The vortical deviation on a stream surface

Let SS be a stream surface in a flow. Let ω3ω3 be the vertical component of the vorticity on SS. In the present work we make an extension of the concept of the vorticity on SS; we define the geodesic vorticity  ΩΩ and the vortical deviation  DD and we present some of its properties. The geodesic vorticity ΩΩ will be Ω=∂u2∂s1−∂u1∂s2 where u⃗=(u1,u2,0) is the velocity field on SS, and s1s1, s2s2 are, respectively, the arc length parameters of the lines of the maximum and minimum normal curvature on the surface SS. The vortical deviation DD will be the difference between the geodesic vorticity ΩΩ and the vorticity ω3ω3, that is D=Ω−ω3D=Ω−ω3. The main results of this work are the relation between DD and the curvatures on SS (total curvature, geodesic curvature).