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Viscosity index improver agents, added to lubricants to improve their viscosity-temperature behavior, as a side effect cause the lubricant to exhibit both shear thinning and moderate elastic effects in simple shear. We here model shear thinning with the power law fluid. First, a theory for the power law fluid is developed for a thin film between rigid walls of arbitrary shape in arbitrary motion under a pressure gradient. This theory forms part of an analysis developed to study the dynamics of piston ring lubrication. The parameters in the model are chosen to match data for two shear thinning lubricants and other data that represents typical piston ring behavior. The temperature dependence of the consistency and power law indices is taken into account. A multiple iterative numerical proceedure is developed to solve the equations and the minimum oil film thickness, piston load and friction force as functions of crank angle are determined.