Viscous heating in large-amplitude oscillatory shear flow

Abstract

When measuring rheological properties in oscillatory shear flow, one worries about experimental error due to the temperature rise in the sample that is caused by viscous heating. For polymeric liquids, for example, this temperature rise causes the measured values of the components of the complex viscosity to be systematically low. For such linear viscoelastic property measurements, we use an analytical solution by Ding et al. [J. Non-Newtonian Fluid Mech. 86, 359 (1999)10.1016/S0377-0257(99)00004-X] to estimate the temperature rise. However, for large-amplitude oscillatory shear flow, no such analytical solution is available. Here we derive an analytical solution for the temperature rise in a corotational Maxwell fluid (a model with just two parameters: η0 and λ) subject to large-amplitude oscillatory shear flow. This result can then be generalized to a superposition of corotational Maxwell models for a quantitative estimate of the temperature rise. We chose the corotational Maxwell model because, when generalized for multiple relaxation times, it gives an accurate prediction for molten plastics in large-amplitude oscillatory shear flow. We identify three relevant pairs of thermal boundary conditions: (i) both plates isothermal, (ii) with heat loss by convection from both plates, and (iii) one plate isothermal, the other with heat loss by convection. We find that the time-averaged viscous heating increases as an even power series of the dimensionless shear rate amplitude (Weissenberg number), and that it decreases with the dimensionless imposed frequency (Deborah number). We distinguish between the dimensionless time-averaged temperature rise, $\bar \Theta $Θ¯, and the oscillating part, $\tilde \Theta $Θ̃, where $\Theta \equiv \bar \Theta + \tilde \Theta $Θ≡Θ¯+Θ̃. We solve analytically for the $\bar \Theta $Θ¯ profile through the sample thickness for all three pairs of thermal boundary conditions. For the worst case, two adiabatic walls, we derive an expression for the oscillating part of the temperature rise, $\tilde \Theta $Θ̃. We find this $\tilde \Theta $Θ̃ to be a Fourier series of even harmonics whose contribution to the temperature rise can be as important as $\bar \Theta $Θ¯. If both plates are adiabatic, then the sample temperature rises without bound. Otherwise, it does not.