Linear stability of viscoelastic Poiseuille flows in cylindrical configurations

The linear stability of annular and pipe Poiseuille flow for a viscoelastic fluid in inertial regime is investigated by considering both linear modal and nonmodal stability properties of infinitesimal disturbances. The viscoelastic fluid is described by the Oldroyd-B model, and the analysis is conducted at moderately high values of the Reynolds number by varying the Weissenberg number and the viscosity ratio between the Newtonian solvent and the polymeric solution. The equations governing both flow and elastic variables are written in polar coordinates and are discretized by an accurate Chebyshev pseudospectral code, suitably adapted in order to mitigate the negative effects of the presence of continuous spectrum on the accuracy of the numerical discretization. The code has been validated by comparing the results against classical and recent numerical studies for pipe and channel viscoelastic Poiseuille flows. In the case of annular Poiseuille flow, the effects of viscoelasticity on marginal curves of the Newtonian case is such that the critical Reynolds number is decreased at low Weissenberg numbers, while high values of polymer relaxation time have a stabilizing effect. This behaviour, which is in agreement with previous findings for channel flow, vanishes as the mean radius of the annulus reduces. The concentration of the polymer has a destabilizing effect. Non modal analysis shows that significant transient growth of kinetic energy is present, as in the case of Newtonian fluids, in both annular and pipe configurations. Viscoelasticity is active in reducing the transient growth for high values of streamwise wavenumber. Energy analysis shows that the reduction of the viscosity ratio produces an increase of the dissipation term due to the interaction of the polymer stress with the flow, while increments of the Weissenberg number reduce the production of energy due to the Reynolds stresses.