Stable and Unstable Periodic Solutions of the Giesekus Model under Homogeneous Large Amplitude Oscillatory Shear Flow

The Giesekus model is a popular constitutive equation for polymer solutions and melts. Under homogeneous large amplitude oscillatory shear (LAOS) flow, it exhibits multiple periodic steady state solutions when solved using spectral methods like harmonic balance, although numerical integration from stress-free initial conditions always yields a unique solution. We investigate this apparent contradiction by performing linear stability analysis via the monodromy matrix method on all the solutions discovered by harmonic balance. For the Giesekus model with anisotropy parameter αG = 0.3, we identify four distinct periodic steady state solutions across four decades of Weissenberg numbers 10–2 ≤ Wi ≤ 102. Stability analysis reveals that only one solution remains stable throughout the parameter space, while the remaining three are unstable. The stable solution corresponds to the unique trajectory obtained via numerical integration. We show that the unstable solutions violate the requirement of symmetric positive definiteness of the underlying conformation tensor, rendering them physically inadmissible. Our findings demonstrate that numerical integration with adaptive time-stepping methods using stress-free initial conditions reliably converges to the unique physical solution.