×

zbMATH — the first resource for mathematics

Nano boundary layers over stretching surfaces. (English) Zbl 1221.76024
Summary: We present similarity solutions for the nano boundary layer flows with Navier boundary condition. We consider viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface. The resulting nonlinear ordinary differential equations are solved analytically by the Homotopy Analysis Method. Numerical solutions are obtained by using a boundary value problem solver, and are shown to agree well with the analytical solutions. The effects of the slip parameter \(K\) and the suction parameter \(s\) on the fluid velocity and on the tangential stress are investigated and discussed. As expected, we find that for such fluid flows at nano scales, the shear stress at the wall decreases (in an absolute sense) with an increase in the slip parameter \(K\).

MSC:
76A05 Non-Newtonian fluids
34E13 Multiple scale methods for ordinary differential equations
65L99 Numerical methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Navier, C.L.M.H., Mémoire sur LES lois du mouvement des fluids, Mém acad R sci inst France, 6, 389, (1823)
[2] Shikhmurzaev, Y.D., The moving contact line on a smooth solid surface, Int J multiphase flow, 19, 589, (1993) · Zbl 1144.76452
[3] Choi CH, Westin JA, Breuer KS. To slip or not to slipwater flows in hydrophilic and hydrophobic microchannels. In: Proceedings of IMECE 2002, New Orleans, LA, Paper No. 2002-33707.
[4] Matthews, M.T.; Hill, J.M., Nano boundary layer equation with nonlinear Navier boundary condition, J math anal appl, 333, 381, (2007) · Zbl 1207.76050
[5] Wang, C.Y., Analysis of viscous flow due to a stretching sheet with surface slip and suction, Nonlinear anal real world appl, 10, 375, (2009) · Zbl 1154.76330
[6] Wang, C.Y., Flow due to a stretching boundary with partial slip – an exact solution of the navier – stokes equations, Chem eng sci, 57, 3745, (2002)
[7] Xu, H.; Liao, S.J.; Wu, G.X., A family of new solutions on the wall jet, Eur J mech B/fluid, 27, 322-334, (2008) · Zbl 1154.76335
[8] Liao, S.J., A new branch of boundary layer flows over a permeable stretching plate, Int J non-linear mech, 42, 819-830, (2007) · Zbl 1200.76046
[9] Liao, S.J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transfer, 48, 2529-2539, (2005) · Zbl 1189.76142
[10] Crane, L.J., Flow past a stretching plate, Z angew math phys, 21, 645, (1970)
[11] Liao SJ. On the proposed homotopy analysis techniques for nonlinear problems and its application. Ph.D. dissertation, Shanghai Jiao Tong University; 1992.
[12] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton
[13] Liao, S.J., An explicit, totally analytic approximation of blasius’ viscous flow problems, Int J non-linear mech, 34, 759, (1999) · Zbl 1342.74180
[14] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl math comput, 147, 499, (2004) · Zbl 1086.35005
[15] Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Stud appl math, 119, 297, (2007)
[16] Liao, S.J., Notes on the homotopy analysis method: some definitions and theorems, Commun nonlinear sci numer simul, 14, 983, (2009) · Zbl 1221.65126
[17] U. Ascher, R. Mattheij, and R. Russell, Numerical solution of boundary value problems for ordinary differential equations, In: SIAM Classics in Applied Mathematics, No. 13; 1995. · Zbl 0843.65054
[18] Ascher, U.; Petzold, L., Computer methods for ordinary differential equations and differential-algebraic equations, (1998), SIAM Philadelphia · Zbl 0908.65055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.