On the theory of periodic layers in incompressible fluid

A conceptual error in the formally closed but physically incomplete Kim–Moin–Moser form of equations for viscous incompressible fluid in a horizontal periodic layer is corrected. This form, which has lately become popular, assumes that the vertical projections of the rotor and the second rotor of the field of accelerations vanish. This assumption considerably simplifies the calculations; however, it is insufficient for the equations of motion. In this paper, the fulfillment of these assumptions is ensured by the additional condition that the vector of the horizontal projection averaged over the period of the acceleration vorticity vanishes, which opens new possibilities. The resulting complete form of the equations with the rotors of three orders admits a reduction to two scalar equations (of the fourth and the sixth orders), which, however, are not less complicated than the equivalent Navier–Stokes equations.