Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-28T20:26:54.624Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  08 February 2013

School of Engineering, Edith Cowan University, Joondalup, WA 6027, Australia
School of Engineering, Edith Cowan University, Joondalup, WA 6027, Australia
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A theoretical investigation of the unsteady flow of a Newtonian fluid through a channel is presented using an alternative boundary condition to the standard no-slip condition, namely the Navier boundary condition, independently proposed over a hundred years ago by both Navier and Maxwell. This boundary condition contains an extra parameter called the slip length, and the most general case of a constant but different slip length on each channel wall is studied. An analytical solution for the velocity distribution through the channel is obtained via a Fourier series, and is used as a benchmark for numerical simulations performed utilizing a finite element analysis modified with a penalty method to implement the slip boundary condition. Comparison between the analytical and numerical solution shows excellent agreement for all combinations of slip lengths considered.

Research Article
Copyright ©2013 Australian Mathematical Society


Batchelor, G. K., An introduction to fluid dynamics (Cambridge University Press, Cambridge, 1967).Google Scholar
Bhushan, B., Israelachvili, J. N. and Landman, U., “Nanotribology: friction, wear and lubrication at the atomic scale”, Nature 374 (1994) 607616; doi:10.1038/374607a0.CrossRefGoogle Scholar
Carslaw, H. S. and Jaeger, J. C., Conduction of heat in solids, 2nd edn (Oxford University Press, New York, 1986).Google Scholar
“COMSOL AB, COMSOL Multiphysics (Version 3.5a)”, 2008; Scholar
Gad-el-Hak, M., “The fluid mechanics of microdevices—The Freeman scholar lecture”, J. Fluids Engng 121 (1999) 533; doi:10.1115/1.2822013.CrossRefGoogle Scholar
Gad-el-Hak, M., “Transport phenomena in microdevices”, Z. Angew. Math. Mech. 84 (2004) 494498; doi:10.1002/zamm.200310118.CrossRefGoogle Scholar
Granick, S., “Motions and relaxations of confined liquids”, Science 253 (1991) 13741379; doi:10.1126/science.253.5026.1374.CrossRefGoogle ScholarPubMed
Hill, J. M. and Dewynne, J. N., Heat conduction (Blackwell Scientific, London, 1987).Google Scholar
John, V., “Slip with friction and penetration with resistance boundary conditions for the Navier–Stokes equations—numerical tests and aspects of the implementation”, J. Comput. Appl. Math. 147 (2002) 287300; doi:10.1016/S0377-0427(02)00437-5.CrossRefGoogle Scholar
Karniadakis, G., Beskok, A. and Aluru, N., Microflows and nanoflows: fundamentals and simulation (Springer, New York, 2005).Google Scholar
Matthews, M. T. and Hill, J. M., “Newtonian flow with nonlinear Navier boundary condition”, Acta Mechanica 191 (2007) 195217; doi:10.1007/s00707-007-0454-8.CrossRefGoogle Scholar
Matthews, M. T. and Hill, J. M., “Effect of slip on the linear stability of flow through a tube”, Z. Angew. Math. Phys. 59 (2008) 360379; doi:10.1007/s00033-007-7116-5.CrossRefGoogle Scholar
Matthews, M. T. and Stokes, Y. M., “Lubrication analysis and numerical simulation of the viscous micropump with slip”, Int. J. Heat Fluid Flow 33 (2012) 2234; doi:10.1016/j.ijheatfluidflow.2011.09.009.CrossRefGoogle Scholar
Maxwell, J. C., “On stresses in rarified gases arising from inequalities of temperature”, Phil. Trans. R. Soc. Lond. 170 (1879) 231256; doi:10.1098/rstl.1879.0067.Google Scholar
Navier, C. L. M. H., “Mémoire sur les lois du mouvement des fluides”, Mémoires de la Classe des Sciences Mathématiques et Physiques de l’Institut de France 6 (1823) 389440.Google Scholar
Qian, T., Wang, X.-P. and Sheng, P., “A variational approach to moving contact line hydrodynamics”, J. Fluid Mech. 564 (2006) 333360; doi:10.1017/S0022112006001935.CrossRefGoogle Scholar
Stokes, Y. M. and Carey, G. F., “On generalised penalty approaches for slip, free surface and related boundary conditions in viscous flow simulation”, Internat. J. Numer. Methods Heat Fluid Flow 21 (2011) 668702; doi:10.1108/09615531111148455.CrossRefGoogle Scholar
Wang, C. Y., “Exact solutions of the unsteady Navier–Stokes equations”, Appl. Mech. Rev. 42 (1989) S269S282; doi:10.1115/1.3152400.CrossRefGoogle Scholar
Wang, C. Y., “Two-fluid oscillatory flow in a channel”, Theor. Appl. Mech. Lett. 1 (2011) 032007; doi:10.1063/2.1103207.CrossRefGoogle Scholar