PRESENTATION ON THE CONTINUITY AND MOMENTUM

1. Continuity & Momentum
Algebraic Equations

2. Continuity Equation
• The principle of mass conservation means that
mass can neither be created nor destroyed.
• The Continuity Equation is a mathematical
expression of that principal.
•

3. Continuity Equation Cont.
• 𝑇ℎ𝑒 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑚𝑎𝑠𝑠 𝑒𝑛𝑒𝑡𝑟𝑠 𝑡ℎ𝑒 𝑟𝑒𝑔𝑖𝑜𝑛
= 𝑇ℎ𝑒 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑚𝑎𝑠𝑠 𝑙𝑒𝑎𝑣𝑒𝑠 𝑡ℎ𝑒 𝑟𝑒𝑔𝑖𝑜𝑛
+ 𝑇ℎ𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑎𝑐𝑐𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑟𝑒𝑔𝑖𝑜𝑛.
• If the flow is steady (i.e. unchanging with time) the rate
at which mass is accumulated within the region is zero.
• ∴ 𝑇ℎ𝑒 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑚𝑎𝑠𝑠 𝑒𝑛𝑡𝑒𝑟𝑠 𝑡ℎ𝑒 𝑟𝑒𝑔𝑖𝑜𝑛
= 𝑇ℎ𝑒 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑚𝑎𝑠𝑠 𝑙𝑒𝑎𝑣𝑒𝑠 𝑡ℎ𝑒 𝑅𝑒𝑔𝑖𝑜𝑛

4. Continuity Equation Cont.

5. Continuity Equation Cont.
• For a control volume with N inlets and outlets:
• When the Density is constant:

6. Momentum Equation
• What causes fluid to move?
• When Pressure is higher in one location than
in another location.
• Gravity can cause fluid to move.
• Shear stresses can cause fluid to move.
• External forces exerted by solid surfaces can
cause fluid to move e.g. changing the jet
direction by an external plate.

7. Momentum Equation Cont.
• Flow in one direction
• How can we find the force on the duct as the fluid
flows through it?
• By applying Newton's second law to the fluid flowing
through the control volume.

8. Momentum Equation Cont.
• The force due to pressure differences.

9. Momentum Equation Cont.
• The resultant force F acting on the CV:

10. Momentum Equation Cont.
• The Change in momentum of the CV.

11. Momentum Equation Cont.
• Since outflows are taken to be positive:

12. Momentum Equation Cont.
• Since the flow is steady, there is no change in
momentum inside the CV.
• The net change of fluid momentum is equal to
the resultant force acting on the fluid.
𝑅𝑒𝑥𝑡 is the force of the duct on the fluid, the force of the fluid on the duct is
opposite to this force in magnitude and opposite in direction.

13. Momentum Equation Cont.
• Flow in two directions

14. Momentum Equation Cont.
• The Change in x-momentum of the CV.

15. Momentum Equation Cont.
• By Newton’s 2nd Law:

16. Momentum Equation Cont.

17. Momentum Equation Cont.

18. Momentum Equation Cont.
• By Newton’s 2nd Law:

19. Momentum Equation Cont.
• Note that, the momentum equations can be
written as:
• 𝐹𝑥 = 𝑚𝑜𝑉
𝑜𝑥 − 𝑚𝑖𝑉𝑖𝑥
• 𝐹𝑦 = 𝑚𝑜𝑉
𝑜𝑦 − 𝑚𝑖𝑉𝑖𝑦
• 𝐹𝑧 = 𝑚𝑜𝑉
𝑜𝑧 − 𝑚𝑖𝑉𝑖𝑧
• Under steady flow conditions.