Similarity solutions Rayleigh problem

Thus the Rayleigh problem is reduced to the solution of the ODE, (3-126), subject to the two boundary conditions, (3-127) and (3-128). When a similarity transformation works, this reduction from a PDE to an ODE is the typical outcome. Although this is a definite simplification in the present problem, the original PDE was already linear, and the existence of a similarity transformation is not essential to its solution. When similarity transformations exist for more complicated, nonlinear PDEs, however, the reduction to an ODE is often a critical simplification in the solution process. 

Necessary conditions for the existence of a self-similar solution are that (1) the governing PDE must reduce to an ODE for F as a function ot// alone, and (2) the original boundary and initial conditions must reduce to a number of equivalent conditions for F that are consistent with the order of the ODE. Of course, a proof of sufficient conditions for existence of a selfsimilar solution would require a proof of existence of a solution to the ODE and boundary conditions that are derived for F. In general, however, the problems of interest will be nonlinear, and we shall be content to derive a self-consistent set of equations and boundary conditions and attempt to solve this latter problem numerically rather than seeking a rigorous existence proof. Let us see how the systematic solution scheme based on the general form (3-135) works for the Rayleigh problem. 

This is precisely the self-similar solution of the Rayleigh problem, (3-134), which was obtained in the previous section. Notice that the length scale d drops completely out of this limiting form of the solution (3-158). This is consistent with our earlier observation that the presence of the upper boundary should have no influence on the velocity field for sufficiently small times 7 <[Pg.151]

In the infinitesimal solid layer at t = 0, the temperature is also the freezing point 6m. Now, in view of the analysis of (3-119) for the Rayleigh problem, it is evident that a general solution of (3-175), expressed in self-similar form, is... 

Problem 10-9. Translating Flat Plate. Consider the high-Reynolds-number laminar boundary-layer flow over a semi-infinite flat plate that is moving parallel to its surface at a constant speed (7 in an otherwise quiescent fluid. Obtain the boundary-layer equations and the similarity transformation for f (r ). Is the solution the same as for uniform flow past a semi-infinite stationary plate Why or why not Obtain the solution for f (this must be done numerically). If the plate were truly semi-infinite, would there be a steady solution at any finite time (Hint. If you go far downstream from the leading edge of the flat plate, the problem looks like the Rayleigh problem from Chap. 3). For an arbitrarily chosen time T, what is the regime of validity of the boundary-layer solution ... 

Even the simple Eq. (8) raises a number of interesting problems. If the shape of the pile is at all complicated— which is almost invariably the case if the chain reacting material is liquid— the solution of (8) could be obtained only by perturbation methods. Spme of these show a remarkable similarity to the Rayleigh-Schrodinger method with which we are familiar from its application to quantum-mechanical problems. We owe many interesting results on (8) to Messrs. F. Murray, L. W. Nordheim, and H. Soodak. 

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