INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 2, NO. 2, 2012 53 Deductive Group Invariance Analysis of Boundary Layer Equations of a Special Non-Newtonian Fluid over a Stretching Sheet R. M. Darji and M. G. Timol Abstract—The general group-theoretic transformations are developed for the solution of highly non-linear partial differential equations governing the boundary layer flow of the special Non- Newtonian fluid so-called Sisko fluid past over stretching sheet. The application of a one-parameter group reduces the number of independent variables by one, and consequently the system of governing non-linear partial differential equations with boundary conditions reduces to a non-linear ordinary differential equation with appropriate boundary conditions. The numerical solution for the present flow situation is derived systematically from similarity requirement using Runge-Kutta scheme with shooting method. Index Terms—Group symmetry, sisko fluid, boundary-layer flow, similarity solution, skin-friction. MSC 2010 Codes – 76A05, 76M55, 54H15 Nomenclature a, b flow parameters f dimensionless stream function G sub-group n power-law index T a group transformation U(x) velocity of main stream u velocity in x direction v velocity in y direction x, y reference coordinate distance Greek symbols α,β similarity conditional constant η similarity variable ∞ condition at infinity ξ invariant conditional function τ yx shear stress Subscript w condition at sheet wall superscript ’ differentiation with respect to η R. M. Darji is with the Department of Mathematics, Sarvajanik College of Engineering and Technology, Surat, Gujarat, 395001 INDIA. E-mail: rmdarji@gmail.com M. G. Timol is with Department of Mathematics, Veer Narmad South Gu- jarat University, Surat, Gujarat, 395007 INDIA. E-mail: mgtimol@gmail.com I. I NTRODUCTION D EDUCTIVE group transformation analysis, also called symmetry analysis, is based on general group of trans- formation that was developed by Sophus Lie to find point transformations that map a given differential equation to itself. This method unifies almost all known exact integration techniques for both ordinary and partial differential equations [1]. Group analysis is the only rigorous mathematical method to find all symmetries of a given differential equation and no adhoc assumptions or a prior knowledge of the equation under investigation is needed. The boundary layer equations are especially interesting from a physical point of view because they have the capacity to admit a large number of invariant solutions, i.e. similarity solutions. In the present context, invariant solutions are meant to be a reduction to a simpler equation such as an ordinary differential equation. Prandtls boundary layer equations admit more and different symmetry groups. Symmetry groups or simply symmetries are invariant transformations, which do not alter the structural form of the equation under investigation [2]. Newtons law of viscosity states that shear stress is proportional to velocity gradient. Fluids that obey this law are known as Newtonian fluids. Amongst Newtonian fluids, we can cite water, benzene, ethyl alcohol, hexane and most solutions of simple molecules. Numerous fluids violate Newtons law of viscosity. On the other hand, fluids that do not obey Newtons law are known as Non-Newtonian fluids. Amongst Non- Newtonian fluids, we can cite some lubricants, whipped cream, some clays, some drilling mud, many paints, synovial fluid, suspensions of corn starch, sand in water, paper pulp in water, latex paint, ice, blood, syrup, molasses, blood plasma, custard etc. The characteristic of Non-Newtonian fluids are defined according to the non-linear stress-strain relationship. Some of the Non-Newtonian fluids are classify according to their stress- strain relationship are [3]; Power-law fluids, Eyring fluids, Sisko fluids, Prandtl - Eyring fluids, Sutterby fluids, Prandtl fluids, Ellis fluids, Williamson fluids, Reiner- Philippoff fluids, Powell-Eyring fluids etcetera. The non- linear character of the partial differential equations governing the motion of a fluid produces difficulties in solving the equations. In the field of fluid mechanics, most of the researchers try to obtain the similarity solutions in such cases. In case of deductive group of transformations, the group-invariant solutions are nothing but the well-known similarity solutions [4]. The most general form of Lie group of transformations, known as deductive group