Stability analysis of a viscoelastic liquid in a channel flow with uniform wall suction/blowing

This paper investigates the linear stability of a pressure-driven plane Poiseuille flow of a viscoelastic second-grade fluid under the influence of uniform transverse cross-flow through porous channel walls. The second-grade fluid model accounts for non-Newtonian effects through the inclusion of nonlinear normal stresses represented by the first and second Rivlin–Ericksen tensors, under the thermodynamically consistent constraints. An exact analytical solution for the base flow is derived and shown to depend sensitively on the elasticity number, E and the cross-flow Reynolds number, Rc, leading to asymmetric velocity profiles with steep gradients near the suction boundary. Two-dimensional disturbances are introduced into the governing equations, leading to a modified Orr–Sommerfeld equation that incorporates both viscoelastic and cross-flow effects. The resulting eigenvalue problem is solved numerically using Chebyshev spectral collocation. Numerical results reveal that the combined effect of elasticity and cross-flow alters the growth rate, phase velocity, and spatial structure of disturbances, and leads to shifts in the critical Reynolds number and dominant mode type. This effect gives rise to nontrivial mode interactions, including stabilizing and destabilizing regimes depending on parameter ranges. The flow becomes linearly unstable above a critical Reynolds number that increases with elasticity. In particular, elasticity delays the restabilization induced by cross-flow and amplifies the growth of wall modes at higher Rc. The interplay between cross-flow and elasticity significantly alters the stability boundaries and the structure of the eigenvalue spectrum which undergoes substantial rearrangement, particularly in the structure of wall and means modes.