Understanding fire sprinkler sprays fills a critical gap in the modeling of fire suppression systems. Previous research has shown that a modeling framework consisting of an instability model coupled with a stochastic transport model can paint most of the sprinkler spray picture, but requires input in the form of the thickness and velocity of unstable fluid sheets. The model outlined forgoes traditional CFD to solve for water jet-deflector interactions, and instead describes the sheet formation as a potential flow boundary value problem, utilizing a free surface formulation and the superposition of the Green's function. The resulting model allows for the determination of the complete flow field over a fire sprinkler head of arbitrary geometry and input conditions. A hypothetical axisymmetric sprinkler is explored to provide insight into the impact of sprinkler head geometry on local fluid as well as complete spray behavior. The resulting flow splits, sheet thicknesses, and sheet velocities are presented for various sprinkler head geometries.