1. Shell Momentum Balances

2. Outline Convective Momentum Transport Shell Momentum Balance
Boundary Conditions
Flow of a Falling Film
Flow Through a Circular Tube

3. Convective Momentum Transport
Recall: MOLECULAR MOMENTUM TRANSPORT
Convective Momentum Transport: transport of momentum by bulk flow of a fluid.

4. Outline Convective Momentum Transport Shell Momentum Balance
Boundary Conditions
Flow of a Falling Film
Flow Through a Circular Tube

5. Shell Momentum Balance
Steady and fully-developed flow is assumed.
Net convective flux in the direction of the flow is zero.

6. Outline Convective Momentum Transport Shell Momentum Balance
Boundary Conditions
Flow of a Falling Film
Flow Through a Circular Tube

7. Boundary Conditions For liquid-gas interfaces:
Recall: No-Slip Condition (for fluid-solid interfaces)
Additional Boundary Conditions:
For liquid-gas interfaces:
“The momentum fluxes at the free liquid surface is zero.”
For liquid-liquid interfaces:
“The momentum fluxes and velocities at the interface are continuous.”

8. Flow of a Falling Film
Liquid is flowing down an inclined plane of length L and width W.
δ – film thickness
Vz will depend on x-direction only
Why? y z x
Assumptions:
Steady-state flow
Incompressible fluid
Only Vz component is significant
At the gas-liquid interface, shear rates are negligible
At the solid-liquid interface, no-slip condition
Significant gravity effects

9. Flow of a Falling Film
τij  flux of j-momentum in the positive i-direction y z x y z L x
τxz ǀ x + δ
τxz ǀ x δ W

10. Flow of a Falling Film
τij  flux of j-momentum in the positive i-direction y z x y z L x
τyz ǀ y=0 δ
τyz ǀ y=W W

11. Flow of a Falling Film
τij  flux of j-momentum in the positive i-direction y z x y
τzz ǀ z=0 z L x δ
τzz ǀ z=L W
ρg cos α

12. Flow of a Falling Film 𝜏 𝑥𝑧| 𝑥+𝛿 − 𝜏 𝑥𝑧| 𝑥 𝛿 =𝜌𝑔 cos 𝛼
P(W∙δ)|z=0 – P(W∙δ)|z=L +
(τxzǀ x )(W*L) – (τxz ǀ x +Δx )(W∙L) +
(τyzǀ y=0 )(L*δ) – (τyz ǀ y=W )(L∙δ) +
(τzz ǀ z=0)(W* δ) – (τzz ǀ z=L)(W∙δ) +
(W∙L∙δ)(ρgcos α) = 0
Dividing all the terms by W∙L∙δ and noting that the direction of flow is along z:
𝜏 𝑥𝑧| 𝑥+𝛿 − 𝜏 𝑥𝑧| 𝑥 𝛿 =𝜌𝑔 cos 𝛼

13. Flow of a Falling Film 𝜏 𝑥𝑧| 𝑥+𝛿 − 𝜏 𝑥𝑧| 𝑥 ∆𝑥 =𝜌𝑔 cos 𝛼
If we let Δx  0,
𝑑(𝜏 𝑥𝑧 ) 𝑑𝑥 =𝜌𝑔 cos 𝛼
Integrating and using the boundary conditions to evaluate,
Boundary conditions:
@ x = 0 𝜏 𝑥𝑧 =0
x = x 𝜏 𝑥𝑧 = 𝜏 𝑥𝑧
𝜏 𝑥𝑧 =𝜌𝑥𝑔 cos 𝛼

14. Flow of a Falling Film 𝜏 𝑥𝑧 =𝜌𝑥𝑔 cos 𝛼 𝜏 𝑥𝑧 =−𝜇 𝑑 𝑣 𝑧 𝑑𝑥
For a Newtonian fluid, Newton’s law of viscosity is
𝜏 𝑥𝑧 =−𝜇 𝑑 𝑣 𝑧 𝑑𝑥
Substitution and rearranging the equation gives
𝑑 𝑣 𝑧 𝑑𝑥 =− 𝜌𝑔 cos 𝛼 𝜇 𝑥

15. Flow of a Falling Film 𝑑 𝑣 𝑧 𝑑𝑥 =− 𝜌𝑔 cos 𝛼 𝜇 𝑥
Solving for the velocity,
𝑣 𝑧 =− 𝜌𝑔 cos 𝛼 2𝜇 𝑥 2 + 𝐶 2
Boundary conditions:
@ x = δ, vz = 0
𝑣 𝑧 = 𝜌𝑔 𝛿 2 cos 𝛼 2𝜇 (1−( 𝑥 𝛿 ) 2 )

16. How does this profile look like?
Flow of a Falling Film
𝑣 𝑧 = 𝜌𝑔 𝛿 2 cos 𝛼 2𝜇 (1−( 𝑥 𝛿 ) 2 )
How does this profile look like?
Compute for the following:
Average Velocity:

17. Flow of a Falling Film 𝑣 𝑧 =− 𝜌𝑔 𝛿 2 cos 𝛼 2𝜇 (1−( 𝑥 𝛿 ) 2 )
Compute for the following:
Mass Flowrate:

18. Flow Between Inclined Platesz x θ δ L
Derive the velocity profile of the fluid inside the two stationary plates. The plate is initially horizontal and the fluid is stationary. It is suddenly raised to the position shown above. The plate has width W.

19. Outline Convective Momentum Transport Shell Momentum Balance
Boundary Conditions
Flow of a Falling Film
Flow Through a Circular Tube

20. Flow Through a Circular Tube
Liquid is flowing across a pipe of length L and radius R.
Assumptions:
Steady-state flow
Incompressible fluid
Only Vx component is significant
At the solid-liquid interface, no-slip condition

21. Recall: Cylindrical Coordinates

22. Flow Through a Circular Tube

23. Flow Through a Circular Tube

24. Flow Through a Circular Tube
BOUNDARY CONDITION!
At the center of the pipe, the flux is zero (the velocity profile attains a maximum value at the center).
C1 must be zero! 

25. Flow Through a Circular Tube
BOUNDARY CONDITION!
At r = R, vz = 0.

26. Flow Through a Circular Tube
Compute for the following:
Average Velocity:

27. Hagen-Poiseuille Equation
Describes the pressure drop and flow of fluid (in the laminar regime) across a conduit with length L and diameter D

28. What if…? The tube is oriented vertically.
What will be the velocity profile of a fluid whose direction of flow is in the +z-direction (downwards)?