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

¹ Laboratory of Mechanics and High Energy Physics, Faculty of Sciences Aïn-Chock, Hassan II University of Casablanca, P.O. Box 5366 Maarif Casablanca 20100, Morocco.
² Higher Institute of Marine Fisheries- Agadir
³ Laboratory of energy Engineering, Materials and Systems, ENSA, Ibn Zoher University, Agadir Morocco

Abstract

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 α_c = 1.021 when transverse flow is absent. We then studied the influence of transverse injection, expressed by the injection Reynolds number R_c. For R_c = 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 R_c, 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.