Generalized Reynolds equations for line contact with double-Newtonian shear-thinning

A proper form of the Reynolds equation is lacking for the shear-thinning of polymer blended oil that displays a second Newtonian plateau. These multigrade lubricants respond linearly to shear stress at low stress, then follow a power-law at intermediate stress followed by linear response again at higher stress. The Rabinowitsch and Ferry models, with a second Newtonian viscosity incorporated in the usual way, can be an accurate description of this behavior and are convenient forms for integration into the Reynolds equation. The following one-dimensional Reynolds-Rabinowitsch equation has been derived for the Rabinowitsch double Newtonian model. d/dx {p'h(3)/12 mu(2) - G(2)-H(2)/2 mu(2)p(')2 [tau(m)ln (H(2)+tau(2)(2)/H(2)+tau(2)(1)) - 2hp' + 2H(tan(-1) tau(2)/H - tan(-1) tau(1)/H]} = (u) over bar dh/dx. Reynolds-Ferry double Newtonian equation has also been derived. Simple Grubin style solutions to the EHL line contact problem show that the new generalized Reynolds equations are accurate forms for double-Newtonian shear-thinning fluids.