1. Chapter 2 Integral Forms of the Conservation Equations for Inviscid Flow
2-1 Problem Solving
1. Physics  2. Approach methods
 3. Tools (mathematics)  4. Solve the problem
Approach methods :
Classical mechanics , no relativity：
1.Mass is conserved
2.Force = ma
3.Energy is conserved

2. A. Finite Control Volume Approach (macroscopic , Integral Form )
FIG Finite control volume approach

3. Statistical averaging → Boltzmann equation from kinetic theory
B. Infinitesimal Fluid Element Approach (Macroscopic , Differential Form)
Eulerian coordinate
Lagrangian coordinate
FIG Infinitesimal fluid element approach
C. Molecular Approach (Microscopic)
Statistical averaging   → Boltzmann equation from kinetic theory
Ref Hirschfelder , Curtiss and Bird
“Molecular Theory of Gases and Liquids”

4. Physical principle : Mass can be neither created nor destroyed.
2-3 Continuity Equation
Physical principle : Mass can be neither created nor destroyed.
denotes the mass flow through dS u
are local velocity and density at B
FIG Fixed control volume

5. The total mass inside the control volume
The net mass flow into the control volume through the entire control surface S
Note : (- )  inflow
(+)  outflow
The total mass inside the control volume
The time rate of change of this mass inside the C.V.
Continuity equation (Integral Form)
Mass is conserved
Applies to all flows , compressible or incompressible , viscous or inviscid

6. Physical principle : The time rate of change of momentum of a
2-4 Momentum Equation
Physical principle : The time rate of change of momentum of a
body equals the net force exerted on it .
Body forces on V , eg. Gravitational and
E.M forces
Total body force =
Forces on the control volume
Body force / mass
2. Surface forces on S , eg. P and τ
Total surface force due to pressure =
We consider inviscid flows only
The total force acting on the C.V. is

7. = the net rate of flow of momentum
The total time rate of change of momentum of the fluid as it flows through a fixed control volume
= the net rate of flow of momentum
across the surface S
+ the change in momentum in V due to
unsteady fluctuations in the local flow
(For fixed C.V.)
Therefore : Momentum Equation

8. change in form. ------- the 1st law of thermodynamics (δQ+δW=dE)
2-5 Energy Equation
Physical principle : Energy can be neither created nor destroyed ; It can only
change in form the 1st law of thermodynamics
(δQ+δW=dE)
B1= rate of heat added to the fluid inside the C.V. from the
surroundings
B2= rate of work done on the fluid inside the C.V.
B3= rate of change of the energy of the fluid as it flows through the
C.V.
B1 + B2 = B3
B1 : volumetric heating of the fluid in V due to
a. Radiation
b. Thermal conduction & diffusion (viscous effect)
: the rate of heat added to C.V./mass

9. B2 = the rate of doing work on a moving body =
= rate of work done on the fluid
inside V due to pressure forces on S
rate of work done on the fluid
inside V due to body forces =
B3 = net rate of flow of energy
across the C.V.
+ time rate of change of energy inside V
due to transient variations of the flow
field variables =

10. Therefore , More general form Energy Equation
(Integral Form , Inviscid)
More general form