Similarity solutions to boundary layer equations for third-grade non-Newtonian fluid in special coordinate system

Two dimensional equations of steady motion for third order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the inviscid flow around an arbitrary object, the streamlines are the $\phi$-coordinates and the velocity potential lines are $ \psi$-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, is then shown to be independent of the body shape immersed into the flow. As the first approximation, it is assumed that the second grade terms are negligible compared to the viscous and third grade terms. The second grade terms spoil scaling transformation which is the only transformation leading to similarity solutions for a third grade fluid. By using Lie's group methods, infinitesimal generators of boundary layer equations are calculated. The equations are transformed into an ordinary differential system. Numerical solutions to the outcoming nonlinear differential equations are found by using a combination of the Runge-Kutta algorithm and a shooting technique.