Splash-free urinals for global sustainability and accessibility: Design through physics and differential equations.

Urinals are a staple of public spaces yet their designs have remained essentially stagnant for over a century. The use of urinals often results in significant splatter (splashback) as urine splashes upon impact with the urinal generating droplets which travel back onto the floor and user, which generates unhygienic environments, high cleaning costs, and adds unpleasant workload for custodial staff. Impinging stream angle is one of many factors that affect splashback. We theoretically predict and experimentally validate that when the impinging angle is below an invariant critical value of $\sim {30}^{\circ }$ , the flow rate of splashback under human urination conditions can be significantly suppressed. We propose novel urinal designs that were generated by solving differential equations derived from the isogonal curve problem to ensure the urine stream impacts at or below this critical angle. Experiments validate that these designs can substantially reduce splashback to only 1.4% of the splash of a common contemporary commercial urinal. The widespread adoption of the urinal designs described in this work would result in considerable conservation of human resources, cost, cleaning chemicals, and water usage, rendering large-scale impacts on modern society by improving sustainability, hygiene, and accessibility.