On the Stability of Double–Diffusive Convection in Rotating Couple–Stress Fluids under Spatially Varying Gravity

The onset of double–diffusive convection in a rotating couple–stress fluid layer subjected to a vertically varying gravitational field is examined through linear and nonlinear analyses. The fluid layer is confined between free–free, rigid–free, and rigid–rigid conducting boundaries, and gravity variation is incorporated via prescribed functional forms to model non-uniform gravitational effects. Linear analysis is carried out using the normal mode technique, while nonlinear analysis is investigated through the energy method, leading to an associated variational eigenvalue problem. The Rayleigh numbers are evaluated using a single–term Galerkin approximation. The analysis focuses on the stationary onset of convection. It is shown that the critical thresholds obtained from linear and nonlinear analyses are found to coincide within the adopted single–term Galerkin approximation for all boundary configurations, thereby establishing nonlinear stability within the adopted Galerkin framework and indicating the absence of subcritical convection in the considered parameter regime. The formulation is further validated through consistency checks with known limiting cases reported in the literature. While gravity variation strongly modulates the onset of convection depending on its spatial profile, rotation, couple-stress effects, and solutal stratification are found to significantly enhance system stability. Boundary rigidity further delays instability, with rigid–rigid boundaries yielding the highest stability thresholds. These results provide further insight into the control of convective instability in rotating complex fluids under spatially varying gravity, with potential relevance to geophysical flows, microgravity environments, and advanced thermal management systems.