Linear stability analysis of Oldroyd-B fluids flow in pipes

The flow of non-Newtonian fluids has recently gained considerable importance because of its applications in various branches of science, engineering and technology: particularly in material processing, chemical and nuclear industries, geophysics, and bioengineering. The study of non-Newtonian fluids flow is also of significant interest in oil reservoir engineering. The inadequacy of the Navier-Stokes equations in describing the dynamics of such complex fluids has led to the formulation of other mathematical models able to predict their behaviour. One of them is the Oldroyd-B fluid model that acquired a particular importance since it can be seen as a conjunction of two special cases: the classical Newtonian fluid and the elastic fluid described by the Upper Convected Maxwell model. Moreover it is able to well describe the dynamics of highly dilute polymer solutions, the so-called Boger's fluids. The aim of this thesis is to investigate the linear stability of fluids described by the Oldroyd-B constitutive equation in cylindrical geometry from the point of view of modal and non-modal analysis. Note that only in recent years non-modal stability analysis of the flow of such kind of fluids, only in two-dimensional channel configuration, has been worked out since of its complexity that does not allow to apply in a straightforward way the classical non-modal analysis tools. Detailed parametric analyses of Oldroyd-B fluids in annular pipe have been performed. This flow configuration has been chosen not only for its wider application in industrial process, but also because it represents an intermediate case between the channel and the Hagen-Poiseuille flow by varying the inner radius of the pipe. This investigation has demonstrated the different behaviour of Oldroyd-B fluid with respect to the classical Newtonian one, and the energy budget analysis of the disturbance energy amplification has allowed to highlight some interesting physical mechanism governing such fluids.