The long-wave instability of a defect in a uniform parallel shear

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Abstract
The stability properties of an inviscid, parallel, incompressible, free shear flow are studied. The shear profile is that of an unbounded, plane Couette flow containing a defect, or transition zone, whose magnitude ε is assumed to be small. The linearized eigenvalue problem is solved first for discretized models. When the defect has a finite thickness, the instability is confined to longitudinal wavenumbers, k [les ] 0(ε), in contrast to the more common 0(1) bandwidth, in units of inverse shear length. This observation motivates the application of a long-wave expansion to a smooth defect profile. A double expansion in both k and ε captures the whole waveband of the instability, and yields convergent expansions for the unstable eigenfunctions and for the dispersion relation describing their growth rate. The fastest growing modes are determined, and their back-reaction on the basic shear is calculated.