Linear stability of a viscoelastic liquid flow on an oscillating plane

Linear stability of a viscoelastic liquid on an oscillating plane is studied for disturbances of arbitrary wavenumbers. The main aim is to extend the earlier study of Dandapat & Gupta (J. Fluid Mech., vol. 72, 1975, pp. 425-432) to the finite wavenumber regime, which has not been attempted so far in the literature. The Orr-Sommerfeld boundary value problem is formulated for an unsteady base flow, and it is resolved numerically based on the Chebyshev spectral collocation method along with the Floquet theory. The analytical solution predicts that U-shaped unstable regions appear in the separated bandwidths of the imposed frequency, and the dominant mode of the long-wave instability intensifies in the presence of the viscoelastic parameter. The numerical solution shows that oblique neutral curves come out from the branch points of the U-shaped neutral curves at finite wavenumber and continue with the imposed frequency until the curves cross the next U-shaped neutral curve. As a consequence, in the finite wavenumber regime, no stable bandwidth of the imposed frequency is predicted by the long-wavelength analysis. Further, in some frequency ranges, the finite wavenumber instability is more dangerous than the long-wave instability.