A new lubrication theory to derive far-field axial pressure difference due to force singularities in cylindrical or annular vessels

Abstract

In simulations of creeping flow, the effect of suspended particles on a fluid is often represented by force singularities responsible for singular Stokesian solutions, which are infinite at the sphere center and decay far from the particle. In this article, we consider such singular fields centered at a point inside a cylindrical or annular conduit containing highly viscous medium. Across an unbounded infinite domain, these singular flows cannot produce a finite pressure difference as they decay to zero far from the center. However, in the presence of bounding cylindrical surfaces, the reflected flow from the walls creates a finite pressure difference between the far fields across the force singularity along the axial direction. To quantify the effect of the reflected flow, we present a new lubrication analysis, which, on the one hand, identifies the specific singular fields capable of producing axial pressure difference and, on the other hand, provides explicit expressions for the far-field pressure. Though we specifically focus on cylindrical or annular geometry, the outlined approach can also be extended to other confinements. Thus, the general formulation can be used in larger context to quantify the effect of small particles on wall-bounded fluid medium.