Unidirectional flows of a Herschel–Bulkley fluid with pressure-dependent rheological moduli

In this paper we study the planar channel flow of a Herschel–Bulkley fluid with pressure-dependent consistency index and yield stress. In particular we consider unidirectional flows in which the velocity is directed along the longitudinal direction of the channel. We show that unidirectional flows are possible only if the dependence on the pressure is linear, since in the nonlinear case secondary flow may appear. We determine constraints that ensure the existence of unyielded regions and constraints that guarantee that the flow does not come to a stop. Analytical solutions for the velocity field and the pressure are determined, and numerical examples that show the dependence on the physical parameters of the model are provided.