Newtonian flow with nonlinear Navier boundary condition Article First Online: 09 May 2007 Received: 14 March 2006 Revised: 08 January 2007 DOI:
Cite this article as: Matthews, M.T. & Hill, J.M. Acta Mechanica (2007) 191: 195. doi:10.1007/s00707-007-0454-8 Summary
The generalized nonlinear Navier boundary condition advocated by Thompson and Troian in the journal
Nature, and motivated from molecular dynamical simulations, is applied to the conventional continuum mechanical description of fluid flow for three simple pressure-driven flows through a pipe, a channel and an annulus, with a view to examining possible non-uniqueness arising from the nonlinear nature of the boundary condition. For the pipe and the channel it is shown that the results with the nonlinear Navier boundary condition may be obtained from a pseudo linear Navier boundary condition but with a modified slip length. For the annulus, two sets of physically acceptable solutions are obtained corresponding to the chosen sign of the normal derivative of the velocity at each solid boundary. Closer examination reveals that although the generalized Navier boundary condition is highly nonlinear, in terms of the assumed form of solution the integration constants obtained are still unique for the three simple pressure-driven flows presented here, provided that care is taken in its application and noting that the multiplicity of solutions obtained for the annulus arise as a consequence of adopting different signs for the normal derivatives of velocity at the boundaries. References Gad-el-Hak, M. 1999 The fluid mechanics of microdevices – the Freeman scholar lecture J. Fluids Engng 121 5 33 Google Scholar Granick, S. 1991 Motions and relaxations of confined liquids Science 253 1374 1379 CrossRef Google Scholar Granick, S. 1999 Soft matter in a tight spot Phys. Today 52 26 31 Google Scholar Bhushan, B., Israelachvili, J. N., Landman, U. 1995 A nanotribology: friction, wear and lubrication at the atomic scale Nature 374 607 616 CrossRef Google Scholar
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