Stability for Two No-Slip, Rigid Boundaries

The linear stability problem when one or both of the boundaries above and below a fluid layer are rigid no-slip surfaces is difficult to solve. Thus the principle of exchange of stabilities, which we have just proven, is a major advantage because we can set a = 0 at the neutral stability point (i.e., at the transition between stable and unstable conditions). 

In view of the symmetry of the problem it is advantageous to assume that the boundaries are at z = 1 /2. Then the boundary conditions are 

the problem (12-225)-( 12-228) is an eigenvalue problem. In this case, given a2, a nonzero solution will exist only a for certain value of Ra. Because we have already set a = 0, the critical value Racrit is then the minimum of Ref for all possible a. 

because the governing equation (12-225) has constant coefficients we can seek a general solution as a superposition of solutions of the form 

To write a solution of this equation, it is convenient to introduce a new variable r such 

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