Introducing a Geometric Potential Theory for two-dimensional steady flows

Abstract

It is demonstrated that the continuity and irrotationality conditions imply geometric constraints for two-dimensional steady flows. This fact leads to the formulation of an exclusively geometric criterion for these kinematical conditions. Thus, flow visualization can be conclusive on whether a flow possesses a potential or not. Moreover, the mutual relationship among the flow geometry, its kinematics and physics, can be expressed mathematically using an eikonal equation. Its analytical solution in the flow-streamline coordinate system enables the formulation of the `Geometric Potential Theory’. Accordingly, the determination of the physical quantities of velocity and static pressure throughout the flow is reduced to the purely geometrical problem of finding the streamline and potential line curvatures. These two functions are combined to introduce the `Global Curvature Vector’, a vector that can be mathematically and physically interpreted. Finally, it is shown that continuity and irrotationality are identically satisfied by the existence of a `Curvature Potential’, that is, the existence of an analytic expression from which the global curvature vector components can be found by partial differentiation.