Topics
Physical World and Measurement
Physical World
Units and Measurements
 International System of Units
 Measurement of Length
 Measurement of Mass
 Measurement of Time
 Accuracy Precision of Instruments and Errors in Measurement
 Significant Figures
 Dimensions of Physical Quantities
 Dimensional Formulae and Dimensional Equations
 Dimensional Analysis and Its Applications
 Need for Measurement
 Units of Measurement
 Fundamental and Derived Units
 Length, Mass and Time Measurements
 Introduction of Units and Measurements
Kinematics
Motion in a Plane
 Scalars and Vectors
 Multiplication of Vectors by a Real Number
 Addition and Subtraction of Vectors — Graphical Method
 Resolution of Vectors
 Vector Addition – Analytical Method
 Motion in a Plane
 Motion in a Plane with Constant Acceleration
 Projectile Motion
 Uniform Circular Motion
 General Vectors and Their Notations
 Motion in a Plane  Average Velocity and Instantaneous Velocity
 Rectangular Components
 Scalar and Vector Product of Vectors
 Relative Velocity in Two Dimensions
 Cases of Uniform Velocity
 Cases of Uniform Acceleration Projectile Motion
 Motion in a Plane  Average Acceleration and Instantaneous Acceleration
 Angular Velocity
 Introduction
Motion in a Straight Line
 Position, Path Length and Displacement
 Average Velocity and Average Speed
 Instantaneous Velocity and Speed
 Kinematic Equations for Uniformly Accelerated Motion
 Acceleration
 Relative Velocity
 Elementary Concepts of Differentiation and Integration for Describing Motion
 Uniform and NonUniform Motion
 Uniformly Accelerated Motion
 Positiontime, Velocitytime and Accelerationtime Graphs
 Motion in a Straight Line  Positiontime Graph
 Relations for Uniformly Accelerated Motion (Graphical Treatment)
 Introduction
Laws of Motion
 Aristotle’s Fallacy
 The Law of Inertia
 Newton'S First Law of Motion
 Newton’s Second Law of Motion
 Newton's Third Law of Motion
 Conservation of Momentum
 Equilibrium of a Particle
 Common Forces in Mechanics
 Circular Motion
 Solving Problems in Mechanics
 Static and Kinetic Friction
 Laws of Friction
 Inertia
 Intuitive Concept of Force
 Dynamics of Uniform Circular Motion  Centripetal Force
 Examples of Circular Motion (Vehicle on a Level Circular Road, Vehicle on a Banked Road)
 Lubrication  (Laws of Motion)
 Law of Conservation of Linear Momentum and Its Applications
 Rolling Friction
 Introduction
Work, Energy and Power
 Introduction of Work, Energy and Power
 Notions of Work and Kinetic Energy: the WorkEnergy Theorem
 Kinetic Energy
 Work Done by a Constant Force and a Variable Force
 Concept of Work
 The Concept of Potential Energy
 The Conservation of Mechanical Energy
 Potential Energy of a Spring
 Various Forms of Energy : the Law of Conservation of Energy
 Power
 Concept of Collisions
 Non  Conservative Forces  Motion in a Vertical Circle
Motion of System of Particles and Rigid Body
System of Particles and Rotational Motion
 Motion  Rigid Body
 Centre of Mass
 Motion of Centre of Mass
 Linear Momentum of a System of Particles
 Vector Product of Two Vectors
 Angular Velocity and Its Relation with Linear Velocity
 Torque and Angular Momentum
 Equilibrium of Rigid Bodies
 Moment of Inertia
 Theorems of Perpendicular and Parallel Axes
 Kinematics of Rotational Motion About a Fixed Axis
 Dynamics of Rotational Motion About a Fixed Axis
 Angular Momentum in Case of Rotation About a Fixed Axis
 Rolling Motion
 Momentum Conservation and Centre of Mass Motion
 Centre of Mass of a Rigid Body
 Centre of Mass of a Uniform Rod
 Rigid Body Rotation
 Equations of Rotational Motion
 Comparison of Linear and Rotational Motions
 Values of Moments of Inertia for Simple Geometrical Objects (No Derivation)
Gravitation
 Kepler’S Laws
 Universal Law of Gravitation
 The Gravitational Constant
 Acceleration Due to Gravity of the Earth
 Acceleration Due to Gravity Below and Above the Surface of Earth
 Acceleration Due to Gravity and Its Variation with Altitude and Depth
 Gravitational Potential Energy
 Escape Speed
 Earth Satellites
 Energy of an Orbiting Satellite
 Geostationary and Polar Satellites
 Weightlessness
 Escape Velocity
 Orbital Velocity of a Satellite
Properties of Bulk Matter
Mechanical Properties of Fluids
 Concept of Pressure
 Pascal's Law
 Variation of Pressure with Depth
 Atmospheric Pressure and Gauge Pressure
 Hydraulic Machines
 STREAMLINE FLOW
 Bernoulli’S Principle
 Viscosity
 Reynolds Number
 Surface Tension
 Effect of Gravity on Fluid Pressure
 Terminal Velocity
 Critical Velocity
 Excess of Pressure Across a Curved Surface
 Introduction to Fluid Machanics
 Archimedes' Principle
 Stokes' Law
 Equation of Continuity
 Torricelli'S Law
Thermal Properties of Matter
 Temperature and Heat
 Measurement of Temperature
 Idealgas Equation and Absolute Temperature
 Thermal Expansion
 Specific Heat Capacity
 Calorimetry
 Change of State  Latent Heat Capacity
 Conduction
 Convection
 Radiation
 Newton’s Law of Cooling
 Qualitative Ideas of Blackbody Radiation
 Wein'S Displacement Law
 Stefan's Law
 Anomalous Expansion of Water
 Liquids and Gases
 Thermal Expansion of Solids
 Green House Effect
Mechanical Properties of Solids
Thermodynamics
 Thermal Equilibrium
 Zeroth Law of Thermodynamics
 Heat, Internal Energy and Work
 First Law of Thermodynamics
 Specific Heat Capacity
 Thermodynamic State Variables and Equation of State
 Thermodynamic Processes
 Heat Engines
 Refrigerators and Heat Pumps
 Second Law of Thermodynamics
 Reversible and Irreversible Processes
 Carnot Engine
 Isothermal Processes
 Adiabatic Processes
Behaviour of Perfect Gases and Kinetic Theory of Gases
Kinetic Theory
 Molecular Nature of Matter
 Behaviour of Gases
 Equation of State of a Perfect Gas
 Work Done in Compressing a Gas
 Introduction of Kinetic Theory of an Ideal Gas
 Kinetic Interpretation of Temperature
 Law of Equipartition of Energy
 Specific Heat Capacities  Gases
 Mean Free Path
 Kinetic Theory of Gases  Concept of Pressure
 Kinetic Theory of Gases Assumptions
 rms Speed of Gas Molecules
 Degrees of Freedom
 Avogadro's Number
Oscillations and Waves
Oscillations
 Periodic and Oscillatory Motions
 Simple Harmonic Motion
 Simple Harmonic Motion and Uniform Circular Motion
 Velocity and Acceleration in Simple Harmonic Motion
 Force Law for Simple Harmonic Motion
 Energy in Simple Harmonic Motion
 Some Systems Executing Simple Harmonic Motion
 Damped Simple Harmonic Motion
 Forced Oscillations and Resonance
 Displacement as a Function of Time
 Periodic Functions
 Oscillations  Frequency
Waves
description
 Streamline and Turbulent Flow
notes
STREAMLINE FLOW:
The flow of a fluid is said to be steady if, at any point, the velocity of each passing fluid particle remains constant within that interval of time.

Streamline is the path followed by the fluid particle.

It means that at any particular instant the velocities of all the particles at any point are the same. But the velocity of all the particles won’t be the same across all the points in the space.

Steady flow is termed as ‘Streamline flow’ and ‘Laminar flow’.
Consider a case when all the particles of fluid passing point A have the same velocity. This means that the first particle will have velocity V1 and second will have velocity V1 and so on. All the particles will have the same velocity V1at point A.
At point B, all particles will have velocity V2.
Similarly, at point C the velocity of all the particles is V3.
We can see that the velocity is changing from point to point but at one particular point, it is the same.

No two streamlines can intersect.

If two streamlines intersect each other, the particles won’t know which path to follow and what velocity to attain. That is why no two streamlines intersect.
Equation of Continuity

According to the equation of continuity Av = constant. Where A =crosssectional area and v=velocity with which the fluid flows.

It means that if any liquid is flowing in streamline flow in a pipe of nonuniform crosssection area, then the rate of flow of liquid across any crosssection remains constant.

Consider a fluid flowing through a tube of varying thickness.
Let the crosssectional area at one end (l) = A_{1} and crosssectional area of other end (ll) = A_{2}
The velocity and density of the fluid at one end (l) = `v_1`,`rho_1` respectively velocity and density of the fluid at other end (ll) = `v_2,rho_2`
Volume covered by the fluid in a small interval of time `Delta"t"`, across left crosssectional area(l) `="A"_1"xv"_1"x"Delta"t"`
Volume covered by the fluid in a small interval of time `Delta"t"`, across left crosssectional area(ll) `="A"_2"xv"_2"x"Delta"t"`
Fluid inside is incompressible (volume of fluid doesnot change by applying pressure) that is density remains same
`rho_1 = rho_2` ...(1)
Along(l) mass = `rho_1A_1v_1Delta"t"` and along second point (ll) mass = `rho_2A_2v_2Delta"t"`
By eq (1), we can conclude that `"A"_1"v"_1="A"_2"v"_2`. This is the equation of continuity.
From Equation of continuity, we can say that Av=constant.
This equation is also termed as “Conservation of mass of incompressible fluids”.
Conclusion:

The volume flux/Flow rate remains constant throughout the pipe. This means the rate of flow of fluid of liquidis is more if the crosssectional area is more, then the velocity will be less, and viceversa.

But the Av will remain constant.

So the volume which is covered by the fluid at any crosssectional area is constant throughout the pipe even if the pipe has different crosssectional areas.

The fluid is accelerated while passing from the wider crosssectional area towards the narrower area. This means if the area is more the velocity is less and viceversa.
Problem: The cylindrical tube of a spray pump has a crosssection of 8.0cm^{2} one end of which has 40 fine holes each of diameter 1.0mm. if the liquid flow inside the tube is 1.5m min^{1}, what is the speed of the ejection of the liquid through the holes?
Answer:
Area of crosssection of the spray pump, `"A"_1`= 8cm^{2} = 8 x 10^{4}m^{2}
number of holes, n=40
Diameter of each hole, d = 1mm = 1 x 10^{3}m
Radius of each hole, r = d/2 = 0.5 x 10^{3}m
Area of cross section of each hole, `a=pir^2 = pi(0.5xx10^3)^2m^2`
Total area of 40 holes, `"A"_2=nxxa=40xxpi(0.5xx10^3)^2m^2`
`=31.41 xx 10^6m^2`
speed of flow of liquid inside the tube , `"V"_1=1.5 m//min = 0.025 m//s`
speed of ejection of liquid through the holes = `"V"_2`
According to the law of continuity we have:
`"A"_1"V"_1="A"_2"V"_2`
`"V"_2=("A"_1"V"_1)/"A"_2`
`=(8xx10^4xx0.025) 31.61xx10^6`
`=0.633 m//s`
Turbulent Flow:

Fluid flow is said to be turbulent if the velocity of the particles varies at any point erratically.

This means fluid particles are moving here and there, they are not moving in an organised manner. They all will have different velocities.

Eddies are generated by this flow. Eddies are the same as ripples.

All the particles are moving here and there randomly.