Time-dependent plane Poiseuille flow of a Johnson–Segalman fluid

Abstract

We numerically solve the time-dependent planar Poiseuille flow of a Johnson–Segalman fluid with added Newtonian viscosity. We consider the case where the shear stress/shear rate curve exhibits a maximum and a minimum at steady state. Beyond a critical volumetric flow rate, there exist infinite piecewise smooth solutions, in addition to the standard smooth one for the velocity. The corresponding stress components are characterized by jump discontinuities, the number of which may be more than one. Beyond a second critical volumetric flow rate, no smooth solutions exist. In agreement with linear stability analysis, the numerical calculations show that the steady-state solutions are unstable only if a part of the velocity profile corresponds to the negative-slope regime of the standard steady-state shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to different stable steady states, depending on the initial perturbation. The asymptotic steady-state velocity solution obtained in start-up flow is smooth for volumetric flow rates less than the second critical value and piecewise smooth with only one kink otherwise. No selection mechanism was observed either for the final shear stress at the wall or for the location of the kink. No periodic solutions have been found for values of the dimensionless solvent viscosity as low as 0.01.