Dynamic bifurcation and instability of Taylor-Dean problem

The main objective of this paper is to address the instability and dynamical bifurcation of the viscous incompressible fluid between two concentric cylinders, which is called Taylor-Dean problem. Compared with traditional numerical analysis methods, our analysis is based on the recently developed nonlinear theory: the bifurcation and stability theory for nonlinear dynamical systems (Ma and Wang in Phase Transition Dynamics, Springer, New York, pp. 80-120, 2014). First, we propose the simplified governing equations in the cylindrical coordinates to describe the Taylor-Dean problem. Second, a nonlinear theory is obtained for the Taylor-Dean problem, leading in particular to rigorous justifications of the linear theory used by physicists, and the vortex structure. It is observed that basic flow becomes unstable and the movement of the flow in r and z directions will occur if the partial derivatives of pressure function exceed a certain critical value. In addition, we have given the structure of flow in r and z directions and the numerical investigation of the critical values of the control parameter.