The motion of fluid through an indented non-symmetric channel or symmetric pipe is considered when the flow ahead of the indentation is fully-developed and the typical Reynolds number, K, is large. The theoretical description, for steady flows and slowly-varying indentations, is founded on a three-region structure, according to which the main core of fluid suffers a small inviscid displacement of its streamlines while the viscous motion close to the walls is nonlinear and forced along by the induced pressure-gradient. The displacement can be shown to be the average of the wall displacements, but the pressure must be calculated together with the viscous problem. Numerical solutions are presented both for linear constrictions or dilatations and for more confined ones, and flow separation, if it occurs, appears to be regular, with, for the more local indentation, a physically sensible eddy and reattachment ensuing downstream. The theory, which is believed to set out a rational approach to the solution, is valid provided the small inclination α of the indentation lies between O(K–1) and O(K–1) for a non-symmetric distortion of the wall, or between O(K–1) and O(K–1) for a symmetric distortion, in which ranges there is no substantial upstream influence. A companion paper (1) considers these limitations further and extends the theory to unsteady flows.