Linear stability analysis of viscoelastic flows overlying permeable surfaces

We present a linear stability analysis of plane Poiseuille flow of an Oldroyd-B fluid confined between rigid, isotropic porous layers at the upper and lower boundaries. The study aims to investigate the interplay between elasticity, inertia and wall permeability on the onset of flow instability. The system is characterized by four dimensionless parameters: the Reynolds number ( $ \textit{Re}$ ), Weissenberg number ( $ \textit{Wi}$ ), permeability parameter ( $\alpha$ ) and solvent viscosity ratio ( $\beta$ ), with porosity held fixed at $\epsilon = 0.6$ . We focus on the low-permeability regime, where inertial effects within the porous layers are negligible. The governing equations are formulated using a modified Darcy–Brinkman–Oldroyd-B model, following the framework of Tan & Masuoka (2005 Phys . Fluids , vol. 17, p. 023101), and are coupled across the fluid–porous interfaces using appropriate interfacial conditions. The formulation recovers the classical results for impermeable viscoelastic channels and Newtonian flows with porous surfaces in the appropriate limits. At low $ \textit{Wi}$ , the dominant mode resembles a viscoelastic Tollmien–Schlichting instability, but porous boundaries significantly alter the spectrum, introducing concentric eigenvalue rings and modal shifts. At higher $ \textit{Wi}$ , an elastic wall mode emerges near the interfaces and dominates at low $ \textit{Re}$ . The critical Reynolds number $ \textit{Re}_{\textit{cr}}$ depends on $ \textit{Wi}$ and $\beta$ , with a secondary instability branch appearing at large $ \textit{Wi}$ . For highly permeable walls ( $\alpha =50$ ), $ \textit{Re}_{\textit{cr}}$ varies in a non-uniform trend with $\beta$ , in contrast to impermeable cases.