Abstract
The primary goal of this paper is to specify sufficient conditions for the inviscid instability of a general class of plane parallel shear flows. For given complex eigenvaluec and real wave number α, and for givenh(y), the real part of the adjoint eigenfunction, Rayleigh's equation is converted into a nonlinear integral equation for the basic velocity profileU(y). Sufficient conditions are deduced for the existence and uniqueness of solutions to this integral equation, subject to appropriate homogeneous boundary conditions on the eigenfunction φ(y); the velocity profilesU(y) so derived are guaranteed to be unstable. Also separately described in this paper is a method to obtain a general class of new, exact neutrally stable solutions of Rayleigh's equation; given any realc and α, and a function Φ(U), the velocity profileU(y) and the eigenfunction Φ(U(y)) may be determined theoretically. A specific example of a class of neutrally stable solutions for jet-like profiles on an unbounded domain is given.
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Srinivasan, R. Exact solutions of Rayleigh's equation and sufficient conditions for inviscid instability of parallel, bounded shear flows. Z. angew. Math. Phys. 45, 615–637 (1994). https://doi.org/10.1007/BF00991900
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DOI: https://doi.org/10.1007/BF00991900