The Rayleigh stability equation

The two boundaries at z = 0 and d can be either rigid walls or a free surface. In fact, we do not need to be so restrictive in our description of the problem. One or both of the boundaries may also be at infinity. The function U(z ) can be considered as an arbitrary function of z. The equations governing U7 and the linear disturbance flow are the dimensional Navier Stokes equations, (12-1) and (12-2), but, in this case, with the viscous terms neglected. 

It is convenient to nondimensionalize. We must identify a characteristic length scale tc and velocity scale uc. For the former we choose d, though we recognize that this would need to be modified if the fluid is unbounded. For the characteristic velocity scale, it is traditional in this field to choose the maximum value of Uf In addition, we assume that the characteristic time is tc = ic/uc, and the characteristic pressure is pc = pu2c. Then the governing equations, which are the continuity equation and the inviscid Navier-Stokes equations (usually called the Euler equations), can be written in the form 3u 

To consider the stability of the basic flow (12 302), we introduce an infmtitesimal perturbation to both the velocity and the pressure fields, 

For convenience, we denote the components of u as u = (it, v, w). Thus, if we substitute (12 303) and (12 304) into (12-301), retaining only the terms that are linear in s and remembering that the basic flow is an exact solution, we obain 

As usual, in order that the solution be well behaved at infinity, ax and ay are real. 

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This equation is known as the Rayleigh stability equation. Together with the boundary conditions... 

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