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Linear stability of Poiseuille flow of viscoelastic fluid in a porous medium

We study the instability of plane Poiseuille flow of the viscoelastic second-order fluid in a homogeneous porous medium. The viscoelastic fluid between two parallel plates is driven by the pressure gradient. The effects of elasticity number E (depends on fluid properties, geometry; E is defined below) and Darcy number Da (gives the permeability of porous medium; Da is defined below) on flow stability are analyzed through the energy method that provides qualitative behavior of flow stability, and the numerical solution of generalized eigenvalue problem that gives the precise upper bound for stability. The plane Poiseuille flow of second-order fluid becomes unstable for increasing elasticity number while preserving Newtonian eigenspectrum up to a certain range of E. For large elasticity number, instability appears as a part of both wall and center modes for all Darcy numbers. We also noticed that along each neutral stability curve, the eigenfunctions are all antisymmetric with a single extremum near the channel walls. When E = 0.0011, we found an additional new elastic mode, which is unstable and also antisymmetric. For E < 0.0011, the neutral curves split into two lobes with different minima. The critical Reynolds number Rec is found to be decreasing (increasing) for higher (lower) values of fluid elasticity (Darcy number). Physical mechanisms are discussed in detail.