FLUID MECHANICS For
Mechanical Engineering By
www.thegateacademy.com
Syllabus
Fluid Mechanics
Syllabus for
Fluid Mechanics Fluid properties; fluid statics, manometry, buoyancy; control-volume analysis of mass, momentum and energy; fluid acceleration; differential equations of continuity and momentum; Bernoulli's equation; viscous flow of incompressible fluids; boundary layer; elementary turbulent flow; flow through pipes, head losses in pipes, bends etc.
Analysis of GATE Papers (Fluid Mechanics) Year
Percentage of marks
2013
6.00
2012
6.00
2011
5.00
2010
8.00
2009
8.00
2008
4.67
2007
10.00
2006
11.33
2005
7.33
Overall Percentage
7.37%
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Contents
Fluid Mechanics
CONTENTS
#1.
#2.
#3.
#4.
Chapters
Page No.
Fluid Properties
1-16
1 1-4 5 5–6 7 – 10 11 – 12 12 – 13 14 14-16
Fluids Properties of Fluids Newton’s Law of Viscosity Types of fluids Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Fluid Statics
17-51
17 – 19 19 – 21 21 – 22 22 – 24 25 – 40 41 – 42 42 – 45 46 46 – 51
Fluid Pressure Forces on Submerged Bodies Buoyancy Floating Bodies Stability Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Fluid Kinematics
52 – 67
52 – 55 55 – 57 58 – 61 62 63 64 64 – 67
Fluid Flow Continuity Equation Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Fluid Dynamics
68 – 89
68 68 – 69
Equations of Motion Euler’s Equations of Motion
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Contents
#5.
#6.
#7.
Bernoulli’s Equation Practical applications of Bernoulli’s theorem Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Fluid Mechanics
69 – 70 70 – 74 75 – 81 82 – 83 83 – 85 86 86 – 89
Boundary Layer
90 – 112
90 – 94 94 – 98 98 – 99 100 – 106 107 – 108 108 – 109 110 110 – 112
Boundary Layer Dimensional Analysis Lift & Drag Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Flow through pipes
113 – 132
113 – 116 117 – 119 120 – 123 124 – 125 125 – 127 128 128 – 132
Losses in pipes Viscous flow Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Hydraulic Machines
Dynamic Force on a Curve Blade Moving In Translation Theory of turbo machines Pelton Wheel Reaction Turbines Specific Speed and Performance of Turbines Cavitation in Turbines Centrifugal Pumps Solved Examples Assignment 1
133 – 167 133 – 134 134 – 135 136 – 140 140 – 142 143 – 144 145 145 – 153 154 – 161 162 – 163
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Contents
Assignment 2 Answer Keys Explanations
Fluid Mechanics
163 – 164 165 165 – 167
Module Test
168 – 187
Test Questions
168 – 177
Answer Keys
178
Explanations
178 – 187
Reference Books
188
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Chapter 1
Fluid Mechanics
CHAPTER 1 Fluid Properties Fluids It is defined as a substance which deforms continuously even with a small amount of shear force exerted on it, whereas a solid offers resistance to the force because very strong intermolecular attraction exists in it. Both liquids and gases come under the fluids. i) ii) iii)
Liquid It has definite volume but no shape for all practical purposes considered incompressible Gas It has no shape and volume highly compressible Vapour A gas whose temperature and pressure are such that it is very near to the liquid phase e.g. Steam
Properties of fluids Mass Density It is defined as mass per unit volume. Unit: kg / m3, Dimension: M / L3 Absolute quantity i.e., does not change with location. As pressure increases mass density increases (as large number of molecules are forced into a given volume) Specific Weight Weight of the substance per unit volume. Also represents force exerted by gravity on a unit volume fluid. Mass density and specific weight of a fluid are related as: ; where g = acceleration due to gravity Units: N/m3, Dimensions: It is not an absolute quantity, varies from place to place, because g is changing from place to place primarily latitude and elevation above M.S.L. Specific weight of water . Specific Volume Volume occupied by a unit mass of fluid, Units: m3/kg
(reciprocal of density)
Specific gravity (G) Specific gravity,
For liquids, standard fluid is water at 40C For gases, standard fluid is hydrogen or air. Units: No units (as it is a ratio of two quantities having same unit) THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 1
Chapter 1
Fluid Mechanics
Specific gravity of water = 1.0, Mercury = 13.6 Since the density of fluid varies with temperature, specific gravity must be determined and specified at a particular temperature. Viscosity A measure of ’ resistance to shear. A property by virtue of which it offers resistance to the movement of one layer of fluid over the adjacent layer. It is due to intermolecular cohesion and transfer of molecular momentum between layers. Dynamic Viscosity Units SI: Pa.sec or N.sec/m2 MKS: kg/(m.sec) CGS: Poise = dyne.sec/cm2 Conversion 1 poise = 0.1 Pa.sec. Dimensions M or F
T
It is dependent on pressure. For liquids dynamic viscosity decreases with gases increase in temperature because density of liquid decreases with increase in temperature for it decreases with increase in temperature because molecular momentum increases and cohesion is negligible in gases. Kinematic Viscosity Units S I: m2/sec CGS: cm2/sec or stokes Dimensions L2 Kinematic viscosity depends on both pressure and temperature Vapour Pressure In a closed vessel at a constant temperature, the liquid molecules break away from the liquid surface and enter the air space in vapour state. When the air above the liquid surface is saturated with liquid vapour molecules then the pressure exerted on liquid surface is called vapour pressure. Vapour pressure increases with temperature. The low vapour pressure of mercury (along with high density) makes it very suitable for use in barometers and other pressure measuring devices.
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Chapter 1
Fluid Mechanics
Cavitation Occurs in a flow system, dissolved gases (vapour bubbles) carried into a region of high pressure and their subsequent collapse gives rise to high pressure, which leads to noise, vibrations and erosion. Cavitation occurs in 1. 2. 3. 4.
Turbine runner Pump impellers Hydraulic structures like spillways and sluice gates Ship propellers.
Compressibility Change in volume (or density) due to change in pressure. Compressibility is inversely proportional to Bulk Modulus K.
(Negative sign indicates a decrease in volume with increase in pressure) Coefficient of compressibility Surface Tension Cohesion: Force of attraction between the molecules of the same liquid. Adhesion: Force of attraction between the molecules of different liquids (or) between the liquid molecules and solid boundary containing the liquid. A liquid forms an interface with a second liquid or gas. This liquid – air interface behaves like a membrane under tension. The surface energy per unit area of interface is called Surface Tension. It can also be expressed as a line force: Force per unit length. Units N/m Dimensions FL-1 or MT-2. Surface tension is due to cohesion between liqu A surface tension decreases (because cohesion decreases) Due to cohesion, surface tension pressure changes occur across a curved surface of
⟶
(i) Liquid jet (ii) Droplet (iii) Soap bubble. A) Liquid jet: Increase in pressure inside and outside of liquid jet where d = dia of jet B) Liquid drop:
where d = dia of drop let
C) Soap bubble:
where d = dia of bubble.
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Chapter 1
Fluid Mechanics
Capillarity The phenomenon of rise or fall of a liquid surface relative to the adjacent general level of liquid in small diameter tubes. The rise of liquid surface is designated as capillary rise and lowering is called capillary depression. Capillary rise happens when adhesive is stronger than cohesive for example in water capillary depression happens when cohesive is stronger than adhesive for example in mercury.
h
Water
Figure. 1.2
h
mercury
Figure. 1.3 for e.g., Mercury depressive with convex upwards capillary (rise or fall) units: cm or mm of liquids . . (1.1) σ θ b d = dia. of tube θ = 00 → W
q
b
θ 3 0→M For tube dia. > 12mm capillary effects are negligible. Hence the dia. of glass tubes used for measuring pressure (manometers, piezometer etc.) should be large enough.
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Chapter 1
N
’
Fluid Mechanics
V moving plate
U F
Gap filled with fluid θ stationary Figure. 1.4 ∝
shear deformation)
∝ Where F is the Force required to move the surface a ∝
τ
‘A’
μ
Differential form:
( )
. . . (1.2)
W τ ; V ;μ D A N ’ v v which fluid deforms(u / y)is inversely proportional to viscosity
v , the rate at .
Types of Fluids Ideal Fluid or Perfect Fluid
Non viscous (frictionless) and incompressible Used in the mathematical analysis of flow problems Does not exist in reality Does not offer shear resistance when fluid is in motion.
Real Fluid
Possess the properties such as viscosity, surface tension and compressibility. Resistance is offered when they are set in motion.
Newtonian Fluid
W
b
N
’
V
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