The main thrust of this proposal is to study the flow between rotating eccentric cylinders. It is expected that the results will contribute to a better understanding of the operating conditions of journal bearings working around the transitional point. Current experimental and theoretical knowledge of flow between eccentric rotating cylinders is limited, and a numerical study, as proposed, is probably the right way to go at this point. Especially if the various operating parameters, i.e. Reynolds number, radius ratio, eccentricity, non-isothermal behavior, temperature dependent viscosity, end-wall effects, etc., are to be investigated with an ultimate goal of determining better design criteria. The approach proposed here is mathematically sophisticated, but practical, well-thoughtout, and appropriate for the task at hand. Support is recommended.