  Viscosity overshoot of entangled polymer melts has been observed under shear
flow and uniaxial elongational flow, but not under biaxial elongational flow
for a long period of time. We confirmed the presence of the viscosity overshoot
under biaxial elongational flows observed in a mixed system of ring and linear
polymers expressed by coarse-grained molecular dynamics simulations. The
overshoot was found to be more pronounced in a weakly entangled melt.
Furthermore, the smallest strain rate showing the strain hardening was found to
be depending on the linear chain length as $\dot{\varepsilon}_{\rm s}(N)\sim
N^{-1/2}$, which is different from the conventional relationship,
$\dot{\varepsilon}_{\rm s}(N) \sim N^{-2}$ or $N^{-3.4}$ for melts of linear
polymers without or with entanglements, respectively. We have concluded that
the cooperative phenomena of ring and linear chains were enhanced under biaxial
elongational flow.
