Non-local model for surface tension in fluid-fluid simulations

• Published in: Journal of computational physics Vol. 421; p. 109732
• Main Authors: Howard, Amanda A., Tartakovsky, Alexandre M.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 15.11.2020; Elsevier Science Ltd
• Subjects: Computational fluid dynamics; Computational physics; Conservation equations; Finite volume; Level set method; Non-local method; Radius of curvature; Spurious currents; Surface tension; Two-phase flows; Surface tension; Non-local method; Finite volume; Two-phase flows; Level set method; Spurious currents
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2020.109732

•A non-local model for calculating the surface tension in multiscale simulations.•Easy-to-implement, partial-differential-equation-based model.•Implemented with the Conservative Level Set method.•Valid at both nano and macroscopic scales. We propose a non-local model for surface tension obtained in the form of an integral of a molecular-force-like function with support 3.5ε added to the Navier-Stokes momentum conservation equation. We demonstrate analytically and numerically that with the non-local model interfaces with a radius of curvature larger than the support length behave macroscopically and microscopically, otherwise. For static droplets, the pressure difference Pε,in−Pε,out satisfies the Young-Laplace law for droplet radius greater than 3.5ε and otherwise deviates from the Young-Laplace law. The latter indicates that the surface tension in the proposed model decreases with decreasing radius of curvature, which agrees with molecular dynamics and experimental studies of nanodroplets. Using the non-local model we perform numerical simulations of droplets under dynamic conditions, including a rising droplet, a droplet in shear flow, and two colliding droplets in shear flow, and compare results with a standard Navier-Stokes model subject to the Young-Laplace boundary condition at the fluid-fluid interface implemented via the Conservative Level Set (CLS) method. We find good agreement with existing numerical methods and analytical results for a rising macroscopic droplet and a droplet in a shear flow. For colliding droplets in shear flow, the non-local model converges (with respect to the grid size) to the correct behavior, including sliding, coalescence, and merging and breaking of two droplets depending on the capillary number. In contrast, we find that the results of the CLS model are highly grid-size dependent.