Linear temporal stability analysis
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Abstract
An infinite flat plate covered with a thin layer of a power-law fluid sheared by an air stream is considered. The equations and boundary conditions governing the temporal linear stability problem are derived assuming small disturbances superimposed on a steady primary flow. The latter consists of a Blasius boundary-layer flow for the air stream and an exact solution of the Navier-Stokes equations for the thin fluid layer. Sources of change in kinematic energy and enstrophy are identified. A spectral collocation method based on Chebyshev polynomials is implemented and the resulting algebraic problem is solved using a QZ-algorithm. An isolated Blasius boundary-layer flow showed instability above a critical Reynolds number (approx. 300) for a range of wave numbers (Blasius mode). The presence of a thin fluid layer introduced, next to the (hardly changed) Blasius mode, an additional unstable mode (interfacial mode) exhibiting smaller amplification rates and a larger range of instability. The Blasius mode instability is driven mainly by the Reynolds stress and the interfacial mode by the action of viscosity. This research has been carried out within the framework of predicting the dynamic behaviour of a thin layer of liquid (e.g. water or anti-icing fluid) sheared by an air flow.