On stability condition for bifluid flows with surface tension: Application to microfluidics

Models for incompressible immiscible bifluid flows with surface tension are here considered. Since Brackbill et al. [J.U. Brackbill, D.B. Kothe, C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100 (1992) 335–354] introduced the Continuum Surface Force (CSF) method, many methods involved in interface tracking or capturing are based on this reference work. Particularly, the surface tension term is discretized explicitly and therefore, a stability condition is induced on the computational time step. This constraint on the time step allows the containment of the amplification of capillary waves along the interface and puts more emphasis on the terms linked with the density in the Navier–Stokes equation (i.e. unsteady and inertia terms) rather than on the viscous terms. Indeed, the viscosity does not appear, as a parameter, in this stability condition.We propose a new stability condition which takes into account all fluid characteristics (density and viscosity) and for which we present a theoretical estimation. We detail the analysis which is based on a perturbation study – with capillary wave – for which we use energy estimate on the induced perturbed velocity. We validate our analysis and algorithms with numerical simulations of microfluidic flows using a Level Set method, namely the exploration of different mixing dynamics inside microdroplets.