ABSTRACT
The chapter is dedicated to the asymptotic analysis of steady heavily loaded isothermal lubricated line contacts under the no-slip condition and two opposite limiting cases of pure rolling and relatively high slide-to-roll ratio. The problem is considered for two general classes of non-Newtonian lubricant rheologies when the shear strain and stress can be expressed as certain explicit functions of shear stress and strain, respectively. Some approximations of the generalized Reynolds equation for non-Newtonian fluids that resemble the Reynolds equation for a Newtonian fluid are obtained. The main idea of the method in the isothermal case is analytical solution of the problem for the sliding shear stress and the consequent reduction the problem to asymptotic and/or numerical solution just for the pressure and gap functions. The procedure for deriving formulas for the isothermal lubrication film thickness is based on the methods outlined in Sections 6.3, 6.6, and Chapter 7 and it is presented for the cases when the influence of the lubrication shear stresses on surface normal and tangential displacements can be neglected. A number of examples illustrating application of the described technique is given. Some general issues as to what is the range of parameters when the proposed approximation provides asymptotically correct solutions and how to define appropriately the pressure viscosity coefficient are sorted out in the case of an isothermal EHL problem for a non-Newtonian lubricant. It is shown that in certain cases the asymptotic procedure described below provides asymptotically correct solutions only in regimes of starved lubrication while in other cases it is valid for both starved and fully flooded lubrication regimes. Let us consider a line lubricated contact for cylindrical solids of radii R1 and
R2 made of elastic materials with Young’s moduli E1 and E2 and Poisson’s ratios ν1 and ν2, respectively. Far from the contact the cylinders’ surfaces are steadily moving with linear velocities u1 and u2 and pressed against each other with a normal force P . The lubricant separating the solids is an incompressible non-Newtonian fluid. Under the classical assumptions [1] of slow motion and
for Line and Point
narrow gap between the surfaces the rheology of the lubricant can be described by the equations
μ∂u ∂z = F (τ) or τ = Φ(μ
∂u ∂z ), (9.1)
where u is the lubricant velocity along the x-axis, u = u(x, z), τ is the shear stress, τ = τ(x, z), and μ is the lubricant viscosity that depends on lubricant pressure p and temperature T , μ = μ(p, T ), F and Φ are functions describing the lubricant rheology. The coordinate system is introduced in such a manner that the x-axis is directed along the contact in the direction of motion, the yaxis is directed along cylinders’ axes, and the z-axis is directed along the line connecting the cylinders’ centers (see Fig. 5.1). In equations (9.1) functions F and Φ are given odd smooth functions, which are inverses of each other and F (0) = Φ(0) = 0.
