Poiseuille flow of soft glasses in narrow channels: From quiescence to steady state

Using numerical simulations, the onset of Poiseuille flow in a confined soft glass is investigated. Starting from the quiescent state, steady flow sets in at a time scale which increases with a decrease in applied forcing. At this onset time scale, a rapid transition occurs via the simultaneous fluidization of regions having different local stresses. In the absence of steady flow at long times, creep is observed even in regions where the local stress is larger than the bulk yielding threshold. Finally, we show that the time scale to attain steady flow depends strongly on the history of the initial state.

Figure 1

(a) Steady-state velocity profiles for ${\sigma }_{\mathrm{w}}/{\sigma }_{\mathrm{d}}=2.65$ (magenta), 2.37 (blue), and 2.09 (green). (b) Transient evolution of $v_x\left(y\right)$ before the onset of steady state, for ${\sigma }_{\mathrm{w}}=2.37{\sigma }_{\mathrm{d}}$. (c) Variation of the spatially averaged flow velocity indicating the sudden outbreak of flow at ${\sigma }_{\mathrm{w}}=2.37{\sigma }_{\mathrm{d}}$. (d) Maps of transverse displacements at $t/{10}^5=1.23$, 1.46, and 1.59. The color bar shows the displacement scales (in units of $d_s)$.

Figure 2

(a) During flow onset, layer-resolved MSD ${\Delta }_y^2\left(t\right)$ for ${\sigma }_{\mathrm{w}}=1.67{\sigma }_{\mathrm{d}}$, marking data for the central layer (dotted) and layers centered $1d_s$ away from the two walls (dashed). (b) For ${\sigma }_{\mathrm{w}}/{\sigma }_{\mathrm{d}}=2.09$ and 2.65, MSD for the latter three layers.

Figure 3

Top: Flow velocity $v_x^{\text{c.m.}}\left(t\right)$ for $m=40$ initial states $\left(t_{\mathrm{age}}={10}^4\right)$, prepared via quench from $T_0=0.20$, at ${\sigma }_{\mathrm{w}}/{\sigma }_{\mathrm{d}}=2.79$ (left) and $2.23$ (right). Bottom left: For the same initial states, the time evolution of the fraction of nonflowing states $n_s\left(t\right)$ for different ${\sigma }_{\mathrm{w}}$. Bottom right: Variation of ${\tau }_o$ with ${\sigma }_{\mathrm{w}}$ for initial states, prepared via quench from $T_0=0.2$ (green triangles) and $0.16$ (purple stars). The dashed line is a fit with $A/{({\sigma }_{\mathrm{w}}/{\sigma }_{\mathrm{d}}-x_c)}^{\beta },x_c=2.01,2.05$ for the two respective $T_0$'s.

Figure 4

(a) For $w=22.66d_s$, flow velocities after quench from ${\sigma }_{\mathrm{w}}=2.79{\sigma }_{\mathrm{d}}$ to ${\sigma }_{\mathrm{w}}/{\sigma }_{\mathrm{d}}=2.51$, 1.95, and 1.67. Dashed lines mark steady-state averages obtained during the onset of flow. (b) Variation of steady-flow velocity $\langle v_x^{\text{c.m.}}\rangle$ with ${\sigma }_{\mathrm{w}}/{\sigma }_{\mathrm{d}}$ for $w/d_s=22.67$ (square), 49.33 (circle), and 102.67 (diamond). Dashed lines are fits to extract the flow thresholds $x_t$. The dotted vertical line marks $x_c$ (see Fig. 3). (c) For ${\sigma }_{\mathrm{w}}=2.09{\sigma }_{\mathrm{d}}$, the time evolution of $n_s\left(t\right)$ for different $w$ [symbols as in (b)]. (d) For $w/d_s=49.33$, layer-resolved MSDs during the transient regime for ${\sigma }_{\mathrm{w}}=1.67{\sigma }_{\mathrm{d}}$, with layers marked similar to Fig. 2.