In this paper, we analyze the local linear stability of plane Poiseuille flow of an upper convected Maxwell (UCM) fluid through a periodic channel under two flow regimes, i.e., inertial (Re \(\ne\) 0) and purely elastic (Re \(\equiv\) 0). The analysis is conducted with respect to the dimensionless control parameters: Reynolds number (Re), elasticity number (E), and Weissenberg number (We). We focus on the stability of two-dimensional perturbations, using spectral methods and Chebyshev collocation to discretize the dispersion equations. For creeping flow, we perform a numerical study to explore the combined effects of periodic modulation ( \(\epsilon\) ), section (x), and control parameters (E, We) on the stability of UCM fluid flow, and to examine the elasto-inertial interplay in flow stability. Our results reveal two key findings: first, the existence of a critical position ( \(x_{c}\) = \(\frac{\pi }{2n}\) ) and ( \(x_{c}\) = \(\frac{3\pi }{2n}\) ) for small wavenumbers (n); and second, insights into the structure of the full elasto-inertial eigenspectrum, consisting of multiple discrete modes influenced by the section (x) and channel amplitude ( \(\epsilon\) ).
