Temperature gradient and heat source/sink impacts on triple-diffusive surface-tension-driven convection

The impact of temperature gradients and internal heat sources/sinks on triple-diffusive surface-tension-driven (Marangoni) convection within a two-layer framework is analytically investigated using an exact solution methodology. A non-Darcy flow model is employed to capture the fluid behavior in the porous medium accurately. The two-layer system, assumed to be horizontally infinite and bounded by adiabatic conditions, exhibits fundamental temperature and concentration gradients. Through the utilization of conventional mode analysis, the ensuing collection of ordinary differential equations is resolved explicitly to derive the eigenvalue. Exact analytical expressions are derived for the temperature distribution considering linear, parabolic, and inverted parabolic profiles as well as for the thermal Marangoni number, with particular emphasis on the roles of the rigid lower and free upper surfaces affected by surface tension. A specialized analytical approach is adopted to address the stability of the system via an eigenvalue formulation. The analysis reveals that system stability is strongly influenced by parameters such as the solutal Marangoni numbers, corrected Rayleigh numbers, and viscosity ratios. In contrast, destabilizing effects are associated with the Darcy number and the corrected Rayleigh number in the porous domain. Among the temperature profiles examined, the inverted parabolic model offers the greatest system stability, whereas the linear profile leads to the highest instability. The study underscores how temperature variations and internal heat generation or absorption significantly affect surface-tension-driven convection, altering surface tension and modifying the interaction of thermal and solutal fields. These phenomena are critically relevant to applications such as microgravity crystal growth, geothermal energy extraction, cryogenic systems, multilayer insulation, and liquid metal batteries. The analytical results align well with existing literature, validating the developed model.