Abstract
The conditions for the emergence and stability of finite amplitude purely elastic (non‐inertial) overstability are examined for axisymmetric Taylor–Couette flow of an Oldroyd‐B fluid in the narrow‐gap limit. The study is a detailed account of the formulation and results published previously [Khayat, Phys. Rev. Lett. 1997; 78: 4918]. The flow field is obtained as a truncated Fourier representation for velocity, pressure and stress in the axial direction, and in terms of symmetric and antisymmetric Chandrasekhar functions along the radial direction. The Galerkin projection of the various modes onto the conservation and constitutive equations leads to a closed low‐dimensional nonlinear dynamical system with 20o of freedom. In contrast to our previous model that was based on the simplifying rigid‐free boundary conditions [Khayat, Phys. Fluids A 1995; 7: 2191], the present formulation incorporates the more realistic rigid–rigid boundary conditions, and is capable of capturing quantitatively the flow sequence observed in the experiment of Muller et al. [J. Non‐Newtonian Fluid Mech. 1993; 46: 315] for a highly elastic (Boger) fluid under conditions of negligible inertia. Existing linear analysis results are first recovered by the present formulation, which predict the exchange of stability between the circular Couette flow and oscillatory Taylor vortex flow via a postcritical Hopf bifurcation as the Deborah number exceeds a critical value. The stability conditions of the limit cycle are determined using the method of multiple scales. The present nonlinear theory predicts, as experiment suggests, the growth of oscillation amplitude of the velocity and the emergence of higher harmonics in the power spectrum as the Deborah number increases. Good agreement is obtained between theory and experiment. Copyright © 2002 John Wiley & Sons, Ltd.