This work presents a parallel finite element solver of incompressible two-phase flow targeting large-scale simulations of three-dimensional dynamics in high-throughput microfluidic separation devices. The method relies on a conservative level set formulation for representing the fluid-fluid interface and uses adaptive mesh refinement on forests of octrees. An implicit time stepping with efficient block solvers for the incompressible Navier-Stokes equations discretized with Taylor-Hood and augmented Taylor-Hood finite elements is presented. A matrix-free implementation is used that reduces the solution time for the Navier-Stokes system by a factor of approximately three compared to the best matrix-based algorithms. Scalability of the chosen algorithms up to 32,768 cores and a billion degrees of freedom is shown.

Motivated by the emerging applications of liquid-infused surfaces (LIS), we study the drag reduction and robustness of transverse flows over two-dimensional microcavities partially filled with an oily lubricant. Using separate simulations at different scales, characteristic contact line velocities at the fluid-solid intersection are first extracted from nano-scale phase field simulations and then applied to micron-scale two-phase flows, thus introducing a multiscale numerical framework to model the interface displacement and deformation within the cavities. As we explore the various effects of the lubricant-to-outer-fluid viscosity ratioμ 2 /μ 1 , the capillary number Ca, the static contact angle θ s , and the filling fraction of the cavity δ, we find that the effective slip is most sensitive to the parameter δ. The effects ofμ 2 /μ 1 and θ s are generally intertwined, but weakened if δ < 1. Moreover, for an initial filling fraction δ = 0.94, our results show that the effective slip is nearly independent of the capillary number, when it is small. Further increasing Ca to about 0.01μ 1 /μ 2 , we identify a possible failure mode, associated with lubricants draining from the LIS, forμ 2 /μ 1 0.1. Very viscous lubricants (e.g.μ 2 /μ 1 > 1), on the other hand, are immune to such failure due to their generally larger contact line velocity.

A main challenge in numerical simulations of moving contact line problems is that the adherence, or no-slip boundary condition leads to a non-integrable stress singularity at the contact line. In this report we perform the first steps in developing the macroscopic part of an accurate multiscale model for a moving contact line problem in two space dimensions. We assume that a micro model has been used to determine a relation between the contact angle and the contact line velocity. An intermediate region is introduced where an analytical expression for the velocity exists. This expression is used to implement boundary conditions for the moving contact line at a macroscopic scale, along a fictitious boundary located a small distance away from the physical boundary.Model problems where the shape of the interface is constant thought the simulation are introduced. For these problems, experiments show that the errors in the resulting contact line velocities converge with the grid size h at a rate of convergence p ≈ 2.