The theory of potential flow is a topic in both the study of fluid mechanics and in mathematics. The mathematical theory treats properties of vector fields generated by gradients of a potential. The curl of a gradient vanishes. The local rotation of a vector field is proportional to its curl so that potential flows do not rotate as they deform. Potential flows are irrotational.

The mathematical theory of potentials goes back to the 18th century (see Kellogg, 1929). This elegant theory has given rise to jewels of mathematical analysis, such as the theory of a complex variable. It is a well-formed or “mature” theory, meaning that the best research results have already been obtained. We are not going to add to the mathematical theory; our contributions are to the fluid mechanics theory, focusing on effects of viscosity and viscoelasticity. Two centuries of research have focused exclusively on the motions of inviscid fluids. Among the 131,000,000 hits that come up under “potential flows” on Google search are mathematical studies of potential functions and studies of inviscid fluids. These studies can be extended to viscous fluids at small cost and great profit.

The fluid mechanics theory of potential flow goes back to Euler in 1761 (see Truesdell, 1954, §36). The concept of viscosity was not known in Euler's time. The fluids he studied were driven by pressures, not by viscous stresses.