1. Chapter 9 Analysis of a Differential Fluid Element in Laminar Flow

2. Main Topics Introduction
Fully Developed Laminar Flow of a Newtonian Fluid in a Circular Conduit of Constant Cross Section
Fully Developed Laminar Flow of a Newtonian Fluid Down an Inclined Plane Surface (Open Channel)
Laminar Flow of a Newtonian Fluid Flow Through an Annulus
Fully Developed Laminar Flow of Newtonian Fluid in Two Parallel Plates
Boundary Conditions
Velocity Distribution in Circular Conduit

3. 9.0 Introduction
In this chapter, we shall direct our attention to elements of fluid as they approach differential size. Our goal is the estimation and description of fluid behaviour from a differential point of view; the resulting expressions from such analyses will be differential equations. The solution to these differential equations will give flow information of a different nature than that achieved from a macroscopic examination.
Most cases considered in this chapter, involved fully developed flow. At the entrance, the fluid enters as one dimensional flow, gradually, after certain length which is known as the ‘entry length’, due to the shear drag, it will be developed into two dimensional flow. The following applications assumed that the consideration of the entry length has been taken care.
The entry length Le can be approximated by the following relations,
Re < 2100: Le /D ≈ ReD
Re > 2100: Le /D ≈ 4.4 ReD1/6

4. 9.1 Fully Developed Laminar Flow of a Newtonian Fluid in a Circular Conduit of Constant Cross Section
Figure Fig.9.1.1
Derivation of equations (9.1.2), (9.1.5) and (9.1.7)

5. 9.2 Fully Developed Laminar Flow of a Newtonian Fluid Down an Inclined Plane Surface (Open Channel)
Figure Fig.9.2.1
Derivation of equations (9.2.2), (9.2.6) and (9.2.7)

6. 9.3 Laminar Flow of a Newtonian Fluid Flow Through an Annulus
Figure Fig.9.3.1
Extra figure for 9.3
Derivation of equations (9.3.3), (9.3.7) and (9.3.8)

7. 9.4 Fully Developed Laminar Flow of Newtonian Fluid in Two Parallel Plates
Figure Fig.9.4.1
Derivation of equations (9.4.5) and (9.4.6)

8. 9.5 Boundary Conditions
The boundary conditions that we may use in fluid mechanics are listed below:
1. At solid - fluid interfaces, the fluid velocity equals the velocity with which the solid surface itself is moving, i.e. the fluid is assumed to cling to any solid surfaces with which it is in contact. - (No-slip condition).
2. At liquid-gas interfaces, the momentum flux (hence the velocity gradient) in the liquid phase is very nearly zero and can be assumed to be zero in most calculations.
3. At liquid-liquid interfaces, the momentum flux perpendicular to the interface, and the velocity are continuous across the interface.

9. 9.6 Velocity Distribution in Circular Conduit
For fully developed turbulent flow in a pipe, the velocity profile can be approximated by the following power law formula,
Table and Figure 9.6.1

10. Points to remember
Before the analysis, one has to have a rough idea on how the velocity profile will look like so that the direction of the transferring of momentum can be determined.
The boundary conditions that we may use in fluid mechanics are listed below:
1. At solid - fluid interfaces, the fluid velocity equals the velocity with which the solid surface itself is moving, i.e. the fluid is assumed to cling to any solid surfaces with which it is in contact. - (No-slip condition).
2. At liquid-gas interfaces, the momentum flux (hence the velocity gradient) in the liquid phase is very nearly zero and can be assumed to be zero in most calculations.
3. At liquid-liquid interfaces, the momentum flux perpendicular to the interface, and the velocity are continuous across the interface.

11. Tutorial
Link to Tutorial 6