Forces

(Book: 5.3, 4, 5)

The forces we will deal with in this class are:

• body forces per unit mass:
  • gravity: ;
• surface forces per unit area:
  • pressure only (inviscid flow): ;
  • pressure and viscous stresses: .

Definition: The total stress tensor is:

Thus R_i = T_ji n_j.

Exercise:

Derive the Archimedes law if the pressure is given by , where the y-axis is vertically upward. Hint: use the divergence theorem to convert the surface integral into a volume integral.

The net force on a finite region is:

Written out in index notation:

We rewrite often in terms of the gravitational potential:

where h is the height above the surface of the earth.

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