Error Estimation for Adaptive Mesh Refinement in Droplet Simulations

We present a one-dimensional shear-force-driven droplet formation model with a flux-based error estimator. The model is derived using asymptotic expansion and a front-tracking method to simulate the droplet interface. The model is then discretized using the Galerkin finite element method in the mixed form. However, the solution gradients exhibit large jumps across element boundaries and can grow rapidly due to the highly convective pinch-off process. This leads to an erroneous droplet interface and incorrect curvature. Therefore, the mesh must be sufficiently refined to capture the interface accurately. The mixed form of the governing equation naturally provides smooth interface gradients that can be used to compute the error estimate. The computed error estimate is then used to drive the adaptive mesh refinement algorithm. The efficacy of the error estimator is illustrated by comparing the droplet profiles obtained with adaptive refinement to those obtained with regular refinement. The adaptive mesh refinement approach reduces the computational cost significantly without compromising accuracy. For an 85% glycerol droplet in co-flowing air, AMR reproduces pinch-off location, surface area, volume, and pinch-off time with only ≈1% accuracy loss compared to the highly refined reference while reducing wall-clock time from 638 s to 153 s (4.17× speedup) and reducing the maximum element count from 800 to 146 (81.75% reduction).