The copper–water nanofluid (40 nm) mixed convection of a static, laminar, incompressible flow of a non-Newtonian Bingham fluid under inertial stresses caused by the tilting of a lid is examined in a square chamber having unit diameter is the main objective of our study. A cold-surfaced, wavy square cylinder is inserted in the core of this chamber, and the isothermal horizontal side walls preserve two different temperatures: The left one is hot (Th), and right one is cold (Tc). Additionally, there is insulation on both upper and lower walls. The flow dynamics of cavities are affected by application of an inclining magnetic field at an angle \(\left( \theta \right)\) to the horizontal and has cold wavy cylinder obstacles added to it. The effect of produced temperature gradients, lid-driven flow, and a magnetic field of magnitude \(B_{0}\) on heat transfer near the obstacle and fluid motion within the enclosure is investigated. Positioning cylinders with different thermal distributions can also enhance or regulate the thermal management in enclosures. Numerical simulations are performed using an open-source application called COMSOL Multiphysics, which is based on the finite element method (FEM). The impacts of isotherms, streamlines and Nusselt number on numerous parameters such as Bingham fluid parameter (Bn) values of (0–10), Hartmann number (Ha) values of (0–60), Reynolds number (Re) values of (1–20), and Richardson number (Ri) values of (1–20) number in a square hollow with wavy square cylinder and an angled magnetic field \(\varphi\) are examined in detail in this article. The local and mean Nusselt numbers rise with increasing Re, suggesting greater convective heat transport as a result of higher inertial forces. Both heat transmission and fluid motion can be enhanced by an increase in Ri, which refers to increased buoyant forces. Also, greater Bn values decrease the fluid's kinetic energy and a local and mean Nusselt numbers. These results have real-world implications for industrial processes involving magnetic field management of non-Newtonian fluids, including metallurgical operations, polymer processing, and the construction of sophisticated cooling systems where precise temperature and flow control are essential. Additionally, cylinders installed in chambers are extensively engaged in a variety of practical settings in civic and industrial contexts, such as indoor energy management, bearing lubrication, cooling shafts, and food processing.
