Theoretical and numerical study of the critical threshold of linear stability for the flow of a weakly viscoelastic fluid in a cylindrical pipe with a horizontal axis

In this paper, we seek to determine the critical Reynolds number of a viscoelastic fluid flowing in a cylindrical pipe with a horizontal axis. The problem obtained is a generalized eigenvalue problem . A Gauss-Lobatto-Tchebyshev method was adopted to discretize this equation and the QZ algorithm combined with the Newton-Raphson method was used to determine this critical value of the Reynolds number. It is obtained by searching for two successive and very close values for which correspond two eigenvalues whose maximum real parts are respectively negative and positive. In other words, the critical value is the smallest value of the Reynolds number for which instability occurs. The code for performing this calculation was written in FORTRAN. The flow is stable if all the real parts of the eigenvalues obtained are negative and unstable if only one of these values is positive.