Instability Investigation of Taylor-Couette Flow with an Axial Velocity by D 4 Chebychev Tau-Method A. Nouri- Borujerdi 1 , A. Kebriaee 2 School of Mechanical Engineering, Sharif University of technology, Tehran, Iran anouri@sharif.ir Abstract Instability of a viscous incompressible flow between two rotating concentric cylinders has been investigated numerically. The outer cylinder is stationary and the inner one rotates and moves axially simultaneously. D 4 Chebychev tau method is used to solve eigenvalue problem governing the investigation of flow stability, In the Chebychev tau method, the eigen functions of governing equations are expressed in terms of Chebychev polynomials and the generalized eigenvalue problem is solved with the QZ algorithm. It should be mentioned that this method is a new approach to solve instability of spiral Couette flow and it is less time consuming and more direct than the methods used by previous workers. The applied method is so efficient and our results are in a very good agreement with pervious works. The critical flow characteristics are calculated at the threshold of instability for various Reynolds number based on the axial velocity. The results point out that the axial velocity increases the flow stability. Also, it is found that the axisymmetric modes are more unstable than the corresponding non-axisymmetric modes for all of Reynolds number. Key Words: instability, Chebyshev tau methods, Taylor-Couette flow Nomenclatures a dimensionless wavelength  angular velocity A, B matrix  kinematic viscosity D differential operator, dd  Superscripts m mode number __ mean L complex differential operator ~ transformed p pressure * dimensionless r radius ' perturb Re Reynolds number, ( ) zi o i u r r   Subscripts t time i inner T Chebyshev polynomials, Taylor number, 2 4 2 1 ( ) o i r r    imag imaginary u velocity o outer Greek Letters c critical, complex  density n index  eigenvalue , , r z  coordinate components Introduction The flow between two concentric cylinders with the inner cylinder rotating and the outer one stationary called Taylor–Couette flow. The stability of the Taylor-Couette flows has been the subject of numerous analytical and experimental investigations. The important of this study is elucidated in different industrial applications such as drilling in oil industries, flows about ship propellers, jet