The vortical deviation on a stream surface

AbstractLet S be a stream surface in a flow. Let ω3 be the vertical component of the vorticity on S. In the present work we make an extension of the concept of the vorticity on S; we define the geodesic vorticity Ω and the vortical deviation D and we present some of its properties. The geodesic vorticity Ω will be Ω=∂u2∂s1−∂u1∂s2 where u⃗=(u1,u2,0) is the velocity field on S, and s1, s2 are, respectively, the arc length parameters of the lines of the maximum and minimum normal curvature on the surface S. The vortical deviation D will be the difference between the geodesic vorticity Ω and the vorticity ω3, that is D=Ω−ω3. The main results of this work are the relation between D and the curvatures on S (total curvature, geodesic curvature)