The Taylor–Couette (TC) geometry is a fundamental configuration in fluid mechanics and rheometry for investigating instabilities in viscoelastic fluids. In this work, we present a combined analytical and numerical study of inertial viscoelastic TC flow with finite aspect ratio using the Oldroyd-B constitutive model. The analytical formulation is developed via a perturbation approach, with the curvature ratio as the perturbation parameter, and solved using the Ritz variational method to determine the velocity field, stream function, secondary flow characteristics, and instability threshold. Numerical simulations based on a finite-difference scheme on a staggered grid validate the analytical predictions. The results show that, for any finite aspect ratio, a pair of Taylor vortices (TVs) forms both in the absence of elasticity, driven by viscous forces, and in the absence of viscosity, induced by gradients of the first normal stress difference. Increasing the Reynolds number shifts vortex centers toward the bulk and smooths corner singularities near the walls. Beyond a critical Reynolds number, the TV mode transitions to disordered vortices, defining a clear instability threshold. The aspect ratio and viscosity ratio strongly influence secondary flow structures, while elasticity suppresses inertial effects and modulates vortex strength. This work establishes, for the first time, an analytical framework based on the Ritz method to predict the onset of instability in finite aspect ratio TC flows. The results provide a basis for future studies of instabilities and applications involving polymer solutions and biofluids in rotational systems.
Yeknomi et al. (Sun,) studied this question.