As far as I can see it now, the buoyancy term looks simpler than the drag-force term.
You first write the equation
J d^2 theta / dt^2 = mgl sin theta -- k(S) l^2 d theta /dt, (1)
where J is rotational inertia of the pendulum about the suspension point; m is mass of the pendulum; theta is its instantaneous deviation from the vertical; l is the distance between the suspension point and the center of mass of the pendulum; k(S) is the proportionality coefficient between the drag force and velocity (v = l d theta/dt); it is a pretty complicated function of the size and shape of the pendulum. The direction of motion towards the vertical is taken as positive.
Then you just replace m -----> m -- rho V, where rho is the fluid density and V - the pendulum's volume. The additional term rho V g will represent the buoyant force (assuming we can neglect the pressure gradient in view of the relatively small l). With the buoyancy term, the equation (1) can be written as
There is still at least one more term missing - that of friction force, but I could not so far figure out the expression for it; it might be considered as absorbed by k(S) were it not for the fact that it does not depend on v in this approximation.
Moses Fayngold,
NJIT
--- On Sun, 1/10/10, Bernard Cleyet <bernardcleyet@redshift.com> wrote
(Sunday, January 10, 2010, 12:30 AM):
bc still wants somene to write the diff. eq. to include the buoyancy