Differential Rotation in Slowly Dissipating Systems

Abstract

If a body does not rotate uniformly it suffers differential rotation. Under certain conditions the existence of differential rotation of a body is a consequence of quite general properties of dissipative processes. Let E be the kinetic energy of rotation, A and I the angular momentum and the moment of inertia along the axis of rotation; then the body cannot rotate rigidly unless the positive semidefinite form K = E - A2/2I vanishes. If this condition is not satisfied, the motion of the system can still be specified provided the relaxation time for establishing a steady state corresponding to a given value of the energy and angular momentum is much less than the relaxation times of the energy and angular momentum. If this be the case, the steady state of the system can be approximated with a state in which the dissipation is a minimum subject to the constraints that E and A have fixed values. This leads to the investigation of states of least dissipation subject to auxiliary conditions. The resulting eigenvalue problem is solved for several simple models in which the dissipation is due solely to viscous friction. If the values of E and A specified through the auxiliary conditions do not make K vanish, the models exhibit differential rotation.