Curl equation in viscous hydrodynamics in a channel of complex geometry

We consider the Navier—Stokes equations for the plane steady motion of a viscous incompressible fluid in an orthogonal coordinate system in which the fluid streamlines coincide with the coordinate lines of one of the families of the orthogonal coordinate system. In this coordinate system, the velocity vector has only the tangential component and the system of three Navier—Stokes equations is an overdetermined system for two functions — the tangential component of velocity and pressure. In the present paper, the system is brought to involution, and the consistency conditions are obtained, which are the equations for the curl of the velocity in this coordinate system. The coefficients of these equations include the curvatures of the coordinate lines and their derivatives up to the second order. The equations obtained are significantly more complicated than the curl equations in a channel of simple geometry.