Isothermal Couette–Poiseuille Flow of a Viscous Incompressible Liquid with Low Vertical Velocity

The paper considers an inhomogeneous solution to the Navier-Stokes equations, which describe a layered, large-scale, isothermal, vortical, Couette-Poiseuille flow of a viscous, incompressible fluid. Tangential stresses are imposed at the free surface of the liquid, simulating the effects of wind. On the solid surface, the sliding conditions of Navier are applied. The paper investigates the influence of small deviations from the compatibility conditions of the redefined set of equations describing this flow. If the compatibility conditions are not satisfied, the vertical component of velocity can be non-zero. The solution to the system of equations is constructed as a series in a small parameter that represents the deviation from the compatibility condition, with coefficients that depend on the vertical coordinate. The coefficients are obtained as polynomials of the vertical coordinate z.