NONSTATIONARY FLOWS OF A VISCOELASTIC FLUID IN THE JOHNSON-SEGALMAN MODEL WITH MULTIPLE RELAXATION TIMES
S.R. Karmushin1,2
1Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: non-Newtonian viscoelastic fluid, rheology, nonlinear hyperbolic model, one-dimensional shear flow, instability, shear banding, relaxation time, hysteresis
Abstract
One-dimensional unsteady flows of an incompressible non-Newtonian viscoelastic fluid between parallel plates are considered within the framework of the Johnson-Segalman model with multiple relaxation times. A distinctive feature of the model is its hyperbolicity over a wide range of flow parameters. A general form of the model with n relaxation times (modes) is obtained, and a change of variables is introduced that allows the governing equations to be written in conservative (divergent) form. A series of unsteady flow simulations in various regimes is performed, demonstrating the occurrence of shear banding with increasing mean flow velocity. The dependence of the wall shear stress on the shear rate is obtained, as well as the dependence of the flow rate on the pressure gradient, for plane steady Couette and Poiseuille flows, respectively. The resulting diagrams are validated by comparison with a range of experimental data. The structure of steady-state solutions exhibiting shear banding, obtained as the numerical limit of unsteady solutions, is investigated. A criterion is formulated for selecting steady-state solutions that are asymptotically realized in numerical unsteady calculations. The phenomenon of hysteresis under cyclic variation of the flow velocity is analyzed.
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