Local linear stability characteristics of viscoelastic periodic channel flow

Linear stability of periodic channel (PC) flow of an upper convected Maxwell (UCM) liquid is investigated in inertial (Reynolds number Re ≫ 0, Weissenberg number We ∼ O(1)) and in purely elastic (Re ≡ 0, We ∼ O(1)) flow regimes. Base state solution is evaluated using O(ε²) domain perturbation analysis where ε denotes the channel wall amplitude. Significant destabilization, i.e. reduction in critical Reynolds number Re_c, with increasing ε is predicted for the Newtonian flow, especially in the diverging section of the channel. Introduction of elasticity E ≡ We/Re, representing the ratio of fluid relaxation time to a time scale of viscous diffusion based on channel half height, leads to further destabilization. However, the minimum in the Re_c-E curve reported for plane channel flow is not observed. Analysis of the budget of perturbation kinetic energy shows that this minimum in the plane channel flow results from two competing contributions to kinetic energy: a normal stress contribution that increases with increasing E and a shear stress contribution that decreases monotonically with increasing E with the two curves intersecting for E ≈ 0.002. This value is approximately equal to the value of E for which the plane shear layer is maximally destabilized. When this happens, the critical Deborah number, defined as the ratio of the fluid relaxation time to time scale of the perturbation, is O(1). Comparison of results obtained for the UCM and second order fluid (SOF) models shows that the latter model does not predict a minimum Re_c for E ≤ 0.003. Moreover, eigenspectrum for the SOF contains a set of eigenvalues with positive real parts equal to 1/We. Results obtained for the eigenspectrum in the purely elastic limit indicate that the PC flow is linearly stable for O(1) axial wavenumbers for We ≤ 10, ε ≤ 0.1 and n ≤ 0.1, although the decay rates of the perturbation are smaller than that of the plane channel flow. The local eigenspectrum could contain spurious eigenvalues with positive real parts that could lead to erroneous predictions of flow instability. Using a contour mapping technique, it is shown that deformation of the flow domain can lead to spurious eigenvalues.