New modified energy-based equations for improving the shape analysis of droplets stains for all impact angles: Theory, experimental validation, and comparison

Shape analysis of stains has always been challenging and inaccurate, especially for sharp impact angles. To compensate for this inaccuracy, this research introduces three new equations for the shape analysis of stains on inclined surfaces. The first equation is formulated based on the conservation of energy for elliptical stains leading to a formula based on equivalent diameter. The second equation deals with stains' width and is derived based on a droplets' lateral spread into a hypothetical circle. Furthermore, an improved version of this equation is derived and experimentally validated. All equations showed superior accuracies over several other existing models, and the errors associated to the first, second and the improved (third) equations were 6%, 11.4% and 3.7%, respectively. The high accuracy of the first equation is due to the use of realistic elliptical shapes, and the accurate results of the third equation imply that there must be a valid physical meaning behind the hypothetical circle. These can also be interesting areas for future investigations.