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This thesis consists of six chapters, and more over it includes the list of recent references, English and Arabic summaries. The first chapter is a general introduction to fluid mechanics and its definition also the governing equation of motion of fluid, temperature equation and concentration equation. We introduce many non-Newtonian fluids such as Sisko fluid, Walter’s B fluid, third order fluid, hyperbolic tangent fluid and Jeffry fluid. The problems have been solved analytically by using perturbation method and numerically by using Rung-Kutta-Merson method and Newton iteration in a shooting and matching technique are used to find the solution of the system equations. . The solutions of these equations such as the velocity, temperature, concentration and nanoparticles distributions are obtained as functions of physical parameters of the problem. The effect of these parameters on those solutions are discussed numerically and illustrated graphically through a set of figures.