Newtonian flow with nonlinear Navier boundary condition

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.