About the Upstream Contamination

This study investigates the upward transport of waterborne pollutants from a lower container to an upper container through vertically falling water streams. While previous analyses have primarily focused on inclined channels, we extend the theoretical framework to consider vertical configurations. Two distinct cases are examined: (i) a vertical flow within a cylindrical tube, and (ii) a free-falling water jet. For the first case, we derive an analytical expression for the critical water flux required to prevent the upward migration of particles. In the second case, we establish a relationship among water flux, particle size, and vertical position along the stream, which determines the feasibility of upward particle transport. Our findings reveal a fundamental difference between the two configurations. In the tubular flow case, surface tension has negligible influence on particle motion. In contrast, in the free-fall scenario, upward particle transport is only possible in the presence of surface tension. Moreover, we demonstrate that for any given water flux, there exists a threshold height difference beyond which contamination of the upper container is not possible. Increasing the water flux further inhibits any upward transport of pollutants