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Abstract

The aim of this work is to introduce, review and discuss similarity and pseudosimilarity solutions to a class of problems in the boundary layer theory. The boundary layer technique is encountered in many aspects of fluid dynamics and aerodynamics. This technique has been created for finding the flow of certain fluids. In 1904 Ludwig Prandtl1 introduced the concept that, at high Reynolds numbers, the flow about a solid can be divided into two regions. A very thin region adjacent to the body in which the viscosity of the fluid exerts an influence on the motion of the fluid. In this region the velocity gradient ∂u/∂y is very large. This region is called the “boundary layer”. In the remaining region the viscosity is negligible. The external flow is determined by the displacement of streamlines about the body and the pressure field is developed. Consequently, the full Navier–Stokes equations are simplified in the boundary layer. In fluid dynamical problems the main questions treated are the study of exact or particular solutions of the PDE approximations – called Prandtl's equations – to the Navier–Stokes equations. A class of these solutions has proved to play an important role for describing the behavior of the fluid in the boundary layer and in the development of mathematical theories as well as in numerical computational schemes. Most remarkable is that certain particular solutions, called similarity solutions, are exhibited by solving problems which are expressed in terms of a lower order PDEs or an ODEs. In fact, the idea of finding similarity or exact solutions is connected with the transformation of the PDEs to a set of equations which are easier to analyze, in general.The model for non-Newtonian fluid are highly non-linear. Indeed, unlike the situation of a Newtonian fluid, which satisfies a linear relationship between its stress and its rate of strain, the elastic effects are relatively unimportant compared to viscous effect, and then a model that deals with these non-linear effects is required. In addition, the constant viscosity of Newtonian fluids is defined as the ratio between a given shear stress and the resultant rate of strain. For a simple unidirectional flow, such as the flow between two parallel plates, this relationship is expressed as: This relation, known as Newtonian law of viscosity, is a mathematical statement, and there is no reason to believe that all real fluids, as polymer melts, paints and foams for example, should obey it. However one may define in a similar way an apparent viscosity for non-Newtonian fluid by Here, however, the apparent viscosity is not constant but depends on the rate of strain. One of the ways to describe the flow behavior of a non-Newtonian fluid is the Ostwald–de Waele power-law model.2 ,3 where n > 0 is called the power-law index. The case n < 1 is referred to as the pseudo-plastic fluid, the case n > 1 is known as dilatant or shear-thickening fluids. The Newtonian fluid is, of course, a special case where the power-law index n is one.The physical origin of a non-Newtonian behavior relates to the microstructure of the material. Polymer materials (solutions and melts) contains molecules having molecular weights of many hundreds of millions. In a fluid at rest, these long chain molecules will tend to exist in a coiled state and will usually entangle each other. During bulk deformation the molecules interact in a complex and a non-linear way. The resulting flow phenomena depend on this interaction, and in general will produce viscoelastic behavior.The range of non-Newtonian fluid behavior exhibited by industrial liquids is very large. A broad description of the behavior in both steady and unsteady flow situations, together with mathematical model, can be found for example in Barnes,4 Bird5 and Tanner.6 In this work we consider some layer models of laminar non-Newtonian fluids, with a power-law viscosity, past a semi-infinite flat plate. With analogy of Blasius, Falkner–Skan, Goldstein and Mangler Methods, partial differential equations are transformed into an autonomous third-order non-linear degenerate equation. We establish the existence of a family of unbounded global solutions. The asymptotic behavior is also discussed. Some properties of solutions depend on the power-law index.This work reflects the scientific interests of the author, during the last three years, on the boundary layer equations of non-Newtonian fluids. Many results presented here have not published before and are obtained with the helpful discussions and remarks of M. Benlahsen, B. Brighi, R. Kersner and A. Gmira. The final version of this book was written during the author's visit to Department of Mathematics of University of A. Essaadi, Tétouan (Maroc). This work has been partially supported in part by PAI No MA/05/116 for France–Maroc scientific cooperation and by Direction des Affaires Internationales, UPJV, Amiens France. The Newtonian case of Chapter 3.4 was presented at the Conference “Self-Similar Solutions in Nonlinear PDE's”, Bedlow, Poland (2005). The author wishes to thank the organizers for the invitation and their kind hospitality.