We study a stationary motion of incompressible viscous fluid with a pressure-dependent viscosity in a thin (or long) straight pipe with variable cross-section. We assume that the flow is governed by the prescribed pressure drop between pipe's ends. Under very general assumption on the viscosity law satisfied by the Barus formula and other empiric laws, the effective behavior of the flow is found via rigorous asymptotic analysis with respect to the pipe's thickness. (C) 2013 Elsevier B.V. All rights reserved.