1. Momentum Equation and its Applications
Moving fluid exerting force
The lift force on an aircraft is exerted by the air moving over the wing.
A jet of water from a hose exerts a force on whatever it hits.
In fluid mechanics the analysis of motion is performed the same way as in solid mechanics - by use of Newton's laws of motion.
Properties of fluids considered when in motion

2. In fluid mechanics not clear mass of moving fluid to use hence the use momentum equation.
Statement of Newton's Second Law relates the sum of the forces acting on an element of fluid to its acceleration or rate of change of momentum.
The Rate of change of momentum of a body is equal to the resultant force acting on the body, and takes place in the direction of the force
Used to analyze solid mechanics to relate applied force to acceleration

3. Consider the streamtube:
Assume flow to be steady and non-uniform

4. In time δt a volume of the fluid moves from the inlet a distance uδt,
Volume entering the stream tube in time δt
Mass entering stream tube in time δt
Momentum of fluid entering stream tube in time δt

5. Similarly, at the exit, the momentum leaving the steam tube is
Calculate force on the fluid
using Newton's 2nd Law
Force = rate of change of momentum
From continuity

6. Assume fluid of constant density
Hence
Analysis assumed inlet and outlet velocities were in the same direction - one dimensional system. or
Force acts in direction of flow of the fluid.

7. Consider two dimensional
At inlet the velocity vector, u1, makes an angle, θ1, with the x-axis.
At the outlet velocity, u2 makes an angle θ2.

8. Resolve forces in the directions of the co-ordinate axes.
The force in the x-direction
Fx = Rate of change of momentum in x-direction
= Rate of change of mass  change in velocity in x-direction

9. The force in the y-direction
Fy = Rate of change of momentum in y-direction
= Rate of change of mass  change in velocity in y-direction

10. The resultant force on the fluid
And the angle which this force acts at is given by
For three-dimensional (x, y, z) system an extra force in the z-direction must be calculated.
This is considered in exactly the same way.

11. This force is made up of three components:
Generally the total force on fluid = rate of change of momentum through the control volume
This force is made up of three components:
Force exerted on the fluid by any solid body touching the control volume
Force exerted on the fluid body (e.g. gravity)
Force exerted on the fluid by fluid pressure outside the control volume

12. Total force, FT, given as the sum of these forces Force exerted by the fluid on the solid body touching the control volume is equal and opposite to
So the reaction force, R, is given by

13. The force due to fluid flow round a pipe bend
Application of the Momentum Equation
Force due to fluid flow round a pipe bend
Force on a nozzle at the outlet of a pipe.
Impact of a jet on a plane surface.
The force due to fluid flow round a pipe bend
Consider a pipe bend with a constant cross section
Lying in the horizontal plane
Turning through an angle of θ°.

15. Step in Analysis:
Draw a control volume
Decide on co-ordinate axis system
Calculate the total force
Calculate the pressure force
Calculate the body force
Calculate the resultant force
Find force on the fluid

16. Control volume
The control volume is drawn with faces at the inlet and outlet of the bend and encompassing the pipe walls.

17. Co-ordinate axis system
Co-ordinate axis chosen such that one axis is pointing in the direction of the inlet velocity.
Calculate the total force
In the x-direction:
In the y-direction:

18. Calculate the pressure force
Pressure force at 1 – pressure at 2
Calculate the body force
There are no body forces in the x or y directions.
The only body force is that exerted by gravity - a direction we do not need to consider.

19. Calculate the resultant force

20. Resultant force on fluid is given as
Direction of application
Force on bend is the same in magnitude as force on fluid, but in opposite direction

21. Example
A 45o reducing bend is connected in a pipe line in on a horizontal floor. The diameters at the inlet and outlet of the bend are 600 and 300 mm respectively. Find the force exerted by water on the bend if the intensity of pressure at the inlet to bend is N/cm2 and rate of flow of water is 600 lit/sec.

22. Decide on co-ordinate axis system Calculate the total force
Force on a pipe nozzle
Step in Analysis:
Draw a control volume
Decide on co-ordinate axis system
Calculate the total force
Calculate the pressure force
Calculate the body force
Calculate the resultant force
Find the force to resist

23. Control volume and Co-ordinate axis1 2 2 1
One dimensional system

24. Calculate the total force
Continuity
Calculate the body force
The only body force is the weight due to gravity in the y-direction - but we need not consider this as the only forces we are considering are in the x-direction.

25. Calculate the pressure force
Apply Bernoulli equation to calculate the pressure
Frictional losses neglected, hf = 0
Nozzle is horizontal, z1 = z2
Pressure outside is atmospheric, p2 = 0

26. Calculate the resultant force
Force to be resisted

27. Example
A nozzle of diameter 20 mm is fitted to a pipe of diameter 60 mm. Find the force exerted by the nozzle on the water which is flowing through the pipe at the rate of 1.2 m3/min