Onset of thermal convection in the Brinkman porous layer under local thermal non-equilibrium with gravity variation and viscous dissipation

This paper aims to study the commencement of convective motion in a Brinkman porous enclosure under local thermal non-equilibrium conditions, and how it is influenced by varying gravity fields, vertical throughflow, and viscous dissipation. Four types of gravity variation—linear, quadratic, cubic, and exponential—are considered, each characterized by the gravity variation parameter (λ). The upper and lower boundaries of the layer are maintained at a constant temperature. Applying the normal mode technique, the resulting eigenvalue problem was solved numerically. The stability characteristics are analyzed through graphs and tables, illustrating the effects of the throughflow parameter (Q), Darcy number (Da), viscous dissipation (Ge), interphase heat transfer parameter (H), and gravity variation parameter (λ). It is observed that the system tends toward local thermal equilibrium for both very high and very small values of the interphase heat transfer parameter. The gravity variation parameter, throughflow, and interphase heat transfer parameter enhance system stability, whereas the porosity-modified conductivity ratio promotes instability; however, for γ≥10, the effect of H becomes negligible. Among the four gravity variations, the exponential form provides the strongest stabilization, followed by linear, quadratic, and cubic variations. In the exponential case, the critical Rayleigh number (Rac) increases markedly from about 450 to nearly 3250 as λ rises from 0 to 1.5. In contrast, viscous dissipation contributes only marginally to instability; the maximum reduction of Rac is merely 1.11% when Ge increases from 0 to 1 under linear gravity variation.