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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 : INCOMPRESSIBLE FLUID FLOW

Second Order Differential equations


Introduction: The simplest equations are constant coefficients such:

Substituting est results in the quadratic equation

If b2 > 4 a c then there are two unique solutions for the quadratic equation and the general solution is

For the case of b2= 4 a c the general solution is

In the case of b2> 4 a c the solution of the quadratic equation is a complex number which means that the solution has exponential and trigonometric functions as

Where the real part is

And the imaginary number is

Non-Homogeneous Equation:

Equation that not equal to zero in this form

The solution of the homogeneous equation is zero so additional solution of l(x) though the operation the right hand side is the total solution as

Where the solution uh is the solution of the homogeneous solution and up is the solution of the particular function l(x). If the function on the right hand side is polynomial than the solution is will

                               

Non-Linear Second Order Equation:

Some of the techniques that were discussed in the previous can be used for the second order differential equation such variable separation. If the following equation

Can be written in the form

Then the equation it is referred that equation is separable. The derivative of u˙ can be treated as new function v and v˙ = u¨. Hence, equation can be integrated

 

The integration results in a first order differential equation which should be dealt with the previous methods. It can be noticed that initial condition of the function is used twice. The physical reason is that the equation represents strong effect of the function at a certain point.

 

 

 

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