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IMPULSE AND MOMENT IMPULSE THEOREM

technical

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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: (d’Alembert 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”.








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