Equilibrium Shape for 2D Asymmetric Cylindrical Droplet on Heterogeneous Surface

We present a theoretical and numerical framework for computing asymmetric two-dimensional droplet shapes on surfaces with a sharp wetting boundary separating regions of distinct contact angles. Through Lagrange multiplier analysis of the constrained Gibbs free energy functional, we derive a simplified spreading condition that relates the contact line position ratio to the ratio of spreading functions encoding unbalanced Young stress at each contact line, reducing to an explicit algebraic relation that eliminates iterative computation. Gravitational effects substantially modify droplet height and curvature distribution across Bond number regimes, yet the contact line position ratio remains invariant, confirming that horizontal partitioning depends exclusively on interfacial energy ratios rather than body forces. Hydrophilic surfaces exhibit intuitive spreading toward regions with better wettability, producing flattened asymmetric profiles, while hydrophobic surfaces display counterintuitive behavior where droplets preferentially occupy regions with poorer wettability, maintaining tall compact geometries. Mixed hydrophilic–hydrophobic boundaries violate equilibrium conditions and drive spontaneous droplet migration. We develop an efficient two-stage computational strategy decoupling shape computation from equilibrium position determination, reducing computational cost by orders of magnitude. These findings provide quantitative design criteria for controlled droplet positioning on patterned substrates, with implications for microfluidic devices and droplet-based technologies.