Critical flow speeds of pipes conveying fluid by the Generalized Differential Quadrature method

The aim of this study is to clarify the discrepancy regarding the critical flow speed of straight pipes conveying fluids that appears to be present in the literature by using the Generalized Differential Quadrature method. It is well known that for a given “mass of the fluid” to the “mass of the pipe” ratio, straight pipes conveying fluid are unstable by a flutter mode via Hopf bifurcation for a certain value of the fluid speed, i.e. the critical flow speed. However, there seems to be lack of consensus if for a given mass ratio the system might lose stability for different values of the critical flow speed or only for a single speed value. In this paper an attempt to answer to this question is given by solving the governing equation following first the practical aspect related to the engineering problem and than by simply considering the mathematics of the problem. The Generalized differential quadrature method is used as a numerical technique to resolve this problem. The differential governing equation is transformed into a discrete system of algebraic equations. The stability of the system is thus reduced to an eigenvalue problem. The relationship between the eigenvalue branches and the corresponding unstable flutter modes are shown via Argand diagram. The transfer of flutter-type instability from one eigenvalue branch to another is thoroughly investigated and discussed. The critical mass ratios, at which the transfer of the eigenvalue branches related to flutter take place, are determined.