Axisymmetric problem of smoothing the surface of a viscous liquid by surface tension forces

This study investigates an analytical solution to the problem of the surface levelling of viscous liquids under the influence of surface tension forces, focusing on the smoothing of plastic surfaces subjected to thermal energy treatment. This study aims to extend Orchard’s formula to axisymmetric surface irregularities and develop an analytical model for predicting levelling time, thereby ensuring efficient process control in thermal treatment applications. The tasks included deriving an analytical solution for axisymmetric levelling, validating it against numerical simulations in LS-DYNA, and incorporating the viscosity variation across the liquid layer. The methods involved analytical formulation and numerical simulation of surface evolution considering different initial surface geometries and viscosity distributions. Validation against numerical results demonstrated high accuracy for moderate and thick liquid layers ( ) and initial surface amplitudes up to 40% of the characteristic radius. Following validation, the model was applied to estimate levelling times for various surface configurations while maintaining simplicity while improving the predictive capabilities. Results showed that the extended formula effectively describes surface smoothing dynamics, including the cases with thickness-dependent viscosity, providing explicit expressions for levelling time. These findings enable precise control of heat input during thermal energy treatment, thereby optimizing the surface quality. In conclusion, the proposed analytical solutions offer a practical tool for surface levelling analysis, expanding the applicability of Orchard’s approach to more complex geometries and viscosity variations. In future work, we will focus on experimental validation and refinements to enhance the accuracy in industrial applications.