A linear stability analysis of flows of transversely isotropic non-Newtonian fluids

Abstract

A linear stability analysis of flows of molten, unidirectionally reinforced composites is presented. The molten composites are modelled using the ideal linear fibre-reinforced fluid model in which the fluid is assumed to be homogeneous and locally transversely isotropic. Two types of plane, steady-state, shearing flow are discussed, one in which the fibres lie along the flow direction (longitudinal flows) and the other in which the fibres lie in the shear planes but perpendicular to the flow direction (transverse flows). Each flow type is analysed using a normal mode linear stability analysis, with perturbations along and transverse to the fibre direction being discussed separately. For both types of flow the transverse mode is always stable. The instability mode along the unperturbed fibre direction causes fibre wrinkling and the stability conditions associated with it are different in longitudinal and transverse flows. In the latter a normal stress difference equivalent to the fibre tension is the stability-determining factor, while in longitudinal Couette flow the difference in shear viscosities is found to be a source of instability.