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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  : associated with force on body in direction due to unit acceleration in direction. For example, for a sphere: Some added masses of simple 2D geometries
• circle figure  • ellipse figure  • plate figure  • square figure  A reasonable estimate for added mass od a 2D body is to use the displaced mass ( of an equivalent cylinder'' of the same lateral dimension or one that rounds off'' the body. For example, we consider a aquare:
1.
inscribed circle: . 2.
circumscribed circle: . Arithmetic mean of 1) + 2) .

General 6 degrees of freedom forces and moments on a rigid body moving
in an unbounded fluid ( at rest at infinity)  Note: fixed in the body.

Then (JNN §4.13)
• forces

• moments

Einstein's notation applies. 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.
Example of Munk Moment - a 2D submarine in steady translation  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 ( g Hsin .
• critical speed given by:  Usually . For small . So, or . Otherwise, control fins are required.  Up: No Title Previous: No Title