There are many coolants frequently used in the industry for controlling not only heat transfer, but also temperature distribution in a confined domain. However, little is known on the thermal properties of sodium droplets. The qualitative analysis of differential equations that model the thermal explosion, nonlinear dynamic of sodium droplet with variable thermophysical properties when thermal radiations are considered as suggested by Cogley et al, Sohrab et al, and P-1 approximation Sazhin et al is deliberated upon in this study. The governing equations, first-order nonlinear ordinary differential equations, are nondimensionalized using the appropriate similarity variables. The existence and uniqueness of the solutions, concavity, and convexity of the temperature distribution, and positivity nature of the solutions of the dimensionless governing equations are established. It is concluded that there exists a solution for a certain range of the admissible parameters and when the reduced activation energy is negative and temperature distribution fits concavity. More so, the major criteria to obtain a positive solution are outlined in this study. K E Y W O R D S concave, convex, existence and uniqueness, explosion, sodium droplet Recently, a comprehensive discussion on concave and convex functions was presented by Kythe. 24 Basic definition, example, and application of convex functions to mathematical programming had also been presented by Jeter. 25 In the same report, basic results concerning the optimization of convex functions, minimization and maximization of a convex function, are outlined. Sequel to the review of reports by Pinchover, 26 Feng et al, 27 Xing, 28 Ge and Xue, 29 and Baotong Cui et al, 30 positivity of the solution can be pointed out as major characteristics of any 1512 | ADEGBIE ET AL.