The instability of double-diffusive convection of an Oldroyd-B fluid in a vertical porous layer caused by temperature and solute concentration differences with the Soret effect is studied using a modified Darcy–Brinkman–Oldroyd model. Under the Oberbeck-Boussinesq approximation, the validity of Squire's theorem is demonstrated, allowing for considering only the two-dimensional linear instability. Based upon perturbation theory, an Orr-Sommerfeld eigenvalue problem is derived and numerically simulated using the Chebyshev collocation method. We examine the effects of the relevant dimensionless parameters on the neutral stability curves. It is discovered that PrD and λ1 have dual effects on the instability, respectively. When PrDc1 < PrD < PrDc2, PrD boosts flow stability. The impact of λ1 on fluid stability is influenced by PrD. Furthermore, Le and η have dual effects on the instability. When η< 0.8, it promotes flow instability. Le enhances the flow in-stability when Le < 0.7 and inhibits the instability when Le > 0.7. In addition, the magnitude of Le influences the role of Sr. There exists a critical value for Le. When Le < Lec2, Sr strengthens the in-stability of the flow, and when Le> Lec2, Sr plays an inhibiting role. Finally, the relaxation pa-rameter λ2 promotes flow stability.

Unidirectional flow of a fluid with pressure-dependent viscosity through a porous structure is considered when the viscosity-pressure relationship is an exponential function of a pressure power function in order to investigate effects of the viscosity-pressure relation on the flow characteristics. The flow is governed by the generalized Brinkman’s equation with constant permeability, and a model flow domain of flow down an inclined porous channel is chosen for the sake of studying flow behaviour. Although the current work considers flow in a constant permeability porous structure, it does represent a first step in studying the more general, flow through a variable permeability porous channel. The arising governing equations are solved numerically using MATLAB and the flow is simulated to illustrate the effects of fluid properties as well as flow and medium parameters on the velocity profiles and shear stress.

In this paper Brinkman type fluid over an infinite plate between side walls is being investigated. The flow is generated by oscillating shear stress of the bottom plate and the solutions are obtained by using Fourier integral transformation. The obtained results are presented in steady and transient states for both sin and cos shear stresses. The general solutions are reduced to some special cases corresponding, namely to the Brinkman type fluid over an infinite plate and flow of a Newtonian viscous fluid. Graphical illustrations are carried out to have in depth analysis of the involved physical parameters