Planar incompressible Navier-Stokes, Euler, and Laplace equations: The angle-function formulation

Abstract

A novel method has recently been proposed for studying planar, incompressible flows from a geometric point of view. The governing equations of motion can accordingly take a special form involving the local curvatures of the streamlines and their orthogonal trajectories, leading to what has been named the “velocity-curvature” formulation of the Navier-Stokes and Euler equations [I. Dimitriou, “Planar incompressible Navier-Stokes and Euler equations: A geometric approach,” Phys. Fluids 29, 117101 (2017)]. The present study extends and, in a way, generalizes the findings of this previous work, further developing on the idea of flow characterization by means of its geometry-physics interplay. In particular, an alternative expression for the momentum and mass-conservation equations, the so-called angle-function formulation, is developed, which can be regarded as the evolution of the aforementioned velocity-curvature description. It is shown that the flow field can be equally described by a pair of scalar equations, in which the concerned variables are the velocity magnitude v and the angle function φ. The latter represents the angle of incidence of the streamlines, and as revealed, it is the geometric analogon of the well-known stream function Ψ. In this geometry-based framework, the Laplace equation characterizing ideal flows has been revised by exchanging these scalar functions Ψ and φ. Finally, the advantages of the developed description of the Navier-Stokes equation and the prospect of its possible evolution towards a pure geometric tool for planar flow analysis are highlighted in this study.