solving single phase fluid flow instability equations using chebyshev tau qz polynomial

In this article the instability of single phase flow in a circular pipe from laminar to turbulence regime has been investigated. to this end, after finding boundary conditions and equation related to instability of flow in cylindrical coordination system, which is called eigenvalue orr sommerfeld equation, the solution method for these equation has been investigated. in this article chebyshev polynomial tau-qz algorithm has been selected for the solution technique to solve the orr sommerfeld equation because in this method some of complex terms in the instability equation in cylindrical coordination will be appeared. after finding orr sommerfeld parameters related to chebyshev polynomial tau-qz algorithm the solution have been done for re=5000 and re=1000, then the results had been compared with the results of valid references where other methods had been used in them. it have been observed that the use of chebyshev tau-qz algorithm has higher accuracy concerning the results and it also has a higher accurate technique to solve the orr sommerfeld instability equations in cylindrical coordination system.