Flow regimes for fiber suspensions in narrow gaps

The flow of fiber suspensions in narrow gaps is examined using rheological theories for suspensions of rigid particles in a Newtonian solvent. A dimensionless particle number N_p describes the intrinsic anisotropy of the suspension. Order-of-magnitude estimates are developed for the velocity, stress and fiber-orientation components in a narrow gap of slowly varying thickness. The estimates reveal four distinct flow regimes. At one extreme the flow is decoupled from fiber orientation and lubrication theory applies. The other extreme exhibits a flat velocity profile with a shear boundary layer of thickness L(N_p^{- 1 2}. The key dimensionless group separating the regimes is N_pδ². Here δ describes the out-of-plane fiber orientation; it equals the greater of ε{lunate} or C_I ^{1 3}, where ε{lunate} is the slenderness of the gap and C_I is an interaction coefficient for fiber orientation. The existence of the first regime supports the practice of decoupling the calculation of flow and fiber orientation, provided that N_pδ² is much smaller than one. The fourth regime generalizes Barone and Caulk's model for compression molding flows of chopped-fiber molding compounds. Limitations due to entrance effects and start-up transients are discussed, along with applications to the modeling of injection and compression molding of fiber-reinforced polymers.