We consider incompressible non-Newtonian fluids reinforced by a single family of inextensible fibres that convect with the fluid. We recall some kinematic results for materials of this type and formulate constitutive equations of the Rivlin-Ericksen type.
In the theory of isotropic non-Newtonian fluids the so-called viscometric flows are of interest because they can be characterized by a small number of response functions. One family of flows of this class comprises helical flow, in which the streamlines are a family of helices about the axis of a system of cylindrical polar coordinates. We show that helical flow is also possible in a fibre-reinforced non-Newtonian fluid, with the fibre trajectories also forming helices. The stress is determined in terms of five response functions, two of which characterize the longitudinal and transverse shear viscosities. It is found that there are two solutions: in the first the streamlines and fibres are helices that wind in opposite senses, and in the second they wind in the same sense. The preferred solution is such that the fibre orientation adjusts so that the flow encounters the least viscous resistance.
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