The Society of Rheology: 70th Annual Meeting (Oct 1998) Paper AN1

STABILITY AND NONLINEAR DYNAMICS OF VISCOELASTIC SHEAR FLOWS SUBJECTED TO SECONDARY FLOW

Session:	Analytical and Numerical Solutions to Flow Problems
Date:	Wednesday - October 7, 1998
Time:	9:45
Room:	Los Angeles

Speaker:	Michael D. Graham, Department of Chemical Engineering, University of Wisconsin-Madison
Authors:	Venkat V. Ramanan, Department of Chemical Engineering, University of Wisconsin-Madison K. A. Kumar, Department of Chemical Engineering, University of Wisconsin-Madison Michael D. Graham, Department of Chemical Engineering, University of Wisconsin-Madison

Abstract

We investigate the effect of superposition of a time-periodic axial Couette flow on the viscoelastic circular Couette and Dean flow instabilities. The analysis, carried out for the Oldroyd-B fluid, generally shows increased stability as compared to when there is no axial flow. However, in the circular Couette case, the flow shows instability - synchronous resonance - for some parameter values. Two general regimes of instability are observed, one when the forcing frequency is close to the inverse of the polymer relaxation time and, surprisingly, another when the frequency is much lower. In the latter regime, the disturbances display quiescent intervals punctuated by periods of large transient growth and subsequent decay. Asymptotic and numerical results indicate that the high frequency, low axial Weissenberg number regime is essentially equivalent to the case without axial flow, implying no stabilization, while at high values of frequency and axial Weissenberg number, the flow is always stable. Consistent with previous results on parametrically forced systems, the zero frequency limit is singular.

We also present stability results for the addition of steady axial Couette and Poiseuille flows to viscoelastic instabilities in azimuthal Dean flows. The qualitative effect of adding a steady axial flow is similar to that of the circular Couette geometry for high axial Weissenberg number, with the critical azimuthal Weissenberg number increasing linearly with the axial Weissenberg number. For low axial Weissenberg number, the flow is stabilized, in constrast to the circular Couette flow. Weakly nonlinear analysis shows that the criticality of the bifurcation depends on the values of axial Weissenberg number and solvent viscosity. Finally, we also show the presence of a codimension-2 Takens-Bogdanov bifurcation point in the linear stability curve of Dean flow. This point represents a transition from one mechanism of instability to another.

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