator 2. Carburetor 3. To measure discharge through pipe. Explain why divergent section has more length than convergent section? In convergent cone because of gradual decrease in diameter there is increase in velocity i.e. kinetic energy there should be decrease in pressure energy. In convergent cone velocity of fluid is increased. This acceleration of flowing fluid may allow to take place rapidly in a relatively small length, without resulting in appreciable loss of energy. In divergent cone due to increase in diameter there is increase in pressure. However if decrease in velocity of flow in divergent section is allowed to take place rapidly in small length, then the flowing fluid will not remain in contact with the boundary of diverging flow passage, flow will separate from walls and eddies are formed. Therefore length of divergent section has more than convergent section and it is kept 2 to 3 times that of convergent section. Page 30

31 Derivation for measurement of discharge Page 31

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35 Orifice meter :- it is used to measure the discharge in pipe. It consist of a circular plate having a sharp edge circular hole known as an orifice. The plate is fixed inside a pipe as shown in figure. Working principle of Orifice meter is similar to that of Venturimeter. Mercury manometer is connected to know pressure difference between pipe and throat.ie Orifice. Working : As the fluid flows through the orifice meter it accelerates thereby increasing velocity and decreasing pressure since orifice diameter is less than the pipe diameter. This pressure difference is measured by the manometer. Orifice meter is cheaper for discharge measurement and requires smaller space as compared with venturimeter. Page 35

36 Derivation for measurement of discharge through Orifice meter : Page 36

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38 Pitot Tube :- This is an instrument used to determine the velocity of flow at a desired section in a pipe or stream. shows as pitot tube which in its simplest form consists of a 900 bent glass tube. The tube is placed in flow such as leg is vertical and the other leg is horizontal. The horizontal leg has the open end facing upstream. The tube is used for measuring the local velocity. At the tip of the tube, the velocity is zero. This point is called the stagnation point. The pressure at the tip of the tube is called the stagnation pressure. OR Page 38

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42 OR Vena-Contracta : Consider a small circular orifice with sharp edges in the side of a tank. Let the centre of the orifice be at a depth below the free surface. Let us assume that the orifice is discharging free into the atmosphere. As the fluid flows through the orifice, it contracts and attains a parallel form at a distance of about d/2 from the plane of orifice. This is due to the fact that the fluid particles cannot change their directions abruptly. The point at which the streamlines first become parallel and get maximum contraction is termed the vena contracta. Significance of vena contracta- 1) To measure the flow rate of fluid, 2) To find out Cd, Cc & Cv (hydraulic coefficients) Hydraulic coefficients : There are four hydraulic coefficients1.coefficient of contraction (Cc): It is the ratio of area of jet at vena contracta to the area of Orifice is known as Coefficient of contraction. 2.Coefficient of velocity(cv): It is the ratio of actual velocity of jet at vena contracta to the theoretical velocity of jet is known as Coefficient of velocity 3.Coefficient of discharge (Cd): It is the ratio of actual discharge through an orifice to the theoretical discharge is known as Coefficient of discharge. 4.Coefficient of Resistance (Cr): It is the ratio of loss of head in the orifice to the head of water available at the exit of orifice is known as Coefficient of resistance. Page 42

43 Relation among Cd, Cc and Cv : We know, Actual Discharge Cd = Theoretical Discharge But by continuity equation Q = Area Velocity Hence Cd = Actual Area Actual velocity Theoretical Area theoretical velocity Cd = Cc Cv (Note : For numericals please refer additional given material at zerox centre in college campus.) Page 43

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