Fluid flow through a partially filled cylinder

The laminar flow of viscous incompressible fluid through a partially filled circular cylinder is studied. The governing equations reduce to Laplace's equation, and the region of solution is the union of two intersecting circles. This region is mapped onto the unit circle by a conformal transformation, so that the problem may be solved by the Poisson integral formula. The integration is performed by a combination of residues and numerical integration. The results agree very well with exact analytic solutions at 50%, 85.4%, and 100% depth to diameter ratios. It is found that by not filling the cylinder completely, the flow rate can be increased by a significant 27.3%.