Flowing granular mixtures in rotating cylindrical drums arise in numerous industrial settings and are of great technological significance worldwide. To date, the development of a robust mathematical model for this process remains an open research problem. However, simple mathematical models may be developed that capture some of the underlying mechanisms, including segregation. The key lies in the analysis of the flowing surface layer whose profile is characteristically S-shaped: past models have shown that capturing this characteristic is an essential ingredient in predicting segregation. Using secondary to first-year undergraduate level mathematics, an analysis is presented of the flowing surface layer, which leads to an S-shaped profile. This problem presents an ideal case study for teaching applied mathematics and modelling. Interesting simulations that aid the visualization of this problem can be downloaded from the web, and simple classroom demonstrations of this process can be assembled with no specialized equipment. (Contains 12 figures.)