## Momentum Equation

**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: p_{1}A_{1}v_{1}= p_{2}A_{2}v_{2}=m

The rate at which momentum exits face CD may be defined as:

p_{2}A_{2}v_{2}v_{2}

The rate at which momentum enters face AB may be as:

p_{1}A_{1}v_{1}v_{1}

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.