The heat transfer in channels that have varying cross sections is fundamentally not the same as that in channels of the same cross section. The heat transfer mechanism is enhanced in comparison with those channels with the channels of constant cross section is based on increased flow intensity. This paper is a numerical exploration of the laminar flow and thermal transport properties of a viscoplastic fluid. The Casson model is a model of viscosity that varies with temperature. The equations of governance include mixed convection driven by buoyancy, with the plastic viscosity being an exponential function of temperature. The proximity of the behavior to the yield surface requires the use of the Casson–Papanastasiou regularization, which guarantees a smooth transition in stress between the yielded zones and the unyielded zones. The finite volume numerical simulations method displays detailed flow fields, temperature fields, and yield surface dynamics depending on the values of Reynolds and Casson numbers. The major finding of this work is the major role contributed by the half-channel geometry to the improvement of flow development and heat transfer. The lopsided form encourages enhanced convectional circulation and speeds up the destruction of unyielded areas with symmetric enclosures, enhancing mixing and thermal diffusion. Findings indicate that the fluid mobility is inhibited by a rise in the Casson number by increasing the area of unyielded flow, whereas increased Reynolds numbers lower the resistance of the flow and pressure drop. The interaction between temperature-dependent viscosity and enclosure geometry provides more insight into the mechanisms of flow enhancement of non-Newtonian systems. These findings can be useful for optimizing industrial and biomedical practices, such as exposure of yield-stress fluids to confined geometries.