Forces

(Book: 5.3, 4, 5)

To figure out fluid motion, we need the forces on the fluid. The two types of forces we will deal with in this class are:

• body forces acting in the interior:
  • gravity: per unit mass;
• surface forces:
  • pressure only (inviscid flow): per unit outside surface;
  • pressure plus viscous stresses (viscous flow): in the i-direction, per unit outside surface.

So, including all these, the total external force on a volume of fluid will be:

Exercise:

Derive the Archimedes law for a submarine of arbitrary shape 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.

To shorten the writing, we define the total stress tensor as:

We alse rewrite often in terms of the gravitational potential:

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

You should now be able to do question 5.3