1. Introduction to Fluid Mechanics
Chapter 5
Introduction to Differential Analysis of Fluid Motion

2. Main Topics Conservation of Mass
Motion of a Fluid Particle (Kinematics)
Momentum Equation

3.  Introduction
In Chapter 4, integral equations for finite control volumes are derived, which reflect the overall balance over the entire control volume under consideration -- A top down approach.
However, only information related to the gross behavior of a flow field is available. Detailed point-by-point knowledge of the flow field is unknown.
Additionally, velocity and pressure distributions are often assumed to be known or uniform in Chapter 4. However, for a complete analysis, detailed distributions of velocity and pressure fields are required.
A bottom-up approach is needed.

4. Conservation of Mass
Basic Law for a System

5. Conservation of Mass Rectangular Coordinate System
The net mass flow rate out of the CV in x direction is:
Differential control volume herein vs. finite control volume in Chapter 4. The differential approach has the ability to attain field solutions. The basic equations from Chapter 4 are still applicable here but with infinitesimal CV in conjunction with coordinate system.

6. Conservation of Mass
Rectangular Coordinate System

7. Conservation of Mass Rectangular Coordinate System
“Continuity Equation”

8. Conservation of Mass
Rectangular Coordinate System
“Del” Operator

9. Conservation of Mass Rectangular Coordinate System
Incompressible Fluid:
Steady Flow:

10. Conservation of Mass
Cylindrical Coordinate System

11. Conservation of Mass
Cylindrical Coordinate System

12. Conservation of Mass
Cylindrical Coordinate System
“Del” Operator

13. Conservation of Mass Cylindrical Coordinate System
Incompressible Fluid:
Steady Flow:

14. Motion of a fluid element (Kinematics)

15. Motion of a fluid element
According to multiple-variable Taylor expansion series
Particle (system) acceleration is expressed in terms of a velocity field (space quantity)

16. Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field

17. Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field

18. Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field

19. Motion of a fluid element

20. Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field (Cylindrical)

21. Momentum Equations (Navier-Stokes Equations)
The force here is that acting on the control volume/surface occupied by the fluid element at time t

22. Momentum Equation Forces Acting on a Fluid Particle
To determine the surface force, the stress condition on the surfaces of the CV element occupied by the fluid element is considered

23. Momentum Equations  

24. Momentum Equations  

25. Momentum Equations  

26. Navier-Stokes Equations
where p is the local thermodynamic pressure, which is related to the density and temperature by the thermodynamic relation usually called the equation of state. Notice that when velocity is zero, all the shear stresses are zero and all the normal stresses reduce to pressure under hydrostatic condition.

27. Navier-Stokes Equations

28. Momentum Equation (incompressible flow)
Navier–Stokes Equations: Cylindrical coordinate

29. Momentum Equation
Special Case: Euler’s Equation

30. c05u011
c05u011

31. c08f001
c08f001

32. c08f004
c08f004