The goal of this study is to examine the thermal performance and the flow of hybrid nanofluid over three types of longitudinal wetted fins encompassing thin tip fin, thick tip fin, and fin with a rectangular shape. It is pertinent to mention that both thin and thick tip fins are of exponential profile. The longitudinal fins are of variable thickness with stagnant, stretching, and shrinking cases in a fully wetted circumstance to study the heat augmentation of a conveyer belt. The previous studies considered constant temperature but, in this analysis, the variable type (temperature-dependent) i.e., exponential and linear forms of thermal conductivities are cogitated for all three cases. A comparison of thermal performance is also elaborated for variables thermal conductivities considering the above-cited three cases. The analysis would be more useful in the designing of fin structures that are helpful to enhance thermal efficiency. This is a pioneering study in this direction. The mathematical model is elucidated numerically. The impact of arising pertinent parameters is arbitrated. It is examined that inverted and rectangular fins possess more heat transfer rate in comparison to the tapered fin. Furthermore, the surface temperature of the stretching fin is higher than the shrinking fin.