The motion of a single small ellipsoidal body immersed in a Newtonian fluid is governed by the equation given by Jeffery (1922), where the inertia and Brownian rotation are neglected. This equation was derived using a no-slip boundary condition at the body surface and matching the velocity field in the inner region near the body to the outer region in the surrounding flow.
In terms of the temporal change of the orientation vector in the major semi-axis direction, the Jeffery equation reads using the tensorial notation for a rotating rod (please see the formula below).
My question is:
How can we express the rotation tensor (the anti-symmetric part of the velocity gradient tensor) knowing instantaneously the positions and velocities of the rod two extremities? Thanks.
Dear Abdallah:
Rotation tensor and velocities of rod two extremities, see the below links:
Rotations by Berkeley at
http://rotations.berkeley.edu/?page_id=575
ROTATIONAL DYNAMICS
http://web.physics.ucsb.edu/~dfolsom/CS32/rotational_dynamics.pdf
The Motion of Rigid Bodies
http://www.damtp.cam.ac.uk/user/tong/dynamics/three.pdf
I hope this help.
http://rotations.berkeley.edu/?page_id=575
http://web.physics.ucsb.edu/~dfolsom/CS32/rotational_dynamics.pdf
http://www.damtp.cam.ac.uk/user/tong/dynamics/three.pdf
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