Hydrodynamic Stability, Second Edition
Containing a large number of problems that provide exercises as well as solutions, this book offers a clear and unparalleled explanation of the fundamentals of hydrodynamic stability.
TABLE OF CONTENTS Hydrodynamic Stability, Second Edition
28: Approximations to The Eigenvalue Relation
28 Approximations to The Eigenvalue Relation
Having now obtained approximations to four solutions of the Orr-Sommerfeld equation, we can now use the properties of these approximations to derive simplified forms of the eigenvalue relation for large values of ?R. The precise form of the eigenvalue relation depends not only on the class of flows considered but also on the viscous approximations that are used, and there are therefore a number of cases to be considered.
28.1 Symmetrical flows in a channel
The most important flow of this type is, of course, plane Poiseuille flow for which U(z)=1 ? z 2 over ?1 ? z ?1 and this flow will be used for illustrative purposes throughout this section. More generally, however, it is convenient to suppose that L has been chosen so that z 1= ?1 and z 2=0, and that U(z) is monotone increasing on this interval with U( z 1)=0. Because of the symmetry of the basic flow we can consider the event [ ] and odd solutions separately. It is found that the odd solutions are stable and we will therefore concentrate our attention on the even solutions which must then satisfy the boundary conditions
The solutions 
3 and 4 are dominant at z 1 and z 2 respectively. But if z c is significantly closer to z 1 than to z 2, as it happens to be...
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