LINEAR-STABILITY OF MULTILAYER PLANE POISEUILLE FLOWS OF OLDROYD-B FLUIDS

The linear stability of plane Poiseuille flows of two and three-symmet rical layers is studied by using both longwave and moderate wavelength analysis. The considered fluids follow Oldroyd-B constitutive equatio ns and hence the stability is controlled by the viscous and elastic st ratifications and the layer thicknesses. For the three symmetrical-lay er Poiseuille flow, competition between varicose (symmetrical) and sin uous (antisymmetrical) mode is considered. In both cases (two and thre e symmetrical layers), the additive character of the longwave formula with respect to viscous and elastic terms is largely used to determine stable arrangements at vanishing Reynolds number. It is found that if the stability of such arrangements is due simultaneously to viscous a nd elastic stratification (the flow is stable for longwave disturbance and the Poiseuille velocity profile is convex), then the Poiseuille f low is also stable with respect to moderate wavelength disturbances an d the critical thickness ratio around which the configurations becomes unstable is given by longwave analysis. Note that a convex velocity p rofile means a positive jump of shear rate at the interface. Finally, the destabilization due to a moderate increase in the Reynolds number is considered and two distinct behaviors are pointed according to the convexity of the Poiseuille velocity profile. Moreover, an important i nfluence of the thickness ratio on the critical wavenumber is found fo r three symmetrical layer case (for two layer case, the critical wave number is of order one and depends weakly on the thickness ratio). (C) 1997 Elsevier Science B.V.