A Reynolds–Ellis equation for line contact with shear-thinning

Abstract

A form of the Reynolds equation is lacking for the general shear-thinning that evolves from linear response at low stress to power-law at high stress. The Ellis model is a reasonable description of the shear-thinning response at elevated pressure of those liquid lubricants with a strong dependence of viscosity on shear, that is, for a power-law exponent less than ½. A one-dimensional Reynolds–Ellis equation has been derived that can be written d d x { p ′ h 3 12 μ + Γ μ p ′ 2 [ ( τ m − ϕ 1 n + 2 + ϕ ) | τ m − ϕ | 1 n + 1 − ( τ m + ϕ 1 n + 2 − ϕ ) | τ m + ϕ | 1 n + 1 ] } = u ¯ d h d x . This reduces to the one-dimensional form of Greenwood's Reynolds–Rabinowitsch equation for n=1/3. Simple Grubin style solutions to the EHL line contact problem show that the new generalized Reynolds equation is an accurate form for an Ellis fluid. Unfortunately, the Ellis model is not ideal, as the transition to shear-thinning becomes extremely broad for n<1/2. The Ellis model does, however, provide an exact solution for a sliding line contact for a liquid that possesses a zero shear viscosity along with a variable power-law exponent, and it may be unique in this respect. Future work should consider a two-dimensional Reynolds equation for more accurate models of moderately shear-thinning lubricants such as Carreau.