On the instability of the fluids of second grade in nearly viscometric motions

Flow of a viscoelastic liquid in a cylindrical cavity, driven by rotating finite disks is investigated. The cylindrical sidewall is fixed and the covers rotate with different angular velocities either in the same or in opposite directions. A regular perturbation in terms of the angular velocity of the caps is used. The flow field is resolved into a primary azimuthal stratified viscometric field and a weaker secondary meridional field. Results are presented for a range of cylinder aspect and cap rotation ratios and viscoelastic parameters. Interesting instabilities of the fluid of second grade are discussed. The controversy concerning the sign of the first Rivlin-Ericksen constant is completely irrelevant to the discussion. It is shown that loss of stability occurs repeatedly and bifurcating flows exist for critical values of an elasticity parameter at fixed aspect and cap rotation ratio. Branching flows also occur at a fixed value of the elasticity parameter for critical values of the cap rotation ratio, when the aspect ratio is fixed.