Couette flow in channels with wavy walls

Summary Three–dimensional Couette flows enclosed by a plane and by a wavy wall are addressed; the wave amplitude is proportional to the mean clearance of the channel multiplied by a small dimensionless parameter ɛ. A perturbation expansion in terms of the powers of ɛ of the full steady Navier–Stokes equations yields a cascade of boundary value problems which are solved at each step in closed form. The supremum value of ɛ for which the expansion converges, is determined as a function of the Reynolds number The analytical-numerical algorithm is applied to compute the velocity in the channel to O (ɛ 4 ). Even in the first order approximation O (ɛ), new results are obtained which complement the triple deck theory and its modifications. In particular, the incipient separation–detachment is discussed using the Prandtl-Schlichting criterion of starting eddies. The value ɛ e for which eddies start in the channel, is analytically deduced as a function of as well as analytical formulas for the coordinates of the separation points. These analytical formulas show that ɛ e in 3D channels is always less than ɛ e in 2D channels. For non-smooth channels, a criterion of infinitesimally small ɛ e is deduced. The critical value of ɛ up to which bifurcation of the solutions can occur is estimated.