An interesting variety of pipe cross-sectional shapes can be generated, for which the Navier-Stokes equations can be solved exactly. The simplest cases include the known solutions for elliptical and equilateral triangle cross-sections. Students can find pipe cross-sections from solutions of Laplace's equation in two dimensions, and then plot the velocity distribution in the pipe. The total flow for a given pressure gradient and pipe area can be readily compared for different pipe shapes. (Contains 5 figures.)