Start-up inertia as an origin for heterogeneous flow

For quite some time nonmonotonic flow curve was thought to be a requirement for shear banded flows in complex fluids. Thus, in simple yield stress fluids shear banding was considered to be absent. Recent spatially resolved rheological experiments have found simple yield stress fluids to exhibit shear banded flow profiles. One proposed mechanism for the initiation of such transient shear banding process has been a small stress heterogeneity rising from the experimental device geometry. Here, using computational fluid dynamics methods, we show that transient shear banding can be initialized even under homogeneous stress conditions by the fluid start-up inertia, and that such mechanism indeed is present in realistic experimental conditions.

Figure 1

The planar Couette flow geometry used in this work. The upper plate induces simple shear to the fluid confined between the two plates, while the lower one remains stationary. Ideally, the stress is uniform over the gap between the plates.

Figure 2

Planar Couette start-up flow: linear velocity profile. The model parameters are $A_s=0.56, B_s=1.0, k=1.6$, and ${\varphi }_0=0.67999$ and the shear rate is fixed at $\dot{\gamma }=0.21/\mathrm{s}$. The stress variations induced by the inertial effects result in the formation of a TSB, seen here as the initial shear localization toward the shearing (outer) plane. Gradually, the profile evolves over time toward the steady-state solution ($t=300$ s) and the TSB vanishes.

Figure 3

Planar Couette start-up flow: linear velocity profile. The model parameters are $A_s=0.56, B_s=1.0, {\varphi }_0=0.67999$, and the shear rate is fixed at $\dot{\gamma }=0.21/\mathrm{s}$ while the exponent of the $\varphi$ model is now reduced to $k=1.0$. Here, the stress variations induced by the inertial effects are unable to produce a TSB due to the highly linear breakdown of the underlying fluid structure.

Figure 4

Planar Couette start-up flow: fluidization time ${\tau }_f$ vs. applied shear rate $\langle \dot{\gamma }\rangle$ on a log-log scale. A sufficient applied shear rate is required for shear localization and nonuniform fluidization, seen as a discrete ramp in the results. The fluidization time decays rapidly by increasing $\langle \dot{\gamma }\rangle$.

Figure 5

Planar Couette start-up flow: the instantaneous stress-time curve ($A_s=0.56, B_s=1.0, k=1.3$, and ${\varphi }_0=0.6799$). As time progresses, the structure breaks down, and a lower stress is required to maintain the applied shear rate $\langle \dot{\gamma }\rangle$. If the magnitude of $\langle \dot{\gamma }\rangle$ is sufficient for forming a TSB, the corresponding stress-time curve exhibits a distinct kink (as seen here for $\langle \dot{\gamma }\rangle$ = and $\langle \dot{\gamma }\rangle$), signaling the beginning of the shear localization. In the event of weak or nonexistent localization, this instability is not observed.

Figure 6

Planar Couette start-up flow: the shear-band edge location (normalized gap) over time ($A_s=0.56, B_s=1.0, k=1.3$, and ${\varphi }_0=0.67999$), calculated for various applied shear rates.

Figure 7

Planar Couette start-up flow: a semilogarithmic phase diagram that classifies the flow as either uniform or localized (a TSB is formed). The boundary is defined by the minimum (triggering) value of the applied shear rate $\langle \dot{\gamma }\rangle$ (dependent on $k$), above which the start-up inertia induces a TSB and the corresponding banded profile is retrieved. Below this value, the flow remains decidedly uniform. As seen here, this value declines rapidly as a function of $k$. Note that the $k$ values for real-world complex fluids are on the order of $k\sim 2$ as easily verified in Ref. [18]. Therefore, extrapolating the minimum value for $\langle \dot{\gamma }\rangle$ using this $k$ suggests that shear localization should occur trivially due to the start-up inertia for any real complex fluids in the experimentally accessible shear rates [typically ${\langle \dot{\gamma }\rangle }_{\text{min}}\sim {10}^{-2}$ (1/s)].