CHAPTER-I
                 HISTORY OF BOUNDARY
                   LAYER THEORY
       Introduction;
            In this chapter in Section 1 we explained the boundary 
       Layer concept due to Prandtl (1904) in detail. In Section 2 
       we gave the major developments in two dimensional boundary 
       Layer theory. Section 3 consists the developments in the 
       boundary Layer theory for axial symmetrical flow. In Section 4 
       we made a survey of three dimensional boundary Layer theory. 
       Lastly in Section 5 we defined some basic concepts which we 
       require for our problems to be discussed in Chapter-II.
       1. Boundary Layer Concept s
            Prandtl was the first person who introduced the 
       concept of the boundary layer in 1904. He gave mathematical 
       formation of boundary layer equations. The complete Navier— 
       stokes equations are elliptic differential equations while 
       the boundary layer equations are parabolic. This transition 
       from elliptic type Navier-Stokes equations to parabolic type 
       boundary layer equation. It is one of the fundamental 
       consequences of the asymptotic transformation conceived by 
       Prandtl. Because of this type transformation, certain aspects 
       of the solution of the full Navier-Stokes equations are lost.
            The. boundary layer theory is the foundation of all 
       modern developments in fluid mechanics, and aerodynamics which
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                      have been clarified by the study of boundary layer flow and 
                       its effects on the general flow around the body, such as in 
                       the study of aircraft response to atmospheric gust, in flutter 
                       phenomenon involving wing etc.
                                      In two-dimensional cases for which the velocity 
                       components depends only two space corordinates. At the same 
                       time the velocity component in the direction of the third 
                       space co-ordinate did not exist. The general three-dimensional 
                       cases of a boundary layer in which the three velocity components 
                       depend on all three coordinates has so far, been hardly 
                       elaborated
                                           because of the enormous mathematical difficulties 
                       associated with the problem.
                                      The mathematical difficulties encountered in the 
                       study of axially symmetrical boundary layer are considerably 
                       smaller and hardly exceed those in the two-dimensional case.
                                      Although more than half a century old the subject of 
                       Boundary
                                       Layer is still receiving considerable interest and 
                       there are still a number of unsolved problems baffling the 
                       investigators. The concept of a thin region of quick transition 
                       near the boundary surface has solved many intricate practical 
                       problems and has enabled deep probing into the non-linear 
                       differential equations. The starting point of this great 
                       physical concept was the well known D'Alembert*s paradox in the
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        late 19th century. D'Alembert observed that when a solid body 
        moved through a fluid the flow pattern based on the imviscial C,,y 
        theory agreed with the experimental results almost everywhere 
        in the flow field; but strangely enough the resistance 
        experienced by the body was found to be zero. Prandtl made an 
        attempt to resolve this dilemma and suggested that the 
        resistance to the body was caused by the viscosity of the 
        fluid and that the flow fields near and away from the body 
        were different in character.
             The boundary layer equations have been well investiga- 
        ted for many engineering problems during the past seventy years.
             The boundary layer equations of motion may be integrated 
        across the boundary layer so that the momentum integral 
        equations are obtained. Some simple methods for evaluating 
        the characteristics of the boundary layer flows, first 
        introduced by Von Karman are based on the integral equation.
        These momentum integral equations for both two and three- ,, __ ^ p 
        dimensional boundary layer flow of incompressible and of 
        compressible fluids.
             The boundary layer equations may be transformed into 
        special forms so that the computation of the numerical results 
        will be simplified or will be easier for special devices.
        The boundary layer equations may be transformed into the