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 Hookean dumbbells as representative, we find that the center-of-mass motion follows: 〈x_{c}^{2}(t)〉∼γ[over ̇]^{2}t^{α+2}, generalizing the earlier result: 〈x_{c}^{2}(t)〉∼γ[over ̇]^{2}t^{3}(α=1). Here 0<α<1 is the coefficient defining the power-law decay of noise correlations in the viscoelastic media. Motion of the relative coordinate, on the other hand, is quite intriguing in that 〈x_{r}^{2}(t)〉∼t^{β} with β=2(1-α), for small α. This implies nonexistence of the steady state, making it inappropriate for addressing tumbling dynamics. We remedy this pathology by introducing a nonlinear spring with FENE-LJ interaction and study tumbling dynamics of the dumbbell. We find that the tumbling frequency exhibits a nonmonotonic behavior as a function of shear rate for various degrees of subdiffusion. We also find that this result is robust against variations in the extension of the spring. We briefly discuss the case of polymers.