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We consider any enclosed volume (&tau) capturing a fluid element. We equate the added momentum throughout &tau over a perid of time dT, with the impact of all the forces of &tau over the same period. This is the equation of motion.

M&tau (T+dT) - M&tau (T) = &int (Du/Dt) &rho d &tau

where u is the velocity of the fluid element, and &rho is its specific gravity, M - is momentum.

The forces that apply are either distance forces (external):

&int Fexternal &rho d &tau

where F is a force per unit mass;

or surface forces.

Surface forces apply to the entire surface (S) of &tau, and at each surface area, the force is may be of any direction.

Accoutning for such direction variety of forces over the direction variety of the surface elements of &tau is a complicated task.

It is handled via a dedicated mathematical construct: the tensor.

Tensor notation allows for identifying the impact of each force of any direction to whatever the direction of the surface element it is associated with. The overall impact is given by:

&intS &sigmaij dSj

where i,j represent all possible directions for ds and for surface forces -- given a cartesian representation.

Applying the surface integral to volume integral identity (Gauss theorem) we get:

&intS &sigmaij dSj = &int&tau (&part &sigmaij /&part xj )d&tau

which allows us to express all integrals with respect to &tau, and hence:

&int &tau &rho (Dui /dt)d &tau = &int &tau &rhoFi d&tau + &int &tau (&part &sigmaij /&part xj )d&tau

which must be true for any arbitrary &tau, and hence;

&rho (Dui /dt)d &tau = &rhoFi d&tau + (&part &sigmaij /&part xj )d&tau

which is the equation of motion.

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