On the determination of yield surfaces in Herschel–Bulkley fluids

Herschel–Bulkley fluids are materials that behave as rigid solids when the local stress τ is lower than a finite yield stress τ0, and flow as nonlinearly viscous fluids for τ>τ0. The flow domain then is characterized by two distinct areas, τ<τ0 and τ>τ0. The surface τ=τ0 is known as the yield surface. In this paper, by using analytic solutions for antiplane shear flow in a wedge between two rigid walls, we discuss the ability of regularized Herschel–Bulkley models such as the Papanastasiou, the bi-viscosity and the Bercovier and Engelman models in determining the topography of the yield surface. Results are shown for different flow parameters and compared to the exact solutions. It is concluded that regularized models with a proper choice of the regularizing parameters can be used to both predict the bulk flow and describe the unyielded zones. The Papanastasiou model predicts well the yield surface, while both the Papanastasiou and the bi-viscosity models predict well the stress field away from τ=τ0. The Bercovier and Engelman model is equivalent to the Papanastasiou model provided a proper choice of the regularization parameter δ is made. It is also demonstrated that in some cases the yield surface can be effectively recovered using an extrapolation procedure based upon an analytical representation of the solution.