This article addresses the analysis of the uniform film thickness and stationary points forthe Carreau thin fluid film flow. The flow of fluid on a vertically upward moving cylinder takesplace in the presence of a surface tension gradient. The resulting non-linear and inhomogeneousordinary differential equation is solved for the series form solution using Adomian decompositionmethods (ADM). Stokes number St, inverse capillary number C, Weissenberg number W e andfluid behavior index n emerged as flow control parameters. The analysis showed that thepositions of stationary points transferred towards the surface of the cylinder by the increase ofSt and C while towards the fluid-air interface by the increase of n. W e delineated vice versaeffects on positions of stationary points for the shear thickening fluid film and shear thinningfluid film. The width of uniform film thickness reduces by an increment in the St and Cwhereas it increases by an increment in the n. The width of shear thickening uniform filmthickness increases whilst shear thinning uniform film thickness decreases as the W e increases. A comparison between the linearly viscous fluid and Carreau fluid is also made.