## Instability modes in viscoelastic Taylor-Couette flows with different rotation regimes

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Different instability modes are predicted and observed in the viscoelastic Taylor-Couette flows for different rotation regimes of the cylindrical annulus. The inner cylinder has the radius $a$, the radius of the outer cylinder $b$, the angular frequency of the inner cylinder is $\Omega_i$, that of the outer cylinder is $\Omega_o$. The basic solvent is a mixture of water and PEG (polyethylene glycol) to which we have added PEO (polyethylene oxyde)in different concentrations up to $1000\, ppm$ to generate working solution of density $\rho$, kinematic viscosity $\nu$. The PEG has been added in order to obtain very viscous solvent to ensure almost constant viscosity of the solution. Viscoelastic solutions with constant viscosity are called Boger fluids and can be described by the Oldrody-B model \cite{bird1977, Larson1992}. Rheological measurements have allowed to determine the viscosity $\nu$ and the relaxation time $\tau$ of the solution \cite{Bai2023}. The control parameters are the rotation ratio $\mu=\Omega_o/\Omega_i$,the Taylor number $Ta=Re\sqrt{d/R}$ where $Re=\Omega Rd/\nu$ with $R\in\left[a,b\right]$ depending on the rotation of the inner or the outer cylinder, the viscosity ratio $S=\nu/\nu_0$ and and elasticity $E=\tau/\tau_{\nu}$. Linear stability analysis has allowed to determine the diagrams of critical states in the plane $(E,Ta$ for different rotation regimes \cite{Bai2015, Bai2022, Bai2023} : $\mu=0$,$\mu=\eta^{3/2}$,$\mu=\eta^{3}$ and $\mu= \infty$. Experiments have confirmed theoretical results in most of cases. Three different states have been confirmed : stationary vortices analog to Taylor-vortices for low values of the elasticity $E$, disordered patterns for large values of $E$ and ribbons states formed of counter-propagating spirals for intermediate values of $E$ \cite{Latrache2016,Latrache2021}. The role of viscosity ratio has been analyzed in more details. In the case of the sole rotating inner cylinder ($\mu=0$), the evolution of the ribbons states and of the disordered has been characterized using PIV measurements.