There are many modern engineering applications of natural convection with free surface square cavity. In this paper analyzes numerically convection that is both natural and Marangoni. Together adiabatic cooling and convective cooling are subject to the upper free surface boundary, whereas a portion of the bottom wall boundary is exposed to constant heat flux and the remaining portions are maintained adiabatically. The unique of this research is the possibility of using its numerical analysis to enhance the applications related to solar energy, which is considered one of the most important applications in the field of renewable energy, in addition to food storage. The boundaries of the vertical walls of square cavity are kept cold. Under the influence of constant heat flux, two physical phenomena thermocapillary and buoyancy forces are considered and statistically depicted. A finite volume method is used to solve the governing equations based on the heat line approach. The current research is supported by other studies that are published in the literature. For different values of the Marangoni number (-103 to +104), Grashof number (104 to 105), Prandtl number (0.054, 0.16, 0.71, 6.2, 100), and Biot number (0 to 80) are produced to investigate the impacts on streamlines, isothermal lines, heat lines, and local entropy generation. The solving of the government equations is depending on numerical solution for the control volume system with assumed that the situation of heat transfer and fluid flow is steady with generating entropy over the all-control volume under investigation and density differences are ignored. It has been noted that the maximum stream function increases as Marangoni numbers increase from 0 to 104. This trend results from the fact that the flow velocity increases with the increasing Marangoni number. However, the Marangoni It has a reciprocal effect with convective heat transfer. The solution and its improvement are within the conditions of this research. The flow pattern is not considerably affected when the Biot number is changed from 40 to 80 as the surface tension is minimal.