Instability of Double-Diffusive Convection in an Oldroyd-B Fluid-Saturated Vertical Brinkman Porous Layer

The instability of double-diffusive convection in an Oldroyd-B fluid in a vertical porous layer is investigated using a modified Darcy–Brinkman–Oldroyd model. Squire's theorem is validated, reducing the problem to two-dimensional linear instability. The Orr–Sommerfeld eigenvalue problem is solved numerically using the Chebyshev collocation method. The effects of dimensionless parameters on the neutral stability curves are examined. It is found that the Darcy–Prandtl number (Pr_D), relaxation time (λ₁), and normalized porosity (η) have dual effects on instability. Pr_D has two critical values: Pr_Dc1 and Pr_Dc2. When Pr_Dc1 < Pr_D < Pr_Dc2, Pr_D inhibits convection; otherwise, Pr_D promotes convection. The effect of λ₁ on fluid stability is influenced by Pr_D. When η > η_c (the critical value of η), it promotes flow instability; when η < η_c, it suppresses instability. For Lewis number Le > 2.31, two instability regions are observed, requiring three critical Darcy–Rayleigh numbers to determine flow instability. For Le < 2.31, the finite unstable region disappears. Finally, the relaxation parameter λ₂ promotes flow stability.