A semi-analytical solution for the lubrication force between two spheres approaching in viscoelastic fluids described by the Oldroyd-B model under small Deborah numbers

Abstract: Viscoelastic fluids play a critical role in various engineering and biological applications, where their lubrication properties are strongly influenced by relaxation times ranging from microseconds to minutes. Although the lubrication mechanism for Newtonian fluids is well-established, its extension to viscoelastic materials - particularly under squeezing flow conditions - remains less explored. This study presents a semi-analytical solution for the lubrication force between two spheres approaching in a Boger fluid under small Deborah numbers. Unlike previous works that assumed a Newtonian velocity field, we derive the velocity profile directly from the mass-momentum conservation and Oldroyd-B constitutive equations using lubrication theory and order-of-magnitude analysis techniques. Under steady-state conditions, viscoelasticity induces a marginal increase in the surface-to-surface normal force as a result of the increased pressure required to overcome the original resistance from the first normal-stress difference. Transient analyses reveal that the normal lubrication force is bounded by two Newtonian plateaus and is nonsymmetric as the spheres approach or separate. Our findings highlight the role of viscoelasticity in improving load capacity and provide new insight for modelling dense particle suspensions in Boger fluids, where short-range interactions dominate.