Modeling polymeric lubricants with non-linear stress constitutive relations

Lubricating oils are used to minimize the friction and wear of mechanical components by virtue of a thin lubricant film separating the sliding surfaces. The film's characteristics, under high pressure, can exhibit non-Newtonian effects, such as viscoelasticity and shear thinning. The strength of these effects are measured using the Weissenberg (Deborah) number Wi (De), i.e., the ratio of the polymer relaxation time to the shear (residence) time scale. Modeling these effects is computationally challenging, especially when relying on the direct numerical simulation (DNS) of the Cauchy momentum and the mass conservation equations. However, the viscoelastic Reynolds (VR) approach (Ahmed & Biancofiore, Journal of Non-Newtonian Fluid Mechanics, 292, 104524, 2021.) has been shown to be effective in modeling (i) the pressure distribution and (ii) the load carrying capacity of a viscoelastic lubricating film for mechanical contacts for the Oldroyd-B constitutive relation, since these contacts operate within the small De limit (but no constraint in Wi). In this work, we have extended the VR approach to the finitely extensible non-linear elastic (FENE) type constitutive relations that account for the (i) finite extension of the polymer chains and (ii) shear thinning. We have validated the VR approach against DNS, showing an excellent agreement over a wide range of the Weissenberg number Wi, and finite extensibility parameter L, using FENE-CR and FENE-P models. Following a thorough validation, the pressure distribution and the load carrying capacity of a journal bearing, whose channel height is governed by the journal eccentricity ratio e, is considered. It is observed that the load carrying capacity of the film portrays a strongly non-linear dependence on Wi, L and e: while it increases for small values of Wi, limited greatly by the capacity of the polymer to stretch, a saturation and a subsequent decline is observed for high Wi regimes. Additionally, a weakly (strongly) eccentric configuration plays an important role in promoting (hindering) the growth in load versus both Wi and L. These effects are significant and have to be considered when modeling thin contacts lubricated with a strongly viscoelastic fluid.