The singularity of the UCM/Oldroyd-B models at a finite Weissenberg number, for the steady sphere translation with Navier slip on the sphere

Abstract

• The steady sphere translation with Navier slip on the sphere is studied. • A singular Weissenberg number is revealed for the UCM/Oldroyd-B models. • The singularity depends on the viscosity ratio and the slip coefficient. • Consistent and convergent approximations of the singularity are derived. For the steady sphere translation in an isothermal viscoelastic fluid which is modelled with the Upper Convected Maxwell and Oldroyd-B models, evidence of a singularity at a finite Weissenberg number is provided, when Navier type linear slip is applied at the surface of the sphere. The singular Weissenberg number depends on the dimensionless slip coefficient which appears in the slip-law, as well as on the polymer viscosity ratio. The analysis is performed following Housiadas et al. (2021) for similar flows of spherical micro-swimmers with prescribed slip velocity at the surface of the swimmer. Although here the exact singularity could not be found, the perturbation solution by Gkormpatsis et al. (2020) is utilized to evaluate consistent approximations of it is magnitude.