×

Small-amplitude oscillatory forcing on two-layer plane channel flow. (English) Zbl 0915.76024

The linear stability of the plane Couette-Poiseuille flow of two superposed fluids against Fourier mode perturbations is studied at low Reynolds numbers. The basic flow, driven by the relative planar time-periodic motion of the upper boundary and streamwise pressure gradient, is written in the closed form. The problem is governed by the nonstationary partial differential Orr-Sommerfeld equation for the perturbation transversal velocity, an ordinary differential equation for the perturbed interface position, and boundary conditions on the walls and the unperturbed interface. Eleven parameters occur. Two of them are supposed to be small and, so, the asymptotic expansions up to second-order terms for the unknown functions and the complex frequency are used to derive an asymptotic approximate model. Then the authors carry out a regular perturbation analysis for very long wavelengths and a singular one for very short wavelengths, providing asymptotic approximate models. Numerical methods are applied separately to the Couette, Poiseuille and Couette-Poiseuille case. The stabilizing/destabilizing effects of boundary and pressure oscillations are found to be relevant only when the two oscillations are considered simultaneously.

MSC:

76E05 Parallel shear flows in hydrodynamic stability
76V05 Reaction effects in flows
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI