1. Outline Time Derivatives & Vector Notation
Differential Equations of Continuity
Momentum Transfer Equations

2. Lagrangian Perspectivez
Lagrangian coordinate system
Motion of a particle (fluid element)
The position of the particle is relative to the position of an observer
pathline 2 1 y x

3. Lagrangian Perspectivez
Local time derivative
pathline 2 1
Local spatial derivative y x

4. Lagrangian Perspective
Total differential/change for any property 
Total time derivative

5. Lagrangian Perspective
Fluid velocity
If the observer follows the fluid motion
Substantial time derivative

6. Eulerian Perspective flow Motion of a fluid as a continuumz
Motion of a fluid as a continuum
flow
Fixed spatial position is being observed rather than the position of a moving fluid particle (x,y,z). y x

7. Equation of Continuity
differential control volume:

8. Differential Equation of Continuity

9. Differential Equation of Continuity
In cylindrical coordinates:
If fluid is incompressible:

10. Equations of Motion Fluid is flowing in 3 directions
For 1D fluid flow, momentum transport occurs in 3 directions
Momentum transport is fully defined by 3 equations of motion

11. Differential Equation of Motion

12. Differential Equation of Motion

13. Navier-Stokes Equations
Assumptions
Newtonian fluid
Obeys Stokes’ hypothesis
Continuum
Isotropic viscosity
Constant density

14. Navier-Stokes Equations

15. Navier-Stokes Equations

16. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Steady state flow

17. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Unidirectional flow

18. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
No viscous dissipation (INVISCID FLOW)
Euler’s equation

19. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
No external forces acting on the system
Inviscid flow:

20. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
No external forces acting on the system
Viscous flow:

21. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Semi-infinite system

22. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Laminar flow (no convective transport)

23. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Laminar flow (no convective transport)

24. Quiz 9 –
Derive the equation giving the velocity distribution at steady state for laminar, downward flow in a circular pipe of length L and diameter D. Neglect entrance and exit effects.
TIME IS UP!!!