The Couette flow of a Jeffrey fluid through an inclined composite duct partly filled with saturated porous layer and partly filled with fluid is studied. It is assumed that thickness of free flow region and thickness of porous layer are different. A fully developed steady flow is assumed to be produced by applying a constant pressure gradient. The flow equations in free flow region are obtained from Navier-Stokes equations for an incompressible fluid. The flow equations in porous region are obtained by using theory of mixtures. The boundary conditions at the solid and fluid interface are formulated and the equations for fluid velocity within the porous region and free flow region are solved analytically. These flow quantities are studied for the effect of flow parameters such as porous layer thickness, volume fraction of fluid, viscosity parameter, drag parameter, angle of inclination. The results are compared with standard literature and show good agreement. The study reveals that under prescribed conditions fluid flux in the porous layer is more than that of in the open channel. © 2019 Elsevier Ltd.