Transport and tumbling of polymers in viscoelastic shear flow

Polymers in shear flow are ubiquitous and we study their motion in a viscoelastic fluid under shear. Employing dumbbells as representative, we find that the center of mass motion follows: $\langle x^2_c(t) \rangle \sim \dot{\gamma}^2 t^{\alpha+2}, ~0< \alpha <1$, generalizing the earlier result: $\langle x^2_c(t) \rangle \sim \dot{\gamma}^2t^3 ~(\alpha = 1)$. Motion of the relative coordinate, on the other hand, is quite intriguing in that $\langle x^2_r(t) \rangle \sim t^\beta$ with $\beta = 2(1-\alpha)$ for small $\alpha$. This implies nonexistence of the steady state. We remedy this pathology by introducing a nonlinear spring with FENE-LJ interaction and study tumbling dynamics of the dumbbell. The overall effect of viscoelasticity is to slow down the dynamics in the experimentally observed ranges of the Weissenberg number. We numerically obtain the characteristic time of tumbling and show that small changes in $\alpha$ result in large changes in tumbling times.