E The Rayleigh Problem - Solution by Similarity Transformation

In this section, we consider a second example of a transient, unidirectional flow. This is the famous problem, first studied in the 1800s by Lord Rayleigh, in which an initially stationary infinite flat plate is assumed to begin suddenly translating in its own plane with a constant velocity through an initially stationary unbounded fluid. 

For convenience, let us adopt a Cartesian coordinate system in which the flat plate is assumed to occupy the xz plane, with the initially stationary fluid occupying the upper half space, v 0. We denote the magnitude of the plate velocity as U so that 

The problem is to determine the velocity distribution in the fluid as a function of time. In this problem, the fluid motion is due entirely to the motion of the boundary - the only pressure gradient is hydrostatic, and this does not affect the velocity parallel to the plate surface. At the initial instant, the velocity profile appears as a step with magnitude Uat the plate surface and magnitude arbitrarily close to zero everywhere else, as sketched in Fig. 3 11. As time increases, however, the effect of the plate motion propagates farther and farther out into the fluid as momentum is transferred normal to the plate by molecular diffusion and a series of velocity profiles is achieved similar to those sketched in Fig. 3-11. In this section, the details of this motion are analyzed, and, in the process, the concept of self-similar solutions that we shall use extensively in later chapters is introduced. 

The second condition is just the no-slip condition at the plate surface, whereas the third condition arises as a consequence of the assumption that the fluid is unbounded. At any finite t 0, a region will always exist sufficiently far from the plate that no significant momentum transfer will yet have occurred, and in this region the fluid velocity will remain arbitrarily small. 

The problem posed by (3-119) with the boundary and initial conditions, (3-120b), is very simple to solve by either Fourier or Laplace transform methods.14 Further, because it is linear, an exact solution is possible, and nondimensionalization need not play a significant role in the solution process. Nevertheless, we pursue the solution by use of a so-called similarity transformation, whose existence is suggested by an attempt to nondimensionalize the equation and boundary conditions. Although it may seem redundant to introduce a new solution technique when standard transform methods could be used, the use of similarity transformations is not limited to linear problems (as are the Fourier and Laplace transform methods), and we shall find the method to be extremely useful in the solution of certain nonlinear DEs later in this book. 

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