Equation if Motion For an Inviscid, Incompressible Fluid

For an incompressible fluid,so

If the body force per unit volume isapplying Newton's Second Law to a cube of unit volume to obtain

We can also writewhereis the pressure in the fluid.

Equating these gives(1)

Consider the motion of a particle in the fluid from positionto position

Let the velocity atbe

The velocity atwill be

Then

Substitute this into (1) to obtain

We can define a vector fieldcalled the vorticity as

Use the identity