The viscous torsional pendulum

A disk that is free to rotate about its axis and connected to a torsional spring behaves as a damped oscillator when twisted and released. The initial elastic energy is periodically turned to kinetic energy and it gets progressively dissipated by the viscous friction exerted by the surrounding fluid. The subsequent oscillating motion is dictated by the fluid solid interaction which is here solved numerically by coupling the second Euler's law, that prescribes the disk's rotation, to the Navier Stokes equations, that govern the fluid's motion. Two different regimes are observed: (i) a low-amplitude regime, where the phase lag between the twisting velocity and the viscous torque is equal to 3 pi/4 and the instantaneous damping rate is constant; (ii) a higher amplitude regime, where the twisting velocity and the viscous torque are out of phase and the damping rate increases proportionally to the square root of the oscillation amplitude. These observations are rationalized through boundary layer theory applied in the vicinity of the disk, thus retrieving analytical expressions of the viscous torque available in the literature. By using a multiple scale technique, an explicit expression for the free decay of the disk torsional pendulum is obtained which well predicts the results of the numerical simulations without any tunable parameter. (C) 2017 Elsevier Ltd. All rights reserved.