In this paper we consider the flow of a Bingham-like fluid driven by a known pressure gradient in a channel of finite width and length. We model the continuum as a linear viscous fluid when the stress is above a certain threshold and as a linear elastic solid when the stress is below such a threshold. We consider a channel of varying width. Moreover, the ratio, denoted by e, between the channel width and length is small. This allows for a two-scale approach. Indeed, we seek a solution performing an asymptotic expansion in powers of E of the main variables. Under specific assumptions on the model characteristic parameters, we prove some analytical results for the zero order approximation. In particular, we show that, when the deformations of the inner elastic core are non-negligible, its width may vary along the longitudinal coordinate. On the other hand, when the core deformations are negligible, the Bingham model is retrieved along with the well known "lubrication paradox". We also show some numerical simulations. (C) 2012 Elsevier B.V. All rights reserved.