Nonmonotonic Models are Not Necessary to Obtain Shear Banding Phenomena in Entangled Polymer Solutions

Recent experiments on entangled polymer solutions may indicate a constitutive instability, and have led some to question the validity of existing constitutive models. We use a modern constitutive model, the Rolie-Poly model plus a solvent viscosity, and show that (i) this simple class of models captures instability, (ii) shear banding phenomena is observable for weakly stable fluids in flow geometries with sufficiently inhomogeneous total stress, and (iii) transient phenomena exhibit inhomogeneities similar to shear banding, even for weakly stable fluids.

Figure 1

(a) Parameters ($\beta$, $ϵ$) where the constitutive curve of the Rolie-Poly model is nonmonotonic (shaded). (b) Section through Couette rheometer showing the flow field and total stress components, $\mathbf{T}$.

Figure 2

Flow curves (solid black) and constitutive curves (dashed red) for a range of stress gradients $q$ and CCR values $\beta$. The shaded area shows the size of the torque difference $\Delta {log}_{10}\Gamma ={log}_{10}{e}^{2q}$. Circles indicate applied shear rates at which the transient state responses are shown in Fig. 3.

Figure 3

Velocity as a function of position $y=q^{-1}\ln (r/R_1)$ for different ($\beta$, $q$), at shear rate ${log}_{10}\dot{\gamma }=1.2$ ($○$ in Fig. 2). The solid (red) lines indicate the steady state profile, and dashed lines are transient profiles at the times shown. The transient torque response $\Gamma (t)$ in the spatially uniform model is shown for the monotonic case ($\beta =0.728$) by a dashed line.

Figure 4

(a) Pipkin diagram (Wi vs De) for $\beta =0.728$ and $q=2\times {10}^{-3}$, showing regions with homogeneous ($○$) and inhomogeneous profiles (●). (b) Parametric shear rate—torque profiles $\mathbf{(}\dot{\gamma }(y),\Gamma (y)\mathbf{)}$ (dashed lines: 3,4) overlaying the constitutive curve (solid). (c) Torque evolution for different De noted on (a). (d) Velocity profiles for ii); profiles (1–5) are at different times in the cycle for $\mathrm{De}\simeq 3$ (ii) [$○$ in (c)].

Figure 5

Relaxation of a step strain of $\gamma =8$, applied using $\dot{\gamma }{\tau }_d=40$ for $t=0.2{\tau }_d$. (Left) Velocity before the shear rate stops and (right) snap shots of recoil velocities.