A theoretical model for nonlinear planar motions of rotors under fluid confinement

Following previous papers by Axisa, Antunes and co-workers, the authors add ress a theoretical model for immersed rotors, under moderate confinement, u sing simplified flow equations on the gap-averaged fluctuating quantities. However, in contrast to our previous efforts, the nonlinear terms of the fl ow equations are here fully accounted. Because such nonlinear analysis is q uite involved, this paper will focus on the simpler case of planar motions, in order to emphasize the main aspects of our approach. A direct integrati on of the continuity and momentum equations leads to extremely lengthy form ulations. Here, in order to solve the flow equations, we perform an exact i ntegration of the continuity equation and an approximate solution of the mo mentum equation, based on a Fourier representation of the azimuthal pressur e gradient. Then, an exact formulation for the dynamic flow force can be ob tained. Our solution is discussed in connection with physical phenomena. Nu merical simulations of the nonlinear rotor-flow coupled system are presente d, showing that the linearized and the fully nonlinear models produces simi lar results when the eccentricity and the spinning velocity are low. Howeve r, if such conditions are not met, the qualitative dynamics stemming from t hese models are quite distinct. Experimental results indicate that the nonl inear flow model leads to better predictions of the rotor dynamics when the eccentricity is significant, when approaching instability and for linearly unstable regimes. (C) 1999 Academic Press.