We study the process of deposition and clogging when a steady flow of a Brownian suspension is introduced into a narrow channel initially containing the liquid only. This exploratory work neglects changes in channel semi-width and hydraulics, and so applies strictly only to small particle concentrations (but also to irreversible adsorption of a solute). A detailed analysis is made of the steady distribution of particle concentration and of particle deposition, which is established relatively rapidly. The mathematics is much simplified by approximating the Poiseuille velocity profile by a power law. A similarity solution is then available for small x (dimensionless downstream distance) with a dominant-mode approximation valid for large x. Particles do not penetrate far down the channel, with the centroids of both suspended and deposited particles being about 0.3 PeZ* (Pe, Peclet number; Z*, channel semi-width) from the channel entrance. A comparison with results for no-shear plug flow reveals the relatively minor effects of shear (so that the approximating velocity profile produces only trivial error). Shear tends to inhibit slightly particle deposition, with mean particle transport 15% less for no-shear flow. Deposition with perfect particle sticking at the wall is in total contrast with Taylor dispersion, governed by the same convection-diffusion equation but with perfect particle reflection. For Taylor dispersion the particle (solute) centroid travels with the liquid centroid : but for deposition its travel distance approaches a relatively small finite value. Shear is absolutely essential to Taylor dispersion, but is almost irrevalent to deposition. A major conclusion is that processes of deposition (or irreversible adsorption) with a reflection coefficient less than 1 cannot be represented adequately as Taylor dispersion with a superposed macroscopic sink term. Finally, the work suggests that the distance of travel of (sticking) particles introduced into a porous medium (at either an external or an internal surface) is O(VL*(2)/D) (V, Darcy flow velocity; L*, characteristic pore length; D, Brownian diffusivity).