Bingham flow development in annular tubes in the presence of wall slip

Abstract

The development of Bingham flow in concentric annular tubes in the presence of wall slip is investigated. It is assumed that slip occurs along both cylinders, following Navier's law, which states that the slip velocity is proportional to the wall shear stress. The open-source finite element software FEniCS is used for the numerical simulations along with the Papanastasiou regularization for the constitutive equation. To correctly determine the entrance region, various definitions of the development length are considered. In addition to the standard definition, which is based on the maximum velocity development, and the global development length, alternative definitions based on the development of the wall shear stresses and of the velocity at the two yield radii are considered. The combined effects of slip, yield stress and inertia on the different development lengths are systematically investigated. The yielded and unyielded zones are also determined using the von Mises criterion. The numerical results show that the standard development length fails to accurately capture the entrance region, even in the case of Newtonian flow with no-slip, and that the inner wall shear stress and yield lengths are also inadequate. The global and the outer wall shear stress and yield development lengths, which can be up to four times bigger than the standard development length, are more reliable. In agreement with previous studies, the development lengths are monotonically increasing with the Reynolds and Bingham numbers. As wall slip becomes stronger these reliable development lengths increase only initially reaching a maximum and then they are abruptly reduced to zero as the slip number approaches the critical value corresponding to sliding (unyielded) motion.