A viscoelastic flow instability near the solid body rotation limit

Global linear stability analysis of the creeping flow of an Oldroyd-B liquid, confined between eccentric cylinders co-rotated at equal angular speeds Ω, is performed using a submatrix-based transformation algorithm [K. Arora, R. Sureshkumar, J. Non-Newtonian Fluid Mech. 104 (2002) 75]. The eccentricity parameter ɛ is defined as the ratio of the distance between the cylinder centers to the average gap width d. In the limit as ɛ → 0 and for narrow gaps, the base flow corresponds to a solid body rotation. A flow instability is predicted even when ɛ ≪ 1. For sufficiently large values of the solvent to total viscosity ratio β, the most dangerous disturbance is time-periodic with frequency ≈ 0.1 Ω/δ, where δ denotes the ratio of d to the inner cylinder radius. The critical Weissenberg number Wec, defined as the product of the fluid relaxation time and the characteristic shear rate at the onset, obeys a scaling law of the form Wecɛ2δ1/2 = K where K is an O(1) constant that is dependent on β. This scaling is explained based on the effect of the variation in ɛ on the convection of the stress perturbations by the base flow. Predictions of Wec are in qualitative agreement with the onset Weissenberg number for a time-dependent secondary flow experimentally reported for non-shear thinning viscoelastic polymer solutions [I.M. Dris, E.S.G. Shaqfeh, J. Non-Newtonian Fluid Mech. 80 (1998) 1].