Modeling lubricants enhanced by finite elasticity polymers

Abstract

Lubrication is essential for the longevity of mechanical and biological surfaces in relative motion and susceptible to friction and wear. A well designed lubricant, for example a base oil enhanced with polymer additives, can effectively reduce both energetic and material losses. However, difficulty arises when modeling these lubricant mixtures exhibiting complex rheological behavior, in particular, a dependence of the viscosity on pressure (piezoviscosity), temperature (thermal thinning), shear rate (shear thinning) and the onset of viscoelasticity. Accurate estimates of the load carrying capacity of the thin lubricating film requires careful modeling of shear thinning. Available models such as the generalized Reynolds equation (GR) and the approximate shear distribution (ASD) have drawbacks such as large computational time and poor accuracy, respectively. In this work, we present a new approach, i.e. the modified viscosity (MV) model. We investigate, for both MV and GR, the load, the maximum pressure and the computational time, for (i) sliding (non-cavitating) contacts, (ii) cavitating and (iii) squeezing contacts. We observe that the computational time is reduced (i) considerably for non-cavitating sliding and rolling contacts and (ii) by several order of magnitudes for cavitating and squeezing contacts. For strongly elastic lubricants, 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. In this work, we have extended the VR approach to the non-linear 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 W i, i.e. the ratio between the polymer relaxation time and the flow time scale, and finite extensibility parameter L, using FENE-CR and FENE-P. 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 W i, L and e: while it increases for small values of W i, limited greatly by the capacity of the polymer to stretch, a saturation and a subsequent decline is observed for highly viscoelastic regimes. Additionally, a weakly (strongly) eccentric configuration plays an important role in promoting (hindering) the growth in load versus both W i and L. These effects are significant and have to be considered when modeling thin contacts lubricated with a strongly viscoelastic fluid. Additionally, we have extended the VR approach towards three-dimensional lubricated contacts (in cartesian and cylindrical coordinate systems) for several non-linear constitutive relations and have provided a linearized model in De. Owing to the increase in computational requirements, a globally fully-implicit numerical technique was adopted for the efficient solution of the equations. The load and friction response for an extruded journal bearing e = 0.9 (and parabolic slider) showed a strong variation versus the channel aspect ratio (otherwise zero for a Newtonian lubricant), i.e. a = ℓx/ℓz, the ratio between the channel streamwise and spanwise lengths. The effects of transient surface motion on the response of an elastic polymer have also been examined, with a specific focus on the load carrying capacity and the friction, via a second-order perturbation model and the VR approach. We find, the perturbed models only offer a matching prediction (i) once the motion has proceeded from some time and, (ii) the De is small. A simplified look into the influence of polymer elasticity on the temperature distribution of the film showed a weak dependency versus De. The film heating owing to the fluid dissipation remained largely unaffected unless the De was large.