Note: A family of analytical solutions of the Navier-Stokes equations in space

Abstract

The author gives exact analytical solutions of the Navier-Stokes equations for steady and unsteady laminar plane flows of the second kind near a plane z = 0, in the form of power series in z (up to a power of 6) of the three Cartesian velocity components and of the piezometric gradient, whose coefficients depend on arbitrary values of the velocity gradients in the x- and z-directions and of the piezometric function and its y-gradient at the boundary. An important application is that of flow in straight flat-bottomed channels or conduits of constant cross-section with transverse flows.