Time-dependent simulations of non-axisymmetric patterns in Taylor-Couette flow of dilute polymer solutions

Nonlinear dynamics that ensue after the inception of viscoelastic flow instabilities in homogeneous, curvilinear shear flows remain largely unexplored. In this work, we have developed an efficient, operator splitting influence matrix spectral (OSIMS) algorithm for the simulation of three-dimensional and transient viscoelastic flows. The OSIMS algorithm is applied to explore, for the first time, the post-critical dynamics of viscoelastic Taylor-Couette flow of dilute polymeric solutions utilizing the Oldroyd-B constitutive equation. Linear stability theory predicts that the flow is unstable to non-axisymmetric and time-dependent disturbances with critical conditions depending on the flow elasticity, E, defined as the ratio of the characteristic time scales of fluid relaxation to viscous diffusion. Two types of secondary flow patterns emerge near the bifurcation point, namely, ribbons and spirals. We have demonstrated via time-dependent simulations for narrow and moderate gap widths, ribbon-like patterns are generally stable at and above the linear stability threshold for 0.05 ≤ E ≤ 0.15. For an inner to outer cylinder radius ratio of 0.8, the bifurcation to ribbons at E = 0.1 and 0.125 occurs through a subcritical transition while the transition is supercritical at smaller E values.