The vortical deviation on a stream surface

Let S be a stream surface in a flow. Let ω 3 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 = ∂ u 2 ∂ s 1 − ∂ u 1 ∂ s 2 where u ⃗ = ( u 1 , u 2 , 0 ) is the velocity field on S , and s 1 , s 2 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 ω 3 , that is D = Ω − ω 3 . The main results of this work are the relation between D and the curvatures on S (total curvature, geodesic curvature).