Transient free convection in a vertical channel with third-kind thermal boundary conditions: A semi-analytical study

This study presents a semi-analytical investigation of transient free convection flow of a viscous, incompressible fluid within a vertical channel subjected to third-kind thermal boundary conditions. These boundary conditions, representing convective heat exchange between the fluid and surroundings, offer a more realistic thermal model for practical systems such as heat exchangers and insulated enclosures. The governing partial differential equations are transformed into the Laplace domain using the Laplace transform technique, and closed-form solutions are obtained. These are subsequently inverted to the time domain via Riemann-sum approximation. To capture memory and hereditary effects inherent in complex fluid behavior, the model incorporates fractional derivatives in the Caputo-Fabrizio and Atangana-Baleanu senses. The study analyzes temperature distribution, velocity profiles, Nusselt number, and skin friction, with results validated numerically using MATLAB. Graphical and tabular analyses reveal the influence of key parameters, including Biot number, buoyancy forces, and various Prandtl numbers. The findings contribute to the broader understanding of transient free convection under realistic thermal conditions and have potential applications in the design and optimization of thermal systems in engineering and industry.