Analytical solutions for coupled orientation–stress Poiseuille flow of short fiber suspensions with wall slip

Abstract We derive analytical solutions for the fully coupled, incompressible, steady, fully developed Poiseuille flow of a suspension of non-Brownian, short, rigid fibers in the presence of wall slip. The flow geometry considered is either a two-dimensional planar channel (slit) or an axisymmetric circular pipe (nozzle). The rheology of the matrix is modeled as a power-law generalized Newtonian fluid. The two-way coupling between the velocity field and the fiber-induced extra stress is incorporated explicitly into the momentum balance. Fiber orientation is described using a second-order orientation tensor formulation that accounts for finite-aspect-ratio fibers and fiber–fiber interactions, together with a hybrid closure approximation for the fourth-order orientation tensor. Closed-form analytical expressions for the velocity and pressure fields are derived for both geometries, while the orientation tensor associated with the hybrid closure is obtained numerically. The analysis shows that wall slip is necessary for the two-way coupling to produce non-trivial modifications to the pressure drop and velocity profiles. Under these conditions, the fiber-induced extra stress leads to a significant increase in the pressure drop required to sustain a prescribed volumetric flow rate, while the velocity profile is only weakly affected. The resulting solutions are also used to investigate parametrically the effects of the material parameters on the pressure and velocity profiles.