Injection of a viscoplastic material inside a tube or between two parallel disks: Conditions for wall detachment of the advancing front

Abstract

The injection of a viscoplastic material, driven by a constant pressure drop, inside a pipe or between two parallel coaxial disks under creeping flow conditions is examined. The transient nature of both flow arrangements requires solving a time-dependent problem and fully accounting for the advancing liquid/air interface. Material viscoplasticity is described by the Papanastasiou constitutive equation. A quasi-elliptic grid generation scheme is employed for the construction of the mesh, combined with local mesh refinement near the material front and, periodically, full mesh reconstruction. All equations are solved using the mixed finite element/Galerkin formulation coupled with the implicit Euler method. For a viscoplastic fluid, the flow field changes qualitatively from that of a Newtonian fluid because the material gets detached from the walls. For small Bingham numbers, the contact line moves in the flow direction, so that initially the flow resembles that of a Newtonian fluid, but even in that case detachment eventually occurs. The distance covered by the contact line, before detachment takes place, decreases as the Bingham number increases. For large enough Bingham numbers, the fluid may even detach from the wall without advancing appreciably. In pipe flow, when detachment occurs, unyielded material arises at the front and the flow changes into one under constant flow rate with pressure distribution that does not vary with time. In the flow between disks, it remains decelerating and the material keeps rearranging at its front because of the increased cross section through which it advances. The wall detachment we predict has been observed experimentally by Bates and Bridgwater [Chem. Eng. Sci. 55, 3003–3012 (2000)] in radial flow of pastes between two disks.