Abstract: In this paper an analytical solution to flow of second order and Maxwell fluid in annular pipe by using Hankel integral transform is presented A derived formula can be used to analyze the behavior of rotatory velocity and shear stress: since the parameters of material and the gap size of annular pipe explicitly appear in the analytical formula one can easily analyze their effection on the flow behavior. This solution can provide a theoretical base to drilling engineering and polymer shaping techniques. In addition, it can be used to analyze the flow characters in concentric cylinder rheometer and obtain material constants with curve fitting procedure.By investigation it is found that when outer cylinder makes uniform rotatory the history curve of velocity and stress of Maxwell fluid exhibit obliquerectangle wave and rawwave oscillation respectively. The wave period and amplitude increase with material constant Ha. This conclusion may be of significance in practice.