Instabilities in Torsional Flows of Polymeric Fluids
Jonathan P. Rothstein and Gareth H. McKinley
Previous experimental measurements and linear stability analyses of curvilinear shearing flows of viscoelastic fluids have shown that the combination of streamwise curvature and elastic normal stresses can lead to flow destabilization. Torsional shear flows of highly elastic fluids with closed streamlines can also accumulate heat from viscous dissipation resulting in nonuniformity in the temperature profile within the flow and nonlinearity in the viscometric properties of the fluid. Recently, it has been shown by Al- Mubaiyedh et al.(Phys. Fluids, 11, 3217 (1999)) that the inclusion of energetics in the linear stability analysis of viscoelastic Taylor-Couette flow can change the dominant mode of the purely elastic instability from a nonaxisymmetric and time-dependent secondary flow to an axisymmetric stationary Taylor-type toroidal vortex that more closely agrees with experimental observations. In this work we present a detailed experimental study of the effect of viscous heating on the torsional steady shearing of elastic fluids between a rotating cone-and-plate and between two rotating coaxial parallel-plates. Elastic effects in the flow are characterized by the Deborah Number, De, while the magnitude of the viscous heating is characterized by the Nahme-Griffiths number, Na. We show that the relative importance of these two competing effects can be quantified by a new dimensionless thermoelastic parameter, é = Na1 2 De, which is a material property of a given viscoelastic fluid independent of the rate of deformation. By utilizing this thermoelastic number, experimental observations of viscoelastic flow stability in three different fluids and two different geometries over a range of temperatures can be rationalized and the critical conditions unified into a single flow stability diagram. The thermoelastic number is a function of the molecular weight of the polymer, the flow geometry and the temperature of the test fluid. The experiments presented here were performed using test fluids consisting of three different high molecular weight monodisperse polystyrene solutions in various flow geometries and over a large range of temperatures. By systematically varying the temperature of the test fluid or the configuration of the test geometry, the thermoelastic number can be adjusted appreciably. When the characteristic timescale for viscous heating is much longer than the relaxation time of the test fluid (é << 1)the critical conditions for the onset of the elastic instability are in good agreement with the predictions of isothermal linear stability analyses. As the thermoelastic number approaches a critical value, the strong temperature gradients induced by viscous heating reduce the elasticity of the test fluid and delay the onset of the instability. At even larger values of the thermoelastic parameter, viscous heating stabilizes the flow completely.
Published Phys. Fluids, 13(2), 2000, p 109-133.