The Lipschitz Picard successive step method in hydrodynamic lubrication problem

The paper shows the successive steps of approximation of Picard unification for the solution of the non-isothermal fluid flow in thin layer including inertia forces and apparent viscosities described by the non-linear dependences. In this paper is presented a unified semi analytical method of solution of the asymmetrical, laminar, steady and unsteady, non-Newtonian lubrication problem flow between two non-rotational in general, convex, differentiable and movable surfaces when the time t depended gap between mentioned surfaces has quite an arbitrary geometry. The presented considerations relate not only to the rotational cooperating surfaces but also to the arbitrary non-rotational surfaces in general. The parallel and longitudinal intersections of mentioned surfaces are curvilinear and non-monotone in general. We consider the non-Newtonian lubricant for non-linear constitutive equations taking into account Reiner Rivlin power law relationship as well Rivlin-Ericksen formula for viscoelastic fluids. The non-Newtonian properties create non-linear dependencies between strain and stress. Moreover, the dynamic viscosity or apparent dynamic viscosity of numerous lubricant liquids with various additions often decreases along with shear rate increasing during motion. Dynamic viscosity of lubricant fluids inside very thin micro and nano boundary layers depends on Young’s modulus of the cell of surface body being in contact with the fluid.