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Differential Analysis of Fluid Flow
2
CHAPTER 9 : DIFFERENTIAL ANALYSIS OF FLUID FLOW
Objectives
- Understand how the differential equation of conservation mass and the differential linear momentum equation are derived and applied
- Calculate the stream function and pressure field,and plot streamlines for a known velocity field
- Obtain analytical solutions of the equations of motion for simple flow fields
1
CONSERVATION OF MASS
2
THE CONTINUITY EQUATION
1
DERIVATION USING THE DIVERGENCE THEOREM
2
DERIVATION USING AN INFINITESIMAL CONTROL VOLUME
3
ALTERNATIVE FORM OF THE CONTINUITY EQUATION
CONTINUITY EQUATION
IN CYLINDRICAL COORDINATES
4
3
THE STREAM FUNCTION
THE STREAM FUNCTION IN CARTESIAN COORDINATES
1
THE STREAM FUNCTION IN CYLINDRICAL COORDINATES
2
THE COMPRESSIBLE STREAM FUNCTION
3
THE DIFFERENTIAL LINEAR MOMENTUM EQUATION - CAUCHY'S EQUATION
4
1
DERIVATION USING THE DIVERGENCE THEOREM
2
DERIVATION USING AN INFINITESIMAL CONTROL VOLUME
3
ALTERNATIVE FORM OF CAUCHY'S EQUATION
4
DERIVATION USING NEWTON'S SECOND LAW
5
THE NAVIER-STOKES EQUATION
1
NEWTONIAN
VS
NON-NEWTONIAN FLUIDS
NEWTONIAN
FLUIDS
DEFINED AS FLUIDS FOR WHICH THE SHEAR STRESS IS LINEARLY PROPORTIONAL TO THE SHEAR STRAIN RATE
EXAMPLE :
1. GASES
2. WATER
3. KEROSENE
4.GASOLINE
NON-NEWTONIAN FLUIDS DEFINES AS FLUIDS FOR WHICH THE SHEAR STRESS IS NOT LINEARLY RELATED TO THE SHEAR STRAIN RATE
EXAMPLE :
1. BLOOD
2. PASTE
3. CAKE BATTER
4. POLYMER SOLUTIONS
NON-NEWTONIAN
FLUIDS
DERIVATION OF THE NAVIER-STOKES EQUATION FOR INCOMPRESSIBLE ISOTHERMAL FLOW
2
CONTINUITY AND NAVIER-STOKES EQUATIONS IN CARTESIAN COORDINATES
3
CONTINUITY AND NAVIER-STOKES EQUATIONS IN CYLINDRICAL COORDINATES
4
DIFFERENTIAL ANALYSIS OF FLUID FLOW PROBLEMS
6
1
CALCULATION OF THE PRESSURE FIELD FOR A KNOWN VELOCITY FIELD
2
EXACT SOLUTIONS OF THE CONTINUITY AND NAVIER-STOKES EQUATIONS
3
BOUNDARY CONDITIONS
AS ITS NAME IMPLIES , THERE IS NO "SLIP" BETWEEN THE FLUID AND THE WALL
NO-SLIP
BOUNDARY CONDITION
INTERFACE
BOUNDARY
CONDITION
APPLIES WHEN 2 FLUIDS MEET AT AN INTERFACE WHERE IN ADDITION TO THE CONDITION THAT THE VELOCITIES OF THE TWO FLUIDS MUST BE EQUAL, THE SHEAR STRESS ACTING ON THE FLUID PARTICLE ADJACENT TO THE INTERFACE IN THE DIRECTION PARALLEL TO THE INTERFACE MUST ALSO MATCH BETWEEN THE TWO FLUIDS
FREE-SURFACE
BOUNDARY
CONDITION
A DEGENERATE FORM OF THE INTERFACE BOUNDARY CONDITION OCCURS AT THE FREE SURFACE OF A LIQUID MEANING THAT FLUID A IS A LIQUID AND FLUID B IS A GAS