The Ganges is India's longest river and flows from the western Himalayas, through the country’s most densely populated regions and eventually out into the Bay of Bengal. While the river serves more than 500 million people for drinking, bathing, and agricultural purposes it is also considered to be one of the most polluted rivers in the entire world. In response, this thesis proposes a mathematical model to monitor the spread of pollutants such as coliform bacteria by sequentially solving the shallow water equations and the advection-diffusion equation. A particular focus was directed towards the Varanasi Bend, a critical area prone to pollution accumulation in the northern parts of the Ganges. Moreover, the general well-posed boundary conditions of the shallow water equations were examined. The obtained results indicate that the general boundary conditions are compatible with Lax-Friedrich's method and that the system is strongly advection dominated. The simulations also suggest that further experimental validation is warranted and that such research could enhance our ability to track pollutants in water bodies such as the Ganges.