Example 1: Force on a sphere accelerating (, unsteady) in an unbounded fluid at rest. (at infinity)

K.B.C on sphere:

Solution: Simply a 3D dipole (no stream)

Check:

Hydrodynamic force:

On r = a:

Finally,

**General 6 degrees of freedom motions **

__Added mass matrix (tensor)__

: associated with force on body in direction due to unit acceleration in direction. For example, for a sphere:

- circle
figure

- ellipse
figure

- plate
figure

- square
figure

- 1.
- inscribed circle:
.

- 2.
- circumscribed circle:
.

Note:

Then (JNN §4.13)

- forces

- moments

Note:

- 1.
- if
0 ,
(as
expected by definition of .
Also if
then
for any ,
no force in
__steady____translation__. - 2.
- ``added momentum'' due to rotation of axes, 2) where is linear momentum. (momentum from 1 coordinate into new direction)
- 3.
- If
.

__Moment__on a body due to pure steady translation - ``Munk'' moment.

Consider steady motion: . Then

For a 2D body, , also . This implies that:

Therefore, for ("Bow up"). Therefore, a submarine under forward motion is unstable in pitch (yaw) (e.g., a small bow-up tends to grow with time), and control surfaces are needed:

- Restoring moment (gHsin.
- critical speed
given by: