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Unformatted text preview: 4.2 The Falkner-Skan equation For the case of a flat plate, we have just shown that the boundary layer solution is self-similar in the sense that two profiles u ( x,y ) at different values of x differ only by a scale factor in y . This simplified the analysis considerably, because the momentum equation reduced to an ordinary differential equation in the single similarity variable . What about more complicated geometries? In this section, we derive the general conditions under which self-similar solutions exist. As usual we work in terms of the stream function so that the continuity equation is automatically satisfied. Consider again the boundary layer transformations 65, which tell us that the basic scales of the y coordinate and the stream function are y = Re 1 / 2 y L and ( x,y ) = LU Re 1 / 2 ( x ,y ) . (106) For a flat plate, we argued that we must actually replace the overall length of the object L by the present distance travelled along it, x . This gave (in an unusual format) = Re 1 / 2 radicalbig 2 x/L y L and = LU radicalbig 2 x/L Re 1 / 2 f ( ) , (107) in which the scaled stream function depends on space only through the single similarity variable . To extend this analysis to more general geometries, we recall that the only way the geometry enters the BL equations is through its influence on u e ( x ). For a flat plate just considered, u e ( x ) = U = constant. More generally, u e = u e ( x ). In what follows, we aim to find the most general form of u e ( x ) that still admits self-similar solutions....
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- Spring '10