ar X iv : 0 90 5 . 31 63 v 1 [ m at h . A P ] 1 9 M ay 2 00 9 Sommerfeld Paradox - A Novel Study

Sommerfeld paradox roughly says that mathematically Couette shear flow is linearly stable for all Reynolds number, but experimentally it is unstable to any size perturbation when the Reynolds number is large enough. Our study here focuses upon a sequence of 2D oscillatory shears which are the Couette linear shear plus small amplitude and high frequency sinusoidal shear perturbations. The sequence of oscillatory shears possesses two intriguing properties: (1). in the fluid velocity variables, the sequence approaches the Couette linear shear, thus it can be viewed as Couette linear shear plus small noises; while in the fluid vorticity variable, the sequence does not approaches the Couette linear shear; (2). unlike the Couette linear shear, the sequence of oscillatory shears has inviscid linear instability; furthermore, with the sequence of oscillatory shears as potentials, the Orr-Sommerfeld operator has unstable eigenvalues when the Reynolds number is large enough, this leads to transient nonlinear growth which manifests as transient turbulence as observed in experiments.