Nonlinear Analysis: Real World Applications 12 (2011) 2919–2930 Contents lists available at SciVerse ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces F. Talay Akyildiz a , Hamid Bellout b,∗ , Kuppalapalle Vajravelu c , Robert A. Van Gorder c a Arts & Sciences, Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates b Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA c Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA article info Article history: Received 10 August 2010 Accepted 28 February 2011 Keywords: Nanofluids Navier boundary condition Existence and uniqueness results Schauder fixed point theorem Stretching surface abstract Solutions for a class of degenerate, nonlinear, nonlocal boundary value problems, arising in nano boundary layer fluid flows over a stretching surface, are obtained. Viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface are considered. Using the Schauder fixed point theorem, existence and uniqueness results are established. The effects of the slip parameter k and the suction parameter a on the fluid velocity and on the tangential stress are investigated and discussed (through numerical results). We find that for fluid flows at nanoscales, the shear stress at the wall decreases (in an absolute sense) with an increase in the slip parameter k. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction and problem formulation In boundary layer theory, the condition of no-slip near solid walls is usually applied. The fluid velocity component is assumed to be zero relative to the solid boundary. This is not true for fluid flows at the micro and nanoscale. Investigations show that the condition of no-slip is no longer valid. Instead, a certain degree of tangential slip must be allowed. To describe the phenomenon of slip, Navier [1] introduced a boundary condition which states the component of the fluid velocity tangential to the boundary walls is proportional to the tangential stress. Later, several researchers [2–4] extended the Navier boundary condition. Recently Wang [5] analyzed the viscous flow due to a stretching sheet with surface slip and suction and brought out interesting results. Very recently, Van Gorder et al. [6], considered the model proposed by Wang [5] describing the viscous flow due to a stretching surface with both surface slip and suction (or injection). As in Wang, they considered two geometrical situations: (i) the two-dimensional stretching surface and (ii) the axisymmetric stretching surface. Using the similarity transform the Navier–Stokes equations are transformed into a nonlinear ordinary differential equation, and approximate analytical and numerical solutions are obtained, with appropriate boundary conditions. Analytical solutions, which could be used to validate the numerical solutions of these nonlinear problems are clearly desirable. Hence, in this paper, we study the existence and behavior of exact solutions of the third order nonlinear differential equations arising in nano boundary layer fluid flows over a stretching sheet. Let (u,v,w) be the velocity components in the (x, y, z ) directions, respectively and let p be the pressure. Then the Navier–Stokes equations for the steady viscous fluid flow can be written as uu x + vu y + wu z =−p x /ρ + ν(u xx + u yy + u zz ), (1.1a) ∗ Corresponding author. Tel.: +1 815 753 6733; fax: +1 815 753 1112. E-mail address: bellout@math.niu.edu (H. Bellout). 1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.02.017