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

2. Main Topics Conservation of Mass
Stream Function for Two-Dimensional Incompressible Flow
Motion of a Fluid Particle (Kinematics)
Momentum Equation
Introduction to Computational Fluid Dynamics

3. Conservation of Mass
Basic Law for a System

4. Conservation of Mass
Rectangular Coordinate System

5. Conservation of Mass
Rectangular Coordinate System

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

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

8. Conservation of Mass
Rectangular Coordinate System

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

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

15. Stream Function for Two-Dimensional Incompressible Flow
Two-Dimensional Flow
Stream Function y
Conservation of mass is automatically satisfied

16. Stream Function for Two-Dimensional Incompressible Flow
Cylindrical Coordinates
Stream Function y(r,q)

17. Stream Function for Two-Dimensional Incompressible Flow
Along a streamline Stream Function y (r,q)= constant

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

19. Motion of a Fluid Particle (Kinematics)

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

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

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

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

24. Motion of a Fluid Particle (Kinematics)
Fluid Rotation
A fluid particle moving in a general three-dimensional flow field may rotate about all three coordinate axes.

25. Motion of a Fluid Particle (Kinematics)
Fluid Rotation

26. Motion of a Fluid Particle (Kinematics)
Fluid Rotation

27. Motion of a Fluid Particle (Kinematics)
Fluid Deformation:
Angular Deformation

28. Motion of a Fluid Particle (Kinematics)
Fluid Deformation:
Angular Deformation

29. Motion of a Fluid Particle (Kinematics)
Fluid Deformation:
Linear Deformation

30. Momentum Equation
Newton’s Second Law

31. Momentum Equation
Forces Acting on a Fluid Particle

32. Momentum Equation Forces Acting on a Fluid Particle
The forces acting on a fluid element may be classified as body forces and surface forces

33. Momentum Equation
Differential Momentum Equation

34. Momentum Equation
Stresses may be expressed in terms of velocity gradients

35. Momentum Equation
Newtonian Fluid: Navier–Stokes Equations

36. Momentum Equation
Special Case: Euler’s Equation

37. Computational Fluid Dynamics
Some Applications

38. Computational Fluid Dynamics
Euler Method

39. Computational Fluid Dynamics
Euler Method – Draining a Tank

40. Computational Fluid Dynamics
A Simple Finite Difference Example

41. Computational Fluid Dynamics
Finite Difference – Flow over a Corner