Introduction: objectives: Introduce the momentum equation for a fluid. Demonstrate how the momentum equation and principle of conservation of momentum is used to predict forces induced by flowing fluids.
MOMENTUM AND FLUID FLOW:
- We have all seen moving fluids exerting forces.
- 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 in the same way as in solid mechanics - by use of Newton’s laws of motion.
- Account is also taken for the special properties of fluids when in motion.
- The momentum equation is a statement of Newton’s Second Law and relates the sum of the forces acting on an element of fluid to its acceleration or rate of change of momentum.
- From solid mechanics you will recognize F = ma
- In fluid mechanics it is not clear what mass of moving fluid, we should use a different form of the equation.
Newton’s 2nd Law can be written: 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.
In mechanics, the momentum of particle or object is defined as:
Momentum = mv
From continuity equation: p1A1v1= p2A2v2=m
The rate at which momentum exits face CD may be defined as:
The rate at which momentum enters face AB may be as:
The rate of change of momentum across the control volume
The rate of change of momentum across the control volume:
And according the Newton’s second law, this change of momentum per unit time will be caused by a force F, Thus:
This is the resultant force acting on the fluid in the direction of motion.
By Newton’s third law, the fluid will exert an equal and opposite reaction on its surroundings.