Acta Mech 202, 205–211 (2009) DOI 10.1007/s00707-008-0106-7 M. Guedda Boundary-layer equations for a power-law shear-driven flow over a plane surface of non-Newtonian fluids Received: 22 February 2006 / Revised: 3 October 2006 / Published online: 14 October 2008 © Springer-Verlag 2008 Abstract We study the boundary-layer similarity flow driven over a semi-infinite flat plate by a power-law shear with asymptotic velocity profile u e ( y ) = β y α ( y →∞,β> 0), for fluids both Newtonian and non- Newtonian. Theoretical analysis is reported to derive a range of exponents α and amplitudes β for which similarity solutions exist. The shear stress parameter f ′′ (0) is determined as a function of α and β. 1. Introduction The purpose of this paper is to present a mathematical analysis for similarity solutions describing laminar flows past a plane surface in a class of non-Newtonian fluids corresponding to an exterior power-law velocity profile of the form u e ( y ) = β y α ( y →∞,β> 0). (1.1) Equation (1.1) is regarded in the following sense: lim y→∞ u (x , y ) y -α = β, (1.2) where the positive x -coordinate is measured along the plate and the positive y -coordinate is measured normal to it, with y = 0 is the plate, the plate origin located at x = y = 0 and u (x , y ) is the x -velocity component. The scaling (power-type) law (1.1), where α and β depend somehow on the flow Reynolds number R e , is proposed by Barenblatt [1] for the mean velocity in fully developed turbulent shear flow. In [2], with the help of experimental data, Barenblatt and Prostokishin confirmed the conjecture proposed in [1] that α = 3 2 ln R e . The case α = 0 coincides with the problem considered by Blasius in his pioneering work [3] for the Newtonian case. Blasius obtained a family of numerical solutions such that the Prandtl velocity profile, u (x , y )/u e , depends only on a single variable η = yx - 1 2 , where u e is assumed to be constant. This results are extended by Falkner and Skan [4] to the class where u e is required to vary with the x coordinate only, which is the essence of similarity reduction reported in a voluminous literature. The problem considered here is structurally quite different from the more familiar Blasius and Falkner– Skan similarity reduction. Surprisingly, in 1987 an important observation was made by Weidman et al. [5]. They noticed that assumption (1.1), under some restriction on α, may underpin the simplification from the M. Guedda (B ) Faculté de Mathématiques et d’Informatique, LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 80039 Amiens, France E-mail: guedda@u-picardie.fr