(441c) On the Critical Conditions of Thermoelastic Instabilities in Curvilinear Shear Flows: A Minimal Model

Relatively small temperature non-homogeneities, on the order of a few degrees Celsius, can lead to significant modifications in the critical conditions of elastic instabilities and ensuing flow patterns in curvilinear shear flows of high molecular weight polymer solutions [1-4]. Such temperature variations could arise even in an experiment presumed to be conducted under isothermal settings due to heat generation caused by viscous energy dissipation. Viscosity and stress relaxation time scales in polymer solutions are sensitive to temperature. Therefore, temperature variations can alter the damping dynamics of flow perturbations, and in turn, the onset and space-time characteristics of the secondary flow. In Taylor-Couette flow, thermoelastic effects give rise to a new branch of instabilities at critical shear rates an order of magnitude lower than those for the isothermal flow [1-3]. For cone and plate flow, the reduction in elastic forces (relaxation time) facilitated by a temperature increase of the test fluid can stabilize the flow, i.e., an increase the critical shear rate is observed experimentally [4].

The most striking effect in the Taylor-Couette flow is a precipitous reduction in the critical Deborah number (De), defined as the ratio of the fluid relaxation time to a characteristic flow time, and a change in the primary mode of instability from a temporal to stationary one, when the Nahme-Griffith number (Na), a measure of the thermal sensitivity of the rheological properties of the fluid, exceeds a threshold value. In this work, we present a minimal model that captures this phenomenon based on a small gap approximation analysis of the Oldroyd-B fluid. The Peclet number, signifying the ratio of heat transfer rates by convection to conduction, for typical viscoelastic polymer solutions used in experiments is on the order of 10,000. This (i.e., the fact that Pe >> 1) allows for further simplification of the governing equations. Scaling laws correlating De, Na, the gap width W and the Peclet number are derived and compared to numerical results. It is shown that the critical value of De is inversely proportional to the product of Pe, Na, and W [5]. This inverse proportionality of the critical Deborah number on the gap width is qualitatively different from that observed for the isothermal elastic instability for which De is inversely proportional to the square root of W [6]. Below a threshold value of Na, a relatively small increase in the critical Deborah number is observed through a rescaling of the fluid relaxation time by the increase in temperature. This effect, which is predominant in curvilinear flows such as the one realized in a cone and plate geometry is explained based on the minimal model. Finally, the effect of the sign of the steady state thermal gradient on the critical Deborah number is discussed.

References

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