Shear localization in large amplitude oscillatory shear (LAOS) flows of particulate suspensions

Strong shear localization effects are observed in large amplitude oscillatory shear (LAOS) simulations of a particulate suspension. Here, the structural response of this complex fluid is completely viscous and governed by the general shear-driven diffusion model by Phillips et al. [Phys. Fluids 4, 30 (1992)]. When coupled to oscillatory shear in LAOS, this model is shown to produce concentration gradients, which imply the existence of regions of disparate viscosities across the simulated measurement gap. This suggests the presence of strong shear localization which is conceived even though the intrinsic flow curve of the model is monotonic, and the simulated geometry is a planar Couette setup, expected to display simple shear flow characteristics. This shear localization is generated due to the oscillatory shear at the shearing plate, which, therefore, induces accelerating motion. The subsequent inertial effects act as perturbations in the nonlinear response of the fluid structure to shear and are sufficient to trigger significant localization in the flow. Due to the ubiquitous nature of shear-driven diffusive mechanisms in complex fluids, these results suggest shear localization to be an integral feature of a LAOS measurement of many complex fluids.

Figure 1

The parallel plates geometry utilized in the LAOS simulations. The fluid is confined between two plates. Here, one plate remains stationary whereas the other plate sustains a sinusoidal velocity profile with respect to time $t$ with an amplitude $v_a$ and angular frequency $\omega$. An ideal fluid experiences this sinusoidal boundary velocity along the moving plate as plotted on the right-hand side of the figure. In the lower figure, the expected concentration (volume fraction) profile across the gap is depicted in the event of shear localization. Due to diffusive nature of the model, the shear localization is accompanied by spatial variations in the volume fraction with no clearly defined interface.

Figure 2

The dimensionless fluid volume fraction ${\varphi }^{*}$ versus the dimensionless gap $r^{*}$. Several initial volume fractions ${\varphi }_0^{*}$ are displayed for the particle size $a=100\mu \mathrm{m}$ at the oscillatory frequency $f=1.0\mathrm{Hz}$. Increasing ${\varphi }_0^{*}$ produces a higher degree of concentration gradients, which, in turn, implies a higher degree of viscosity disparity between the moving ($r^{*}=0$) and the stationary ($r^{*}=1.0$) planes of the planar Couette setup.

Figure 3

The dimensionless fluid volume fraction ${\varphi }^{*}$ versus the dimensionless gap $r^{*}$. Several frequencies $f$ of the oscillatory shear are simulated and plotted for the particle size $a=100\mu \mathrm{m}$. As the oscillatory frequency $f$ at the shearing plate of the planar Couette geometry increases, the unsteady inertial terms increase as well, resulting in a strongly localized shear in the gap.

Figure 4

The dimensionless fluid volume fraction ${\varphi }^{*}$ versus the dimensionless gap $r^{*}$ plotted for several particle sizes $a$ whereas the oscillatory frequency $f$ remains constant (0.1 Hz). Larger particles are more susceptible to mechanical interactions and less influenced by thermal diffusion, implying a stronger tendency to shear localization due to diffusive fluxes.

Figure 5

The coefficient of variation of the Reynolds number ${\sigma }_{\mathrm{Re}}/\langle \mathrm{Re}\rangle$, which quantifies the strength of the shear localization in each case, is given in this contour plot as a function of the oscillatory frequency $f$ and the particle size $a$. High $f$ and $a$, subsequently, produce higher shear localization, as expected. However, the degree of localization has a maximum at around $f=0.1\mathrm{Hz}$ for the largest particle sizes. This rather puzzling result is discussed more in conjunction with Fig. 8.

Figure 6

The coefficient of variation of the dimensionless viscosity ${\sigma }_{\mathrm{Re}}/\langle \mathrm{Re}\rangle$ is depicted in this contour plot as a function of the simulation time $t$ and the particle size $a$ whereas the oscillation frequency is fixed at $f=0.1\mathrm{Hz}$. With time, the degree of localization $C_v$ becomes more pronounced. From a theoretical point of view, this is expected since the oscillatory perturbation keeps forcing the nonlinear structural response in the diffusion equation and the subsequent shear localization effect.

Figure 7

The coefficient of variation $C_v$ versus the simulation time $t$ plotted for several particles sizes $a$ (with fixed $f=0.1\mathrm{Hz}$). Especially for larger particles, the degree of localization increases exponentially, beginning well below the typical timescale of observation $O\left(1\right)$.

Figure 8

The inverse of the averaged viscous timescale $\langle 1/{\tau }_{\mathrm{visc}}\rangle$ versus the oscillatory frequency $f$ for several particle sizes $a$. Here, $\langle 1/{\tau }_{\mathrm{visc}}\rangle$ represents the inverse timescale of the Navier-Stokes equation, whereas $f$ represents the inverse timescale of the oscillation. Judging by this figure, the localization begins to increase markedly only when these timescales coincide.