1. Chapter 10 Approximate Solutions of the Navier-Stokes Equation
Introduction
Nondimensionalization equation of motion
The creeping flow approximation

2. Objectives
Appreciate why approximations are necessary, and know when and where to use.
Understand effects of lack of inertial terms in the creeping flow approximation.
Understand superposition as a method for solving potential flow.
Predict boundary layer thickness and other boundary layer properties.

3. About the whole chapter
N-S equation is difficult to solve, therefore approximations are often used for practical engineering analyses.
Be sure that the approximation is appropriate in the region of flow being analysed.
Several approximation will be examined according to the flow situations.

4. About the whole chapter
First, nondimensionalize the N-S eq which later yielding several nondimensional parameters:
Stroudal number (St)
Froude number (Fr)
Euler number (Eu)
Reynolds number (Re)

5. Introduction
In Chap. 9, we derived the NSE and developed several exact solutions.
In this Chapter, we will study several methods for simplifying the NSE, which permit use of mathematical analysis and solution
These approximations often hold for certain regions of the flow field.
Figure 10.2

6. Nondimensionalization of the NSE
Purpose: Order-of-magnitude analysis of the terms in the NSE, which is necessary for simplification and approximate solutions.
We begin with the incompressible NSE
Each term is dimensional, and each variable or property ( V, t, , etc.) is also dimensional.
What are the primary dimensions of each term in the NSE equation?
Eq (10.2)

7. Nondimensionalization of the NSE
To nondimensionalize, we choose scaling parameters as follows

8. Nondimensionalization of the NSE
Next, we define nondimensional variables, using the scaling parameters in Table 10-1
To plug the nondimensional variables into the NSE, we need to first rearrange the equations in terms of the dimensional variables
Eq. (10.3)

9. Nondimensionalization of the NSE
Now we substitute into the NSE to obtain
Every additive term has primary dimensions {m1L-2t-2}. To nondimensionalize, we multiply every term by L/(V2), which has primary dimensions {m-1L2t2}, so that the dimensions cancel. After rearrangement,
Eq (10.5)

10. Nondimensionalization of the NSE
Terms in [ ] are nondimensional parameters

11. Nondimensionalization of the NSE

12. Creeping Flow
Also known as “Stokes Flow” or “Low Reynolds number flow”
Occurs when Re << 1
, V, or L are very small, e.g., micro-organisms, MEMS, nano-tech, particles, bubbles
 is very large, e.g., honey, lava

13. Creeping flow

14. Creeping flow

15. Creeping flow
Solution of Stokes flow is beyond the scope of this course.
Analytical solution for flow over a sphere gives a drag coefficient which is a linear function of velocity V and viscosity m.

16. Example of application of general transport equations: viscous flows

17. Viscously dominated flows
Low Reynolds numbers.
Sometimes called as CREEPING FLOWS.
Assumptions are:
Incompressible.
viscosity constant
Gravitational forces negligible or driving flow
Steady flow
Fully developed(velocity profile does not change
with position).

18. Creeping flow in a circular pipe: control volume approach

19. How to solve a problem???

20. Flow development

21. Consequences of fully developed flow

22. Creeping flow between flat plates

23. Creeping flow in a film on a wall

24. Hydrodynamic lubrication

25. Slipper bearing

27. Journal bearing

28. Creeping flow in a circular pipe