Flow in a Channel with a Second-Order Fluid Under Transverse Flow Conditions

We examine the linear stability of a second-order fluid flow between two parallel, porous plates, with uniform transverse flow. An analytical approach was developed to obtain the base solution for stationary flows of a slightly viscoelastic fluid, which was then perturbed around the equilibrium state. The governing problem is formulated as a modified Orr–Sommerfeld equation and solved numerically using the Chebyshev collocation technique. Our numerical code was validated by reproducing classical results for a Newtonian fluid, with a critical Reynolds number Rec = 5772.22 and critical wave number αc = 1.021 when transverse flow is absent. We then studied the influence of transverse injection, expressed by the injection Reynolds number Rc. For Rc = 0.2, 0.4, and 0.6, the flow shows increasing stabilization, with critical Reynolds numbers rising accordingly. When fluid elasticity is included, with K = - 10–4, the growth rate of the most unstable mode decreases by roughly 15%, delaying the onset of instability. At high Rc, elasticity and transverse flow combine to shift the flow to more stable regimes, highlighting how even slight viscoelasticity can significantly modify the transition to instability. These results provide a clearer understanding of how transverse flow and fluid elasticity interact to influence channel flow stability, with potential applications in polymer processing, filtration, and porous media flows.