The effects of small-amplitude, two-dimensional grooves on pressure losses in a laminar channel flow have been analysed. Grooves with an arbitrary shape and an arbitrary orientation with respect to the flow direction have been considered. It has been demonstrated that losses can be expressed as a superposition of two parts, one associated with change in the mean positions of the walls and one induced by flow modulations associated with the geometry of the grooves. The former effect can be determined analytically, while the latter has to be determined numerically and can be captured with an acceptable accuracy using reduced-order geometry models. Projection of the wall shape onto a Fourier space has been used to generate such a model. It has been found that in most cases replacement of the actual wall geometry with the leading mode of the relevant Fourier expansion permits determination of pressure losses with an error of less than 10 %. Detailed results are given for sinusoidal grooves for the range of parameters of practical interest. These results describe the performance of arbitrary grooves with the accuracy set by the properties of the reduced-order geometry model and are exact for sinusoidal grooves. The results show a strong dependence of the pressure losses on the groove orientation. Longitudinal grooves produce the smallest drag, and oblique grooves with an inclination angle of
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