An important problem in rheology is the prediction of entry losses when a viscous fluid flows through a contraction at the junction of two tubes of different diameters, and of the exit losses when the fluid leaves the tube and enters a jet. The problem is important in rheometry as well as in technological applications, where contractions are present in most forming devices [7], [17].
Figure below shows the geometry of the problem and the boundary conditions. We consider an axisymmetric 4:1 contraction, where 4 and 1 are the ratio between the radius of the upstream and downstream tubes, respectively. The boundary ABCD is a fixed and impenetrable wall on which the velocity components of the fluid are assumed zero, while EF is an axis of symmetry. The velocity profile adopted as upstream boundary condition may be found in Bose and Carey [5].