Newtonian limit: viscosity renormalization

In the limit $\tau \to 0$ the elastic force originated from the thermal motion keeps the molecules coiled near their equilibrium configuration, and the polymer solution is supposed to behave like a Newtonian fluid. Indeed, a perturbative expansion in $\tau$ for the conformation tensor

$$\sigma_{ij} = \sigma_{ij}^{(0)} + \tau \sigma_{ij}^{(1)} + O(\tau^2)$$ (3.26)

plugged in the equation Eq. (3.20) gives for the zeroth and first order terms:

$$\sigma_{ij}^{(0)} = \delta_{ij} \;, \sigma_{ij}^{(1)} = 1/2 (\nabla_j u_i + \nabla_i u_j) = e_{ij}$$ (3.27)

We observe that at first order in $\tau$ the elastic stress tensor is proportional to the deformation tensor

$$\mathbb{T}_{ij}^P = n K_0 R_0^2 \tau e_{ij} + O(\tau^2)\:,$$ (3.28)

which means that the fluid is Newtonian up to higher order corrections $O(\tau^2)$. Anyway the presence of polymers changes the properties of the fluid also in the Newtonian limit, because the fluid is partially entrapped in the coiled polymers, producing a change in the total viscosity of the solution $\mu_{t}$ which is renormalized as:

$$\mu_{t}= \mu + {1 \over 2} n K_0 R_0^2 \tau = \mu(1+\eta)$$ (3.29)