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  • BASIC CONCEPTS AND PROPERTIES
    • Fluids and continuum
    • Fluid Statics
    • FLUID KINEMATICS
    • Laminar and Turbulent flow
    • Laminar and Turbulent flow-1
    • Physical Properties of Fluids
    • VISCOSITY
    • Newtonian and Non Newtonian Fluids
    • SURFACE TENSION
    • HYDROSTATIC FORCES ON SURFACES
    • Buoyancy
    • Dimensional Analysis
    • Units and Dimensions
    • Rayleigh’s method
    • Buckingham’s π theorem
    • Dimensionless Numbers
    • Ordinary Differential Equations
    • Non–Linear Equation
    • Second Order Differential equations

  • FLIUD KINEMATICS AND FLUID DYNAMICS
    • Capillary Rise or Depression
    • COMPRESSIBILITY AND BULK MODULUS
    • VAPOUR PRESSURE
    • Cavitation
    • Rheology
    • Pascal's law
    • Relations between height, pressure, density and temperature
    • Pressure measurement
    • stability of submerged and floating bodies
    • SOLID PARTICLES IN A CARRYING LIQUID
    • CONTROL VOLUME APPROACH AND CONTINUITY PRINCIPLE
    • The Control Volume and Mass Conservation
    • CONTINUITY EQUATION
    • Reynolds Transport Theorem
    • Momentum Conservation for Control Volume
    • Momentum For Steady State and Uniform Flow
    • Momentum Equation Application
    • Energy Conservation
    • The First Law of Thermodynamics
    • Approximation of Energy Equation

  • INCOMPRESSIBLE FLUID FLOW
    • Multi–Phase Flow
    • Classification of Liquid-Liquid Flow Regimes
    • Solid–Liquid Flow
    • Ordinary Differential Equations
    • Non–Linear Equation
    • Second Order Differential equations
    • CONTROL VOLUME APPROACH AND CONTINUITY PRINCIPLE
    • The Reynolds Transport Theorem
    • Gauss Theorem
    • Cavitation
    • Navier-Stokes Equations
    • Euler’s equation
    • Bernoulli's Equation
    • Pitot tube
    • Venturi Meter
    • Flow through Orifices
    • Flow Through Mouthpieces
    • Nozzles
    • Notches
    • Weirs
    • Submerged Flow Below Sluice Gate
    • Submereged flow
    • The Flow through Pipes
    • PIPE FLOW
    • Variation of Resistance Coefficients
    • The Hydraulic Gradient
    • Momentum equation application
    • Compressibility Effects in Pipe Flow
    • Pressure Wave Transmission along theHuman Aorta
    • Elementary concept of the uniform flow
    • Flow rate.
    • Continuity
    • The Bernoulli Equation - Work and Energy
    • Bernoulli’s Equation
    • Flow over submerged bodies
    • Drag Force and its Coefficient
    • Drag on sphere

  • HYDRAULIC PUMPS
    • PIPE OR TUBE BENDING
    • MASS, MOMENTUM , AND ENERGY EQUATIONS
    • Flow Measurements
    • DETERMINATION OF COEFFICIENT OF DISCHARGE
    • Head Losses
    • Laminar Flow
    • Sudden Changes To Pipe Size Or Shape
    • Sudden Contraction
    • Flow Between Parallel Plates
    • Introduction to Boundary Layer Analysis
    • Boundary-layer thickness
    • Two-dimensional Boundary Layer along a Flat Plate
    • Laminar Boundary Layer Theory
    • Mathematical Formulation of Laminar Boundary Layer
    • Application of Von-Karman Integral Momentum Equation
    • Turbulent Boundary Layer Theory
    • The Laminar Sublayer
    • Total drag

  • Fluid Dynamics
    • Impulse-Momentum Principle
    • Moment of Momentum Equation
    • Momentum Equation
    • Kinetic energy and Momentum correction factors
    • Stokes' law
    • Darcy's law
    • Fluidization
    • Viscosity Measurement
    • Transition from laminar to turbulent flow
    • introduction of Turbulent Flow
    • Equation for turbulent flow
    • Reynolds stress
    • BOUNDARY LAYER SEPARATION CONTROL
    • Turbulent flow in pipes
    • Velocity distribution over smooth and rough surface
    • WATER HAMMER
    • ANALYSIS AND DESIGN OF A SIMPLE SURGE TANK
    • Flow in a sudden expansion
    • Diffuser and Nozzle
    • Introduction to Compressible Flow
    • Ideal Fluid
    • Free Vortex Flow
    • Drag Classification
    • Magnus effect
    • Turbulence

Branch : Civil Engineering
Subject : Fluid Mechanics
Unit : HYDRAULIC PUMPS

Application of Von-Karman Integral Momentum Equation


Introduction: An absolute estimate of flat-plate laminar boundary layer properties for

Application of method A: Self-similarity of boundary layer velocity profiles signifies that

Where , the boundary layer thickness, = f(x), the distance from the leading edge the boundary layer thickness, = f(x), the distance from the leading edge
The boundary conditions to be satisfied for a boundary layer over a flat plate are:
At y = 0: u = 0, (since ), ,The last condition is imposed by the B L equation :


since LHS = 0 at y = 0.)

and at : u = uo, (since at
Simplest polynomial form of satisfying these is


Now it is found that this form of velocity profile does indeed correspond quite well with measurements for laminar B.L.s over flat plates: so what follows can represent the behavior of such a laminar B.L.Then, by definition, Displacement Thickness


Similarly, for Momentum Thickness :


Therefore

 

Substituting in Momentum Equation for zero pressure gradient



Local friction coefficient,

 

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