IMPULSE AND MOMENT IMPULSE THEOREM

We take into consideration a volume of fluid. This fluid is homogeneous, incompressible, of density , bordered by surface . The elementary volume   has the speed .

The elementary impulse will be:

(6.1)

(6.2)

(6.3)

At the same time

(6.4)

But: (dAlembert principle). (6.5)

Therefore:

(6.6)

The total derivative, of the impulse with respect to time, is equal to the resultant   of the exterior forces, or

, (6.7)

where   are the mass flows through entrance/ exit surfaces.

Under permanent flow conditions of ideal fluid, the vectorial sum of the external forces which act upon the fluid in the volume , is equal with the impulse flow through the exit surfaces (from the volume ), less the impulse flow through the entrance surfaces (to the volume ) .

- the position vector of the centre of volume with respect to origin of the reference system.

The elementary inertia moment with respect to point O (the origin) is:

(6.8)

since

(6.9)

then

(6.10)

If:

the elementary impulse, (6.11)

the moment of elementary impulse, (6.12)

(6.13)

(6.14)

The derivative of the resultant moment of impulse with respect to time is equal with the resultant moment of inertia forces with reversible sign.

(6.15)

where

- the moment of mass forces,

- the moment pressure forces,

- the moment of external forces.

- the position vector of the centre of gravity for the exit /entrance surfaces.

(6.16)

Under permanent flow conditions of ideal fluids, the vectorial addition of the moments of external forces which act upon the fluid in the volume , is equal to the moment of the impulse flow through the exit surfaces less the moment of the impulse flow through the entrance surfaces.