An implicit surface tension model for the analysis of droplet dynamics

Abstract

A Lagrangian incompressible fluid flow model is extended by including an implicit surface tension term in order to analyze droplet dynamics. The Lagrangian framework is adopted to model the fluid and track its boundary, and the implicit surface tension term is used to introduce the appropriate forces at the domain boundary. The introduction of the tangent matrix corresponding to the surface tension force term ensures enhanced stability of the derived model. Static, dynamic and sessile droplet examples are simulated to validate the model and evaluate its performance. Numerical results are capable of reproducing the pressure distribution in droplets, and the advancing and receding contact angles evolution for droplets in varying substrates and inclined planes. The model is stable even at time steps up to 20 times larger than previously reported in literature and achieves first and second order convergence in time and space, respectively. The present implicit surface tension implementation is applicable to any model where the interface is represented by a moving boundary mesh.

A particle finite element-based model for droplet spreading analysis

“…The fluid is considered incompressible, viscous, and Newtonian. Thus, the governing equations read 12,34 :…”

“…However, in the present section, it is illustrated using the Backward-Euler scheme for sake of simplicity 43 . A more complete algorithm for a two-phase flow model can be found in 34,37 . The problem statement is the following: given a known velocity and pressurev n andp n at time t n , find the values of these variables at the next time step t n+1 (v n+1 and p n+1 ) by solving:…”

“…However, this approach cannot account for droplet pinning, and therefore, cannot predict the dynamics of spreading droplets. A dynamic contact angle condition is needed to account for this phenomenon 9,34 . The most basic dynamic contact line condition imposes a slip boundary condition, i.e., contact line velocity normal to wall surface is zero, when a critical con-tact angle is reached 16 .…”

“…In addition, most of the aforementioned models were applied to fixed-mesh methods. Our previous work has shown that a moving-grid particle finite element method (PFEM)-based model is well suited to droplet modeling 9,10,34,37 . Application of a PFEM model has proven to be particularly advantageous when dealing with surface tension dominated problems, as it allows to accurately track the evolution of the liquid domain boundary and to account for surface tension, without introducing enrichment shape functions or smearing the surface tension via continuous force approach 38,39 .…”

“…However, neither a dissipative force term on the contact line nor a viscous dissipation term on the solid-liquid interface has been implemented and tested in PFEM-based droplet models. The objective of this article is to identify an appropriate dynamic contact line condition for the PFEM-based fluid flow solver proposed in 34,37 and to verify that it provides a mesh-independent solution. In the present work, our proposed droplet dynamic model is equipped with i) an effective slip contact angle boundary condition that balances the induced Young's stress, and ii) a solid-liquid slip boundary condition, away from the contact line, that accounts for the viscous dissipation along the solid-liquid interface 15,22 .…”