Effect of a surface tension imbalance on a partly submerged cylinder

We perform a force analysis of a circular cylinder which lays between a liquid--gas interface and acts as a barrier between a surfactant-free surface and a surfactant-loaded surface. The respective surfaces have uniform surface tensions $\gamma_a$ and $\gamma_b$ which generate a surface tension imbalance $\Delta \gamma=\gamma_a-\gamma_b$, also referred to as surface pressure. In addition to the general force analysis, we determine the effect of $\Delta\gamma$ on the load-bearing capacity of a floating cylinder upon sinking for a specific set of parameters. Moreover, we demonstrate that $\Delta \gamma$ induces a horizontal force component which in magnitude is equal to $\Delta \gamma$, when measured per unit length cylinder, and use an energetic argument to prove that this relation applies to prismatic bodies in general.