Viscoelastic Effects in Modeling Web Handling Systems: Unsteady-State Analysis

Nonlinear evolution of a continuous spectrum of unstable waves near the first bifurcation point in circular Couette flow has been investigated. The disturbance is represented by a Fourier integral over all possible axial wave numbers, and an integrodifferential equation for the amplitude-density function of a continuous spectrum is derived. The equations describing the evolution of monochromatic waves and slowly varying wave packets of classical weakly nonlinear instability theories are shown to be special limiting cases. Numerical integration of the integrodifferential equation shows that the final equilibrium state depends on the initial disturbance, as observed experimentally, and it is not unique. In all cases, the final equilibrium state consists of a single dominant mode and its harmonics of smaller amplitudes. The predicted range of wave numbers for stable supercritical Taylor vortices is found to be narrower than the span of the neutral curve from linear theory. Taylor-vortex flows with wave numbers outside this range are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The results also show that a linearly stable long wave can excite a short unstable wave through nonlinear wave interaction. An important implication of the existence of nonunique equilibrium states is that the torque induced by the fluid motion cannot be determined uniquely. The numerical results show that the uncertainty, associated with nonuniqueness, of using any accurately measured Taylor-vortex torque slightly above the first bifurcation point in engineering practice can be as large as ten percent. The presence of multiple solutions at a fixed Reynolds number for a given geometry in Taylor-Couette flows has been known since Coles' monumental contribution in 1965. A theoretical confirmation has come only 30 years later. It is worthwhile to point out that the existence of multiple solutions, found by Coles, differs from current popular bifurcation theories. The current study indicates that the state of flows on a stable bifurcation branch can involve any wave number within a finite band and can not be determined uniquely. The multiple solutions in Coles' sense have also been found for mixed-convection flows (Yao and Ghosh Moulic, 1993, 1994) besides the Taylor-Couette flows. We believe that the nonuniqueness of Coles sense, which complements the bifurcation theories, is a generic property for all fluid flows.