Contents
1 Brief Outline of the Equations of Fluid Flow . . . . . . . . . . . . . . . . . . .
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Eulerian and Lagrangian Form of the Equations . . . . . . . . . . . . .
2
1.3
Some Elements of Thermodynamics . . . . . . . . . . . . . . . . . . . . . .
3
1.3.1
Ideal Gas Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3.2
The First Law of Thermodynamics . . . . . . . . . . . . . . . .
4
1.3.3
Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3.4
Isothermal Expansion or Compression of an
Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3.5
Reversible Adiabatic Process for an Ideal Gas . . . . . . . .
8
1.3.6
Work Done by an Ideal Gas During an
Adiabatic Expansion . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3.7
Alternate Form of the Equations for Specific
Internal Energy and Enthalpy . . . . . . . . . . . . . . . . . . . . 10
1.3.8
Ratio of the Specific Heats for Air . . . . . . . . . . . . . . . . . 10
1.3.9
The Second Law of Thermodynamics . . . . . . . . . . . . . . 11
1.4
Conservation Equations in Plane Geometry . . . . . . . . . . . . . . . . 13
1.4.1
Equation of Mass Conservation: The Continuity
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2
Equation of Motion: The Momentum Equation . . . . . . . 15
1.4.3
Energy Balance Equation . . . . . . . . . . . . . . . . . . . . . . . 17
1.5
Constancy of the Entropy with Time for a Fluid Element . . . . . . . 21
1.6
Entropy Change for an Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7
Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7.1
Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.7.2
Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7.3
Equation of Energy Conservation . . . . . . . . . . . . . . . . . 32
1.8
Small Amplitude Disturbances: Sound Waves . . . . . . . . . . . . . . . 33
xi
1 Brief Outline of the Equations of Fluid Flow . . . . . . . . . . . . . . . . . . .
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Eulerian and Lagrangian Form of the Equations . . . . . . . . . . . . .
2
1.3
Some Elements of Thermodynamics . . . . . . . . . . . . . . . . . . . . . .
3
1.3.1
Ideal Gas Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3.2
The First Law of Thermodynamics . . . . . . . . . . . . . . . .
4
1.3.3
Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3.4
Isothermal Expansion or Compression of an
Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3.5
Reversible Adiabatic Process for an Ideal Gas . . . . . . . .
8
1.3.6
Work Done by an Ideal Gas During an
Adiabatic Expansion . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3.7
Alternate Form of the Equations for Specific
Internal Energy and Enthalpy . . . . . . . . . . . . . . . . . . . . 10
1.3.8
Ratio of the Specific Heats for Air . . . . . . . . . . . . . . . . . 10
1.3.9
The Second Law of Thermodynamics . . . . . . . . . . . . . . 11
1.4
Conservation Equations in Plane Geometry . . . . . . . . . . . . . . . . 13
1.4.1
Equation of Mass Conservation: The Continuity
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2
Equation of Motion: The Momentum Equation . . . . . . . 15
1.4.3
Energy Balance Equation . . . . . . . . . . . . . . . . . . . . . . . 17
1.5
Constancy of the Entropy with Time for a Fluid Element . . . . . . . 21
1.6
Entropy Change for an Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7
Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7.1
Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.7.2
Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7.3
Equation of Energy Conservation . . . . . . . . . . . . . . . . . 32
1.8
Small Amplitude Disturbances: Sound Waves . . . . . . . . . . . . . . . 33
xi