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 spreading condition that relates the contact line position ratio to the ratio of spreading functions, which encode the unbalanced Young stress at each contact line. Under geometric self-similarity assumptions valid for moderate gravitational effects, this condition reduces to an explicit algebraic relation. Hydrophilic surfaces exhibit intuitive spreading toward regions with better wettability, producing flattened asymmetric profiles. Conversely, hydrophobic surfaces display counterintuitive behavior where droplets preferentially occupy regions with poorer wettability, maintaining tall compact geometries. Bond number variations from capillary-dominated to gravity-influenced regimes demonstrate systematic gravitational flattening, yet the contact line position ratio remains invariant across gravitational conditions, confirming that horizontal partitioning depends exclusively on interfacial energy ratios rather than body forces. Mixed hydrophilic-hydrophobic boundaries violate equilibrium conditions and drive spontaneous droplet migration. These findings provide quantitative design criteria for applications requiring controlled droplet positioning on patterned substrates.