Linear stability of three-layer Poiseuille flow for Oldroyd-B fluids

The linear stability of three-layer plane Poiseuille flow is studied in the longwave limit and for moderate wavelengths. The fluids are assumed to fol low Oldroyd-B constitutive equations with constant viscosities and elastici ties. We find that the jumps of the Poiseuille shear rate at both interface s which give the convexity of the Poiseuille velocity profile, allow us to determine the longwave stability for Newtonian fluids. On the other hand, t he stability of viscoelastic fluids is analyzed by using the additive chara cter of the longwave eigenvalues with respect to viscous and elastic terms. The stability with respect to moderate wavelength disturbances has to deal with two different modes called 'shortwave' (SW) and 'longwave' (LW), acco rding to their values at zero wavenumber. The SW eigenvalues can become the most dangerous modes for large Weissenberg numbers and their influences ca n be studied by means of shortwave analysis. Moreover, we point out that th e longwave stability analysis and convexity of the Poiseuille velocity prof ile allow us to determine the LW eigenvalues which are stable with respect to order one wavelength disturbances. (C) 1999 Elsevier Science B.V. All ri ghts reserved.