# Solving lubrication problems at the nanometer scale

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## Abstract

Lubrication problems at lengthscales for which the traditional Navier–Stokes description fails can be solved using a modified Reynolds lubrication equation that is based on the following two observations: first, classical Reynolds equation failure at small lengthscales is a result of the failure of the Poiseuille flowrate closure (the Reynolds equation is derived from a statement of mass conservation, which is valid at all scales); second, averaging across the film thickness eliminates the need for a constitutive relation providing spatial resolution of flow profiles in this direction. In other words, the constitutive information required to extend the classical Reynolds lubrication equation to small lengthscales is limited to knowledge of the flowrate as a function of the gap height, which is significantly less complex than a general constitutive relation, and can be obtained by experiments and/or offline molecular simulations of pressure-driven flow under fully developed conditions. The proposed methodology, which is an extension of the generalized lubrication equation of Fukui and Kaneko to dense fluids, is demonstrated and validated via comparison to molecular dynamics simulations of a model lubrication problem.

## Notes

### Acknowledgements

The authors would like to thank Mathew Swisher and Gerald J. Wang for help with the computations and many helpful comments and suggestions during the course of this research. This work was sponsored by the Consortium on Lubrication in Internal Combustion Engines with additional support from Argonne National Laboratory and the US Department of Energy. The current consortium members are Daimler, Mahle, MTU Friedrichshafen, PSA Peugeot Citroën, Renault, Shell, Toyota, Volkswagen, Volvo Cars, Volvo Trucks and Weichai Power. The use of the Center for Nanoscale Materials, an Office of Science user facility, was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

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